Wikiversity enwikiversity https://en.wikiversity.org/wiki/Wikiversity:Main_Page MediaWiki 1.47.0-wmf.24 first-letter Media Special Talk User User talk Wikiversity Wikiversity talk File File talk MediaWiki MediaWiki talk Template Template talk Help Help talk Category Category talk School School talk Portal Portal talk Topic Topic talk Collection Collection talk Draft Draft talk TimedText TimedText talk Module Module talk Event Event talk 4 28 2842508 2842412 2026-10-25T02:01:35Z Jtneill 10242 /* Discussion */ Neutral 2842508 wikitext text/x-wiki {{Wikiversity:Colloquium/Header}} <!-- MESSAGES GO BELOW --> == Paid part-time Administrative Officer position with WikiJournal == [[WikiJournal User Group|WikiJournal]] is an open-access, nonprofit scholarly publisher hosted on Wikiversity. It uses the wiki format to allow authors to write and develop scholarly articles directly online. Submitted articles then undergo formal academic peer review and, when accepted, are published as citable journal articles while remaining freely accessible. WikiJournal is seeking expressions of interest in a new part-time '''Administrative Officer''' position. The officer would support the project's day-to-day organizational work, including: * Setting up and administering a system for tracking contractor hours and issuing payments. * Coordinating the contracting, onboarding, timekeeping and payment of technical editors and other contractors. * Maintaining financial, contractor and administrative records. * Assisting with grant administration, budget tracking and reporting. * Following up on board action items, deadlines and recurring obligations. * Supporting recruitment and other operational tasks as needed. This is a remote, flexibly scheduled contractor role averaging approximately '''6 hours per week'''. Compensation is proposed at '''US$25 per hour''', with up to approximately '''320 hours (US$8,000) available over the year'''. Applicants may be based internationally, although appointment will depend on WikiJournal being able to establish a practical and legally appropriate contracting and payment arrangement in the applicant's country of residence. Relevant experience may include administrative organization, bookkeeping or financial administration, contractor or personnel coordination, nonprofit or grant administration, Wikimedia participation, academic publishing, and open-access work. Applicants are not expected to have experience in every area. For the complete role description and further details, see [[WikiJournal User Group/Administrative officer]]. Those interested should reply in the '''[[Talk:WikiJournal User Group#Call for interest: Administrative Officer|Call for interest: Administrative Officer]]''' discussion by '''30 September 2026''', briefly describing their relevant experience, any connection to WikiJournal or related communities, and their general availability. Expressions of interest received later may still be considered if the position has not been filled. Recommendations of potentially suitable candidates are also welcome. [[User:Mikael Häggström|Mikael Häggström]] ([[User talk:Mikael Häggström|discuss]] • [[Special:Contributions/Mikael Häggström|contribs]]) 20:00, 10 September 2026 (UTC) == Server switch - Your wiki will be read-only for a short time soon == <section begin="server-switch"/><div class="plainlinks"> [[:m:Special:MyLanguage/Tech/Server switch|Read this message in another language]] • [https://meta.wikimedia.org/w/index.php?title=Special:Translate&group=page-Tech%2FServer+switch&language=&action=page&filter= {{int:please-translate}}] The [[foundation:|Wikimedia Foundation]] will switch the traffic between its data centers. This will make sure that Wikipedia and the other Wikimedia wikis can stay online even after a disaster. All traffic will switch on '''{{#time:j xg|2026-09-23|en}}'''. The switch will start at '''[https://zonestamp.toolforge.org/{{#time:U|2026-09-23T14:00|en}} {{#time:H:i e|2026-09-23T14:00}}]'''. Unfortunately, because of some limitations in [[mw:Special:MyLanguage/Manual:What is MediaWiki?|MediaWiki]], all editing must stop while the switch is made. We apologize for this disruption, and we are working to minimize it in the future. A banner will be displayed on all wikis 30 minutes before this operation happens. This banner will remain visible until the end of the operation. You can contribute to the [https://meta.wikimedia.org/w/index.php?title=Special%3ATranslate&group=Centralnotice-tgroup-read_only_banner&task=view&language=&filter=&action=translate translation or proofreading] of this banner text. '''You will be able to read, but not edit, all wikis for a short period of time.''' * You will not be able to edit for up to an hour on {{#time:l j xg Y|2026-09-23|en}}. * If you try to edit or save during these times, you will see an error message. We hope that no edits will be lost during these minutes, but we can't guarantee it. If you see the error message, then please wait until everything is back to normal. Then you should be able to save your edit. But, we recommend that you make a copy of your changes first, just in case. ''Other effects'': * Background jobs will be slower and some may be dropped. Red links might not be updated as quickly as normal. If you create an article that is already linked somewhere else, the link will stay red longer than usual. Some long-running scripts will have to be stopped. * We expect the code deployments to happen as any other week. However, some case-by-case code freezes could punctually happen if the operation require them afterwards. * [[mw:Special:MyLanguage/GitLab|GitLab]] will be unavailable for about 90 minutes. This project may be postponed if necessary. You can [[wikitech:Switch_Datacenter|read the schedule at wikitech.wikimedia.org]]. Any changes will be announced in the schedule. '''Please share this information with your community.'''</div><section end="server-switch"/> [[m:user:Trizek (WMF)|Trizek (WMF)]] 12:58, 15 September 2026 (UTC) <!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Non-Technical_Village_Pumps_distribution_list&oldid=30513874 --> == Your feedback wanted: Remove ''(Language link added:) and similar'' edits from Special:RecentChanges == ''(Apologies for posting in English. You can help by translating it)'' Hello, I’m Danny from [[:en:w:Wikimedia_Deutschland|Wikimedia Deutschland’s]] [[m:Wikidata_For_Wikimedia_Projects|Wikidata For Wikimedia Projects]] team.<br/> We have been investigating and examining how Wikidata’s use in the [[Special:RecentChanges]] and [[Special:Watchlist]] pages can be made to be more useful, or less ‘noisy’ (fewer edits that have little value to the reader). We are currently exploring to make a change in the volume and the type of Wikidata edits that appear in these pages, and '''we need your feedback''' to help us ensure this is a positive change for the Wikis and won’t disrupt your workflows.<br/> Language links are also known as sitelinks or interlanguage links and describe when a Wikidata item is connected to a Wiki, adding it to the language dropdown tool for other connected Wikis. [[File:Types of Lang Link changes.png|x350px|frame|A variety of "Language Link" edits as seen in the Recent Changes feed]] The request for feedback is whether you support the removal of these language link edits from the Recent Changes and Watchlist feed, or would you oppose such a change and if so, why? Please share you thoughts on the dedicated '''[[m:Talk:Wikidata_For_Wikimedia_Projects/Clearer_Wikidata_Edit_Summaries/Hide_Language_Link_edits|metawiki Talk page]]'''. Further information about this change: [[m:Wikidata_For_Wikimedia_Projects/Clearer_Wikidata_Edit_Summaries/Hide_Language_Link_edits|m:Wikidata_For_Wikimedia_Projects/Hide_Language_Link_edits]] Thank you for any participation or feedback you give, -- [[m:User:Danny_Benjafield_(WMDE)|User:Danny Benjafield (WMDE)]] 12:23, 25 September 2026 (UTC) <!-- Message sent by User:Danny Benjafield (WMDE)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Danny_Benjafield_(WMDE)/MassMessage_test_list&oldid=31080985 --> == Wikidata Sitelink changes in Recent Changes and Watchlist == ''(Apologies for cross-posting in English, translating it will help spread the message)'' Editors on group 1 wikis, including Wikiquote, Wikibooks, Wikiversity, Wikivoyage, and Wikisource, will no longer see notifications on [[Special:RecentChanges|Recent Changes]] and [[Special:Watchlist|Watchlist]] when a link to ''another'' language version of a page is added, removed, or changed in its Wikidata item. This change will begin on '''October 5 at 07:00 UTC''' as a two-week trial to reduce high-volume, low-value notifications and make higher-value changes easier to identify. Editors who rely on these notifications should be aware of the change and can share feedback or concerns on the [[m:Wikidata For Wikimedia Projects/Clearer Wikidata Edit Summaries/Hide Language Link edits|project]] or [[m:Talk:Wikidata_For_Wikimedia_Projects/Clearer_Wikidata_Edit_Summaries/Hide_Language_Link_edits|talk page]].<br/>Please reach out if you have any questions, thank you! -- [[User:Danny_Benjafield_(WMDE)|Danny Benjafield (WMDE)]] ([[User_talk:Danny_Benjafield_(WMDE)|talk]]) sent via [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 12:42, 1 October 2026 (UTC) <!-- Message sent by User:Danny Benjafield (WMDE)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Danny_Benjafield_(WMDE)/MassMessage_test_list&oldid=31106967 --> == Demote [[Wikiversity:CheckUser policy]]? == I am proposing that we demote that policy to a standard page, mostly for two reasons: # Considering the few amount of custodians and the size of this wiki, I'd say that this project is considered too small to have any CUs, and relying on the stewards would be a more logical choice. # We haven't elected two CUs in a few years. [[User:Codename Noreste|<span style="color: blue">Codename Noreste</span>]] /// [[User talk:Codename Noreste|⭐️✨]] 20:53, 1 October 2026 (UTC) :Reasonable. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 17:09, 4 October 2026 (UTC) :I don't see much point in demoting it. Even if we are not currently (or recently) using the policy, it is there should a need ever arise. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 18:24, 23 October 2026 (UTC) == General Questions and Reference Questions == <p>Hey everyone,</p> <p> My apologies in advance if these questions are oblivious or their responses readily apparent, but I figured I'd ask just to be safe.</p> <p> '''Questions'''</p> <p> 1. Should pages have inline citations and a reference section? A [[Fundamentals of Neuroscience/What is Neuroscience|majority of pages]] I have explored have neither, while [[Neurobiology of emotion|others have a more traditional approach to references]].</br> 2. Are there widely-used templates for weasel-words, references, etc.? </p> '''CONTEXT''': I migrated from Wikipedia, so I'm not entirely sure what the norms/roc are here.</br> Thanks in advance,</br> [[User:SamLaxarox|SamLaxarox]] ([[User talk:SamLaxarox|discuss]] • [[Special:Contributions/SamLaxarox|contribs]]) 18:40, 12 October 2026 (UTC) :You're asking fine questions. I posted {{tl|welcome}} to your talk page which I think will help you. To answer your direct questions: :#This project spun out of [[:b:]] and both here and there, you can post original research. This does not mean that you can post just anything, so if you want to post an educational resource and you want to justify claims you make with inline citations, that's great. If you want to include a bibliography, that's also quite helpful for readers. Generally, the way that Wikiversity works in practice is that individual users make a resource and then tend to work on it solo and have broad latitude about what you can do and how it looks. :#You can take al look at [[:Category:Templates]] and the subcat [[:Category:Maintenance templates]]. At Wikipedia, it's very common to point out issues with tone or uncited claims. That's not really as common here. We [[:d:Q7780480|do not have an equivalent]] of [[:w:en:Template:weasel]], but we do have a [[Template:Citation needed|local version]] of [[:w:en:Template:Citation needed]]. :―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 20:31, 12 October 2026 (UTC) ::Thanks for the heads up on both accounts; I'm happy to join forces with all y'all! ::[[User:SamLaxarox|SamLaxarox]] ([[User talk:SamLaxarox|discuss]] • [[Special:Contributions/SamLaxarox|contribs]]) 23:39, 12 October 2026 (UTC) ==Agent editors== What about inviting agents to edit this wiki? And they would need a human being responsible for them. [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|discuss]] • [[Special:Contributions/Michael Ten|contribs]]) 06:31, 17 October 2026 (UTC) == October 2026 Wikimedia Café: legal issues on Commons, and WMF Board selection process == <div class="border-box" style="background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)"> <div class="box" style="float:left; padding-top: 10px; padding-right: 10px; padding-left: 10px; padding-bottom: 10px;">[[File:Wikimedia Café logo in plain SVG format.svg|60px|alt=The logo for the Wikimedia Café]]</div> Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities in October. In a change from the usual Café practice, this month's sessions will have different topics. Anyone who meets the Café attendance criteria is welcome to attend either or both sessions. Session A will happen on Saturday, 24 October 2026 at 15:00 UTC ([https://zonestamp.toolforge.org/1792854000 timestamp converter]), at a time friendly to the Americas, Africa, and Europe. This session will feature [https://meta.wikimedia.org/wiki/User:PBradley-WMF Wikimedia Foundation lawyer Phil Bradley-Schmieg]. We will discuss legal issues on Wikimedia Commons, including complexities of international law. (I am hoping that in a future month, a similar Café session will be offered at a time friendly to Asia and the Pacific.) Session B will happen on Sunday, 25 October 2026 at 03:00 UTC ([https://zonestamp.toolforge.org/1792897200 timestamp converter]), at a time friendly to Asia and the Pacific, we will discuss [https://meta.wikimedia.org/wiki/Wikimedia_Foundation_elections/2027/Proposed_eligibility_criteria proposed changes to how Wikimedia Foundation Board candidates are selected]. The latest proposals are much different from the earlier July 2026 proposals. The due date for feedback on current proposals is 27 October, and Wikimedians are welcome to participate on the [https://meta.wikimedia.org/wiki/Talk:Wikimedia_Foundation_elections/2027/Proposed_eligibility_criteria proposal talk page] regardless of Café attendance. Please see the [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Café] page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]! To subscribe or unsubscribe for Café notices on your talk page please [https://meta.wikimedia.org/wiki/Global_message_delivery/Targets/Wikimedia_Caf%C3%A9 go here]. <br /> [[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]</div> <span style="white-space:nowrap;">[[User:Pine|<span style="color:#01796f; text-shadow:#00BFFF 0 0 1.0em">༄ᨒ𖠰 Pine</span>]] [[User talk:Pine|<span style="color:DeepSkyBlue">(<b style="color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em">✉</b>)</span>]]</span> 06:22, 18 October 2026 (UTC) == A new cosmological model == I would like to publish independent research that challenges the standard cosmological model. Does this fall within the scope of the "Original research" space? And what precautions should I take to ensure the page is not considered an inadmissible personal work, and is not open to modification by other users? [[User:Wb525|Wb525]] ([[User talk:Wb525|discuss]] • [[Special:Contributions/Wb525|contribs]]) 11:53, 20 October 2026 (UTC) : This sounds like it could be [[Wikiversity:Original research|original research]]. Some things to note: :* Sometimes what an author considers original research turns out to be original theory :* Some qualities that may help to ensure viability: objectivity, use of simple language, careful use of peer-reviewed citations, and minimal use of genAI with careful attribution/acknowledgement if AI is used per [[Wikiversity:Artificial intelligence]] :* Consider starting with a sub-page in user space e.g., [[User:Wb525/Subpage name]] :* As an open knowledge project, Wikiversity content is openly editable by others : -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 23:05, 20 October 2026 (UTC) == Interface administrators policy== This is a discussion of potential changes to [[Wikiversity:Interface administrators]] policy. Please refer to: "For now, Wikiversity does not have a need for permanent or long term interface administrators. Accordingly, interface administratorship may only be granted temporarily by bureaucrats (between 14 and 120 days based on discussion, at the discretion of that bureaucrat and perceived need)." --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 05:14, 23 October 2026 (UTC) === Temporary rights duration === I propose that temporary rights be granted for a duration of from one day to one year. (to conform with selections in [[Special:UserRights]] in the pull down menus.) --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 05:14, 23 October 2026 (UTC) ==== Discussion ==== *{{support}} ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 13:56, 23 October 2026 (UTC) * {{support}}. [[User:Codename Noreste|<span style="color: blue">Codename Noreste</span>]] /// [[User talk:Codename Noreste|⭐️✨]] 14:56, 23 October 2026 (UTC) * {{support}} as proposer.--[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 16:30, 23 October 2026 (UTC) * {{neutral}}; if we're changing the time period again, this time to align with the drop-down options, then I suggest that the policy require that nominators request one of the following time-frames supported by a corresponding plan of work: ** 1 day ** 1 week ** 1 month ** 3 months ** 6 months ** 1 year : -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 02:01, 25 October 2026 (UTC) === Temporary rights extention === I propose that we clarify that temporary rights can be extended. If the community approves of an extension, it can be granted more than once. The temporary rights can repeatedly be extended, for new temporary durations. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 05:14, 23 October 2026 (UTC) ==== Discussion ==== *{{support}} ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 13:56, 23 October 2026 (UTC) * {{support}} for more than one extension. [[User:Codename Noreste|<span style="color: blue">Codename Noreste</span>]] /// [[User talk:Codename Noreste|⭐️✨]] 14:56, 23 October 2026 (UTC) * {{support}} as proposer. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 16:30, 23 October 2026 (UTC) === Permanent IA === There has been a suggestion of permanent IA rights. Another suggestion is that there should be at least two members with such rights. Please express your preference. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 05:14, 23 October 2026 (UTC) ==== Discussion ==== *{{support}} ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 13:56, 23 October 2026 (UTC) * {{support}} for having at least two members because there shouldn't be only one. [[User:Codename Noreste|<span style="color: blue">Codename Noreste</span>]] /// [[User talk:Codename Noreste|⭐️✨]] 13:59, 23 October 2026 (UTC) * Conditional {{support}} with the understanding that there be a minimum of 2 (if only 1 is remaining, the other perm is set to temp until a 2nd becomes active) and perm can only be granted after having a temp. I also think a perm nom should have a requirement to demonstrate need - for example show activity logs so that the community can assess that IA is needed long term. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 17:55, 23 October 2026 (UTC) *: You mean that we may have at least 2 permanent IAs, but if one is removed, the other one's is set to temporary until another IA is appointed, makes sense? [[User:Codename Noreste|<span style="color: blue">Codename Noreste</span>]] /// [[User talk:Codename Noreste|⭐️✨]] 20:34, 23 October 2026 (UTC) *::Yes, I meant that if the second IA is no longer active, the remaining IA would be reset to 1 year temporary. That should give enough time to find a second perm IA. That 1 year could be extended, if it takes more time to find a second. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 04:15, 24 October 2026 (UTC) *:::Once the community approves a perm IA, there is no need to renominate. A crat can just reset the time to temp until the conditions are met, and reset to perm when we have two or more. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 04:25, 24 October 2026 (UTC) gkjsfvgesadu6qhd4cdpzwbw65yqvtt 2842509 2842508 2026-10-25T02:04:06Z Jtneill 10242 /* Discussion */ reply ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]]) 2842509 wikitext text/x-wiki {{Wikiversity:Colloquium/Header}} <!-- MESSAGES GO BELOW --> == Paid part-time Administrative Officer position with WikiJournal == [[WikiJournal User Group|WikiJournal]] is an open-access, nonprofit scholarly publisher hosted on Wikiversity. It uses the wiki format to allow authors to write and develop scholarly articles directly online. Submitted articles then undergo formal academic peer review and, when accepted, are published as citable journal articles while remaining freely accessible. WikiJournal is seeking expressions of interest in a new part-time '''Administrative Officer''' position. The officer would support the project's day-to-day organizational work, including: * Setting up and administering a system for tracking contractor hours and issuing payments. * Coordinating the contracting, onboarding, timekeeping and payment of technical editors and other contractors. * Maintaining financial, contractor and administrative records. * Assisting with grant administration, budget tracking and reporting. * Following up on board action items, deadlines and recurring obligations. * Supporting recruitment and other operational tasks as needed. This is a remote, flexibly scheduled contractor role averaging approximately '''6 hours per week'''. Compensation is proposed at '''US$25 per hour''', with up to approximately '''320 hours (US$8,000) available over the year'''. Applicants may be based internationally, although appointment will depend on WikiJournal being able to establish a practical and legally appropriate contracting and payment arrangement in the applicant's country of residence. Relevant experience may include administrative organization, bookkeeping or financial administration, contractor or personnel coordination, nonprofit or grant administration, Wikimedia participation, academic publishing, and open-access work. Applicants are not expected to have experience in every area. For the complete role description and further details, see [[WikiJournal User Group/Administrative officer]]. Those interested should reply in the '''[[Talk:WikiJournal User Group#Call for interest: Administrative Officer|Call for interest: Administrative Officer]]''' discussion by '''30 September 2026''', briefly describing their relevant experience, any connection to WikiJournal or related communities, and their general availability. Expressions of interest received later may still be considered if the position has not been filled. Recommendations of potentially suitable candidates are also welcome. [[User:Mikael Häggström|Mikael Häggström]] ([[User talk:Mikael Häggström|discuss]] • [[Special:Contributions/Mikael Häggström|contribs]]) 20:00, 10 September 2026 (UTC) == Server switch - Your wiki will be read-only for a short time soon == <section begin="server-switch"/><div class="plainlinks"> [[:m:Special:MyLanguage/Tech/Server switch|Read this message in another language]] • [https://meta.wikimedia.org/w/index.php?title=Special:Translate&group=page-Tech%2FServer+switch&language=&action=page&filter= {{int:please-translate}}] The [[foundation:|Wikimedia Foundation]] will switch the traffic between its data centers. This will make sure that Wikipedia and the other Wikimedia wikis can stay online even after a disaster. All traffic will switch on '''{{#time:j xg|2026-09-23|en}}'''. The switch will start at '''[https://zonestamp.toolforge.org/{{#time:U|2026-09-23T14:00|en}} {{#time:H:i e|2026-09-23T14:00}}]'''. Unfortunately, because of some limitations in [[mw:Special:MyLanguage/Manual:What is MediaWiki?|MediaWiki]], all editing must stop while the switch is made. We apologize for this disruption, and we are working to minimize it in the future. A banner will be displayed on all wikis 30 minutes before this operation happens. This banner will remain visible until the end of the operation. You can contribute to the [https://meta.wikimedia.org/w/index.php?title=Special%3ATranslate&group=Centralnotice-tgroup-read_only_banner&task=view&language=&filter=&action=translate translation or proofreading] of this banner text. '''You will be able to read, but not edit, all wikis for a short period of time.''' * You will not be able to edit for up to an hour on {{#time:l j xg Y|2026-09-23|en}}. * If you try to edit or save during these times, you will see an error message. We hope that no edits will be lost during these minutes, but we can't guarantee it. If you see the error message, then please wait until everything is back to normal. Then you should be able to save your edit. But, we recommend that you make a copy of your changes first, just in case. ''Other effects'': * Background jobs will be slower and some may be dropped. Red links might not be updated as quickly as normal. If you create an article that is already linked somewhere else, the link will stay red longer than usual. Some long-running scripts will have to be stopped. * We expect the code deployments to happen as any other week. However, some case-by-case code freezes could punctually happen if the operation require them afterwards. * [[mw:Special:MyLanguage/GitLab|GitLab]] will be unavailable for about 90 minutes. This project may be postponed if necessary. You can [[wikitech:Switch_Datacenter|read the schedule at wikitech.wikimedia.org]]. Any changes will be announced in the schedule. '''Please share this information with your community.'''</div><section end="server-switch"/> [[m:user:Trizek (WMF)|Trizek (WMF)]] 12:58, 15 September 2026 (UTC) <!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Non-Technical_Village_Pumps_distribution_list&oldid=30513874 --> == Your feedback wanted: Remove ''(Language link added:) and similar'' edits from Special:RecentChanges == ''(Apologies for posting in English. You can help by translating it)'' Hello, I’m Danny from [[:en:w:Wikimedia_Deutschland|Wikimedia Deutschland’s]] [[m:Wikidata_For_Wikimedia_Projects|Wikidata For Wikimedia Projects]] team.<br/> We have been investigating and examining how Wikidata’s use in the [[Special:RecentChanges]] and [[Special:Watchlist]] pages can be made to be more useful, or less ‘noisy’ (fewer edits that have little value to the reader). We are currently exploring to make a change in the volume and the type of Wikidata edits that appear in these pages, and '''we need your feedback''' to help us ensure this is a positive change for the Wikis and won’t disrupt your workflows.<br/> Language links are also known as sitelinks or interlanguage links and describe when a Wikidata item is connected to a Wiki, adding it to the language dropdown tool for other connected Wikis. [[File:Types of Lang Link changes.png|x350px|frame|A variety of "Language Link" edits as seen in the Recent Changes feed]] The request for feedback is whether you support the removal of these language link edits from the Recent Changes and Watchlist feed, or would you oppose such a change and if so, why? Please share you thoughts on the dedicated '''[[m:Talk:Wikidata_For_Wikimedia_Projects/Clearer_Wikidata_Edit_Summaries/Hide_Language_Link_edits|metawiki Talk page]]'''. Further information about this change: [[m:Wikidata_For_Wikimedia_Projects/Clearer_Wikidata_Edit_Summaries/Hide_Language_Link_edits|m:Wikidata_For_Wikimedia_Projects/Hide_Language_Link_edits]] Thank you for any participation or feedback you give, -- [[m:User:Danny_Benjafield_(WMDE)|User:Danny Benjafield (WMDE)]] 12:23, 25 September 2026 (UTC) <!-- Message sent by User:Danny Benjafield (WMDE)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Danny_Benjafield_(WMDE)/MassMessage_test_list&oldid=31080985 --> == Wikidata Sitelink changes in Recent Changes and Watchlist == ''(Apologies for cross-posting in English, translating it will help spread the message)'' Editors on group 1 wikis, including Wikiquote, Wikibooks, Wikiversity, Wikivoyage, and Wikisource, will no longer see notifications on [[Special:RecentChanges|Recent Changes]] and [[Special:Watchlist|Watchlist]] when a link to ''another'' language version of a page is added, removed, or changed in its Wikidata item. This change will begin on '''October 5 at 07:00 UTC''' as a two-week trial to reduce high-volume, low-value notifications and make higher-value changes easier to identify. Editors who rely on these notifications should be aware of the change and can share feedback or concerns on the [[m:Wikidata For Wikimedia Projects/Clearer Wikidata Edit Summaries/Hide Language Link edits|project]] or [[m:Talk:Wikidata_For_Wikimedia_Projects/Clearer_Wikidata_Edit_Summaries/Hide_Language_Link_edits|talk page]].<br/>Please reach out if you have any questions, thank you! -- [[User:Danny_Benjafield_(WMDE)|Danny Benjafield (WMDE)]] ([[User_talk:Danny_Benjafield_(WMDE)|talk]]) sent via [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 12:42, 1 October 2026 (UTC) <!-- Message sent by User:Danny Benjafield (WMDE)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Danny_Benjafield_(WMDE)/MassMessage_test_list&oldid=31106967 --> == Demote [[Wikiversity:CheckUser policy]]? == I am proposing that we demote that policy to a standard page, mostly for two reasons: # Considering the few amount of custodians and the size of this wiki, I'd say that this project is considered too small to have any CUs, and relying on the stewards would be a more logical choice. # We haven't elected two CUs in a few years. [[User:Codename Noreste|<span style="color: blue">Codename Noreste</span>]] /// [[User talk:Codename Noreste|⭐️✨]] 20:53, 1 October 2026 (UTC) :Reasonable. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 17:09, 4 October 2026 (UTC) :I don't see much point in demoting it. Even if we are not currently (or recently) using the policy, it is there should a need ever arise. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 18:24, 23 October 2026 (UTC) == General Questions and Reference Questions == <p>Hey everyone,</p> <p> My apologies in advance if these questions are oblivious or their responses readily apparent, but I figured I'd ask just to be safe.</p> <p> '''Questions'''</p> <p> 1. Should pages have inline citations and a reference section? A [[Fundamentals of Neuroscience/What is Neuroscience|majority of pages]] I have explored have neither, while [[Neurobiology of emotion|others have a more traditional approach to references]].</br> 2. Are there widely-used templates for weasel-words, references, etc.? </p> '''CONTEXT''': I migrated from Wikipedia, so I'm not entirely sure what the norms/roc are here.</br> Thanks in advance,</br> [[User:SamLaxarox|SamLaxarox]] ([[User talk:SamLaxarox|discuss]] • [[Special:Contributions/SamLaxarox|contribs]]) 18:40, 12 October 2026 (UTC) :You're asking fine questions. I posted {{tl|welcome}} to your talk page which I think will help you. To answer your direct questions: :#This project spun out of [[:b:]] and both here and there, you can post original research. This does not mean that you can post just anything, so if you want to post an educational resource and you want to justify claims you make with inline citations, that's great. If you want to include a bibliography, that's also quite helpful for readers. Generally, the way that Wikiversity works in practice is that individual users make a resource and then tend to work on it solo and have broad latitude about what you can do and how it looks. :#You can take al look at [[:Category:Templates]] and the subcat [[:Category:Maintenance templates]]. At Wikipedia, it's very common to point out issues with tone or uncited claims. That's not really as common here. We [[:d:Q7780480|do not have an equivalent]] of [[:w:en:Template:weasel]], but we do have a [[Template:Citation needed|local version]] of [[:w:en:Template:Citation needed]]. :―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 20:31, 12 October 2026 (UTC) ::Thanks for the heads up on both accounts; I'm happy to join forces with all y'all! ::[[User:SamLaxarox|SamLaxarox]] ([[User talk:SamLaxarox|discuss]] • [[Special:Contributions/SamLaxarox|contribs]]) 23:39, 12 October 2026 (UTC) ==Agent editors== What about inviting agents to edit this wiki? And they would need a human being responsible for them. [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|discuss]] • [[Special:Contributions/Michael Ten|contribs]]) 06:31, 17 October 2026 (UTC) == October 2026 Wikimedia Café: legal issues on Commons, and WMF Board selection process == <div class="border-box" style="background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)"> <div class="box" style="float:left; padding-top: 10px; padding-right: 10px; padding-left: 10px; padding-bottom: 10px;">[[File:Wikimedia Café logo in plain SVG format.svg|60px|alt=The logo for the Wikimedia Café]]</div> Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities in October. In a change from the usual Café practice, this month's sessions will have different topics. Anyone who meets the Café attendance criteria is welcome to attend either or both sessions. Session A will happen on Saturday, 24 October 2026 at 15:00 UTC ([https://zonestamp.toolforge.org/1792854000 timestamp converter]), at a time friendly to the Americas, Africa, and Europe. This session will feature [https://meta.wikimedia.org/wiki/User:PBradley-WMF Wikimedia Foundation lawyer Phil Bradley-Schmieg]. We will discuss legal issues on Wikimedia Commons, including complexities of international law. (I am hoping that in a future month, a similar Café session will be offered at a time friendly to Asia and the Pacific.) Session B will happen on Sunday, 25 October 2026 at 03:00 UTC ([https://zonestamp.toolforge.org/1792897200 timestamp converter]), at a time friendly to Asia and the Pacific, we will discuss [https://meta.wikimedia.org/wiki/Wikimedia_Foundation_elections/2027/Proposed_eligibility_criteria proposed changes to how Wikimedia Foundation Board candidates are selected]. The latest proposals are much different from the earlier July 2026 proposals. The due date for feedback on current proposals is 27 October, and Wikimedians are welcome to participate on the [https://meta.wikimedia.org/wiki/Talk:Wikimedia_Foundation_elections/2027/Proposed_eligibility_criteria proposal talk page] regardless of Café attendance. Please see the [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Café] page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]! To subscribe or unsubscribe for Café notices on your talk page please [https://meta.wikimedia.org/wiki/Global_message_delivery/Targets/Wikimedia_Caf%C3%A9 go here]. <br /> [[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]</div> <span style="white-space:nowrap;">[[User:Pine|<span style="color:#01796f; text-shadow:#00BFFF 0 0 1.0em">༄ᨒ𖠰 Pine</span>]] [[User talk:Pine|<span style="color:DeepSkyBlue">(<b style="color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em">✉</b>)</span>]]</span> 06:22, 18 October 2026 (UTC) == A new cosmological model == I would like to publish independent research that challenges the standard cosmological model. Does this fall within the scope of the "Original research" space? And what precautions should I take to ensure the page is not considered an inadmissible personal work, and is not open to modification by other users? [[User:Wb525|Wb525]] ([[User talk:Wb525|discuss]] • [[Special:Contributions/Wb525|contribs]]) 11:53, 20 October 2026 (UTC) : This sounds like it could be [[Wikiversity:Original research|original research]]. Some things to note: :* Sometimes what an author considers original research turns out to be original theory :* Some qualities that may help to ensure viability: objectivity, use of simple language, careful use of peer-reviewed citations, and minimal use of genAI with careful attribution/acknowledgement if AI is used per [[Wikiversity:Artificial intelligence]] :* Consider starting with a sub-page in user space e.g., [[User:Wb525/Subpage name]] :* As an open knowledge project, Wikiversity content is openly editable by others : -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 23:05, 20 October 2026 (UTC) == Interface administrators policy== This is a discussion of potential changes to [[Wikiversity:Interface administrators]] policy. Please refer to: "For now, Wikiversity does not have a need for permanent or long term interface administrators. Accordingly, interface administratorship may only be granted temporarily by bureaucrats (between 14 and 120 days based on discussion, at the discretion of that bureaucrat and perceived need)." --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 05:14, 23 October 2026 (UTC) === Temporary rights duration === I propose that temporary rights be granted for a duration of from one day to one year. (to conform with selections in [[Special:UserRights]] in the pull down menus.) --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 05:14, 23 October 2026 (UTC) ==== Discussion ==== *{{support}} ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 13:56, 23 October 2026 (UTC) * {{support}}. [[User:Codename Noreste|<span style="color: blue">Codename Noreste</span>]] /// [[User talk:Codename Noreste|⭐️✨]] 14:56, 23 October 2026 (UTC) * {{support}} as proposer.--[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 16:30, 23 October 2026 (UTC) * {{neutral}}; if we're changing the time period again, this time to align with the drop-down options, then I suggest that the policy require that nominators request one of the following time-frames supported by a corresponding plan of work: ** 1 day ** 1 week ** 1 month ** 3 months ** 6 months ** 1 year : -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 02:01, 25 October 2026 (UTC) === Temporary rights extention === I propose that we clarify that temporary rights can be extended. If the community approves of an extension, it can be granted more than once. The temporary rights can repeatedly be extended, for new temporary durations. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 05:14, 23 October 2026 (UTC) ==== Discussion ==== *{{support}} ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 13:56, 23 October 2026 (UTC) * {{support}} for more than one extension. [[User:Codename Noreste|<span style="color: blue">Codename Noreste</span>]] /// [[User talk:Codename Noreste|⭐️✨]] 14:56, 23 October 2026 (UTC) * {{support}} as proposer. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 16:30, 23 October 2026 (UTC) * {{Neutral}} because I'm not sure what the practical difference is between applying for an extension and applying for the right. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 02:04, 25 October 2026 (UTC) === Permanent IA === There has been a suggestion of permanent IA rights. Another suggestion is that there should be at least two members with such rights. Please express your preference. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 05:14, 23 October 2026 (UTC) ==== Discussion ==== *{{support}} ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 13:56, 23 October 2026 (UTC) * {{support}} for having at least two members because there shouldn't be only one. [[User:Codename Noreste|<span style="color: blue">Codename Noreste</span>]] /// [[User talk:Codename Noreste|⭐️✨]] 13:59, 23 October 2026 (UTC) * Conditional {{support}} with the understanding that there be a minimum of 2 (if only 1 is remaining, the other perm is set to temp until a 2nd becomes active) and perm can only be granted after having a temp. I also think a perm nom should have a requirement to demonstrate need - for example show activity logs so that the community can assess that IA is needed long term. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 17:55, 23 October 2026 (UTC) *: You mean that we may have at least 2 permanent IAs, but if one is removed, the other one's is set to temporary until another IA is appointed, makes sense? [[User:Codename Noreste|<span style="color: blue">Codename Noreste</span>]] /// [[User talk:Codename Noreste|⭐️✨]] 20:34, 23 October 2026 (UTC) *::Yes, I meant that if the second IA is no longer active, the remaining IA would be reset to 1 year temporary. That should give enough time to find a second perm IA. That 1 year could be extended, if it takes more time to find a second. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 04:15, 24 October 2026 (UTC) *:::Once the community approves a perm IA, there is no need to renominate. A crat can just reset the time to temp until the conditions are met, and reset to perm when we have two or more. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 04:25, 24 October 2026 (UTC) bli9sbdpp4x8csjqs98jfwgssc9bqe2 2842511 2842509 2026-10-25T02:09:54Z Jtneill 10242 /* Discussion */ reply ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]]) 2842511 wikitext text/x-wiki {{Wikiversity:Colloquium/Header}} <!-- MESSAGES GO BELOW --> == Paid part-time Administrative Officer position with WikiJournal == [[WikiJournal User Group|WikiJournal]] is an open-access, nonprofit scholarly publisher hosted on Wikiversity. It uses the wiki format to allow authors to write and develop scholarly articles directly online. Submitted articles then undergo formal academic peer review and, when accepted, are published as citable journal articles while remaining freely accessible. WikiJournal is seeking expressions of interest in a new part-time '''Administrative Officer''' position. The officer would support the project's day-to-day organizational work, including: * Setting up and administering a system for tracking contractor hours and issuing payments. * Coordinating the contracting, onboarding, timekeeping and payment of technical editors and other contractors. * Maintaining financial, contractor and administrative records. * Assisting with grant administration, budget tracking and reporting. * Following up on board action items, deadlines and recurring obligations. * Supporting recruitment and other operational tasks as needed. This is a remote, flexibly scheduled contractor role averaging approximately '''6 hours per week'''. Compensation is proposed at '''US$25 per hour''', with up to approximately '''320 hours (US$8,000) available over the year'''. Applicants may be based internationally, although appointment will depend on WikiJournal being able to establish a practical and legally appropriate contracting and payment arrangement in the applicant's country of residence. Relevant experience may include administrative organization, bookkeeping or financial administration, contractor or personnel coordination, nonprofit or grant administration, Wikimedia participation, academic publishing, and open-access work. Applicants are not expected to have experience in every area. For the complete role description and further details, see [[WikiJournal User Group/Administrative officer]]. Those interested should reply in the '''[[Talk:WikiJournal User Group#Call for interest: Administrative Officer|Call for interest: Administrative Officer]]''' discussion by '''30 September 2026''', briefly describing their relevant experience, any connection to WikiJournal or related communities, and their general availability. Expressions of interest received later may still be considered if the position has not been filled. Recommendations of potentially suitable candidates are also welcome. [[User:Mikael Häggström|Mikael Häggström]] ([[User talk:Mikael Häggström|discuss]] • [[Special:Contributions/Mikael Häggström|contribs]]) 20:00, 10 September 2026 (UTC) == Server switch - Your wiki will be read-only for a short time soon == <section begin="server-switch"/><div class="plainlinks"> [[:m:Special:MyLanguage/Tech/Server switch|Read this message in another language]] • [https://meta.wikimedia.org/w/index.php?title=Special:Translate&group=page-Tech%2FServer+switch&language=&action=page&filter= {{int:please-translate}}] The [[foundation:|Wikimedia Foundation]] will switch the traffic between its data centers. This will make sure that Wikipedia and the other Wikimedia wikis can stay online even after a disaster. All traffic will switch on '''{{#time:j xg|2026-09-23|en}}'''. The switch will start at '''[https://zonestamp.toolforge.org/{{#time:U|2026-09-23T14:00|en}} {{#time:H:i e|2026-09-23T14:00}}]'''. Unfortunately, because of some limitations in [[mw:Special:MyLanguage/Manual:What is MediaWiki?|MediaWiki]], all editing must stop while the switch is made. We apologize for this disruption, and we are working to minimize it in the future. A banner will be displayed on all wikis 30 minutes before this operation happens. This banner will remain visible until the end of the operation. You can contribute to the [https://meta.wikimedia.org/w/index.php?title=Special%3ATranslate&group=Centralnotice-tgroup-read_only_banner&task=view&language=&filter=&action=translate translation or proofreading] of this banner text. '''You will be able to read, but not edit, all wikis for a short period of time.''' * You will not be able to edit for up to an hour on {{#time:l j xg Y|2026-09-23|en}}. * If you try to edit or save during these times, you will see an error message. We hope that no edits will be lost during these minutes, but we can't guarantee it. If you see the error message, then please wait until everything is back to normal. Then you should be able to save your edit. But, we recommend that you make a copy of your changes first, just in case. ''Other effects'': * Background jobs will be slower and some may be dropped. Red links might not be updated as quickly as normal. If you create an article that is already linked somewhere else, the link will stay red longer than usual. Some long-running scripts will have to be stopped. * We expect the code deployments to happen as any other week. However, some case-by-case code freezes could punctually happen if the operation require them afterwards. * [[mw:Special:MyLanguage/GitLab|GitLab]] will be unavailable for about 90 minutes. This project may be postponed if necessary. You can [[wikitech:Switch_Datacenter|read the schedule at wikitech.wikimedia.org]]. Any changes will be announced in the schedule. '''Please share this information with your community.'''</div><section end="server-switch"/> [[m:user:Trizek (WMF)|Trizek (WMF)]] 12:58, 15 September 2026 (UTC) <!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Non-Technical_Village_Pumps_distribution_list&oldid=30513874 --> == Your feedback wanted: Remove ''(Language link added:) and similar'' edits from Special:RecentChanges == ''(Apologies for posting in English. You can help by translating it)'' Hello, I’m Danny from [[:en:w:Wikimedia_Deutschland|Wikimedia Deutschland’s]] [[m:Wikidata_For_Wikimedia_Projects|Wikidata For Wikimedia Projects]] team.<br/> We have been investigating and examining how Wikidata’s use in the [[Special:RecentChanges]] and [[Special:Watchlist]] pages can be made to be more useful, or less ‘noisy’ (fewer edits that have little value to the reader). We are currently exploring to make a change in the volume and the type of Wikidata edits that appear in these pages, and '''we need your feedback''' to help us ensure this is a positive change for the Wikis and won’t disrupt your workflows.<br/> Language links are also known as sitelinks or interlanguage links and describe when a Wikidata item is connected to a Wiki, adding it to the language dropdown tool for other connected Wikis. [[File:Types of Lang Link changes.png|x350px|frame|A variety of "Language Link" edits as seen in the Recent Changes feed]] The request for feedback is whether you support the removal of these language link edits from the Recent Changes and Watchlist feed, or would you oppose such a change and if so, why? Please share you thoughts on the dedicated '''[[m:Talk:Wikidata_For_Wikimedia_Projects/Clearer_Wikidata_Edit_Summaries/Hide_Language_Link_edits|metawiki Talk page]]'''. Further information about this change: [[m:Wikidata_For_Wikimedia_Projects/Clearer_Wikidata_Edit_Summaries/Hide_Language_Link_edits|m:Wikidata_For_Wikimedia_Projects/Hide_Language_Link_edits]] Thank you for any participation or feedback you give, -- [[m:User:Danny_Benjafield_(WMDE)|User:Danny Benjafield (WMDE)]] 12:23, 25 September 2026 (UTC) <!-- Message sent by User:Danny Benjafield (WMDE)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Danny_Benjafield_(WMDE)/MassMessage_test_list&oldid=31080985 --> == Wikidata Sitelink changes in Recent Changes and Watchlist == ''(Apologies for cross-posting in English, translating it will help spread the message)'' Editors on group 1 wikis, including Wikiquote, Wikibooks, Wikiversity, Wikivoyage, and Wikisource, will no longer see notifications on [[Special:RecentChanges|Recent Changes]] and [[Special:Watchlist|Watchlist]] when a link to ''another'' language version of a page is added, removed, or changed in its Wikidata item. This change will begin on '''October 5 at 07:00 UTC''' as a two-week trial to reduce high-volume, low-value notifications and make higher-value changes easier to identify. Editors who rely on these notifications should be aware of the change and can share feedback or concerns on the [[m:Wikidata For Wikimedia Projects/Clearer Wikidata Edit Summaries/Hide Language Link edits|project]] or [[m:Talk:Wikidata_For_Wikimedia_Projects/Clearer_Wikidata_Edit_Summaries/Hide_Language_Link_edits|talk page]].<br/>Please reach out if you have any questions, thank you! -- [[User:Danny_Benjafield_(WMDE)|Danny Benjafield (WMDE)]] ([[User_talk:Danny_Benjafield_(WMDE)|talk]]) sent via [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 12:42, 1 October 2026 (UTC) <!-- Message sent by User:Danny Benjafield (WMDE)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Danny_Benjafield_(WMDE)/MassMessage_test_list&oldid=31106967 --> == Demote [[Wikiversity:CheckUser policy]]? == I am proposing that we demote that policy to a standard page, mostly for two reasons: # Considering the few amount of custodians and the size of this wiki, I'd say that this project is considered too small to have any CUs, and relying on the stewards would be a more logical choice. # We haven't elected two CUs in a few years. [[User:Codename Noreste|<span style="color: blue">Codename Noreste</span>]] /// [[User talk:Codename Noreste|⭐️✨]] 20:53, 1 October 2026 (UTC) :Reasonable. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 17:09, 4 October 2026 (UTC) :I don't see much point in demoting it. Even if we are not currently (or recently) using the policy, it is there should a need ever arise. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 18:24, 23 October 2026 (UTC) == General Questions and Reference Questions == <p>Hey everyone,</p> <p> My apologies in advance if these questions are oblivious or their responses readily apparent, but I figured I'd ask just to be safe.</p> <p> '''Questions'''</p> <p> 1. Should pages have inline citations and a reference section? A [[Fundamentals of Neuroscience/What is Neuroscience|majority of pages]] I have explored have neither, while [[Neurobiology of emotion|others have a more traditional approach to references]].</br> 2. Are there widely-used templates for weasel-words, references, etc.? </p> '''CONTEXT''': I migrated from Wikipedia, so I'm not entirely sure what the norms/roc are here.</br> Thanks in advance,</br> [[User:SamLaxarox|SamLaxarox]] ([[User talk:SamLaxarox|discuss]] • [[Special:Contributions/SamLaxarox|contribs]]) 18:40, 12 October 2026 (UTC) :You're asking fine questions. I posted {{tl|welcome}} to your talk page which I think will help you. To answer your direct questions: :#This project spun out of [[:b:]] and both here and there, you can post original research. This does not mean that you can post just anything, so if you want to post an educational resource and you want to justify claims you make with inline citations, that's great. If you want to include a bibliography, that's also quite helpful for readers. Generally, the way that Wikiversity works in practice is that individual users make a resource and then tend to work on it solo and have broad latitude about what you can do and how it looks. :#You can take al look at [[:Category:Templates]] and the subcat [[:Category:Maintenance templates]]. At Wikipedia, it's very common to point out issues with tone or uncited claims. That's not really as common here. We [[:d:Q7780480|do not have an equivalent]] of [[:w:en:Template:weasel]], but we do have a [[Template:Citation needed|local version]] of [[:w:en:Template:Citation needed]]. :―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 20:31, 12 October 2026 (UTC) ::Thanks for the heads up on both accounts; I'm happy to join forces with all y'all! ::[[User:SamLaxarox|SamLaxarox]] ([[User talk:SamLaxarox|discuss]] • [[Special:Contributions/SamLaxarox|contribs]]) 23:39, 12 October 2026 (UTC) ==Agent editors== What about inviting agents to edit this wiki? And they would need a human being responsible for them. [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|discuss]] • [[Special:Contributions/Michael Ten|contribs]]) 06:31, 17 October 2026 (UTC) == October 2026 Wikimedia Café: legal issues on Commons, and WMF Board selection process == <div class="border-box" style="background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)"> <div class="box" style="float:left; padding-top: 10px; padding-right: 10px; padding-left: 10px; padding-bottom: 10px;">[[File:Wikimedia Café logo in plain SVG format.svg|60px|alt=The logo for the Wikimedia Café]]</div> Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities in October. In a change from the usual Café practice, this month's sessions will have different topics. Anyone who meets the Café attendance criteria is welcome to attend either or both sessions. Session A will happen on Saturday, 24 October 2026 at 15:00 UTC ([https://zonestamp.toolforge.org/1792854000 timestamp converter]), at a time friendly to the Americas, Africa, and Europe. This session will feature [https://meta.wikimedia.org/wiki/User:PBradley-WMF Wikimedia Foundation lawyer Phil Bradley-Schmieg]. We will discuss legal issues on Wikimedia Commons, including complexities of international law. (I am hoping that in a future month, a similar Café session will be offered at a time friendly to Asia and the Pacific.) Session B will happen on Sunday, 25 October 2026 at 03:00 UTC ([https://zonestamp.toolforge.org/1792897200 timestamp converter]), at a time friendly to Asia and the Pacific, we will discuss [https://meta.wikimedia.org/wiki/Wikimedia_Foundation_elections/2027/Proposed_eligibility_criteria proposed changes to how Wikimedia Foundation Board candidates are selected]. The latest proposals are much different from the earlier July 2026 proposals. The due date for feedback on current proposals is 27 October, and Wikimedians are welcome to participate on the [https://meta.wikimedia.org/wiki/Talk:Wikimedia_Foundation_elections/2027/Proposed_eligibility_criteria proposal talk page] regardless of Café attendance. Please see the [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Café] page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]! To subscribe or unsubscribe for Café notices on your talk page please [https://meta.wikimedia.org/wiki/Global_message_delivery/Targets/Wikimedia_Caf%C3%A9 go here]. <br /> [[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]</div> <span style="white-space:nowrap;">[[User:Pine|<span style="color:#01796f; text-shadow:#00BFFF 0 0 1.0em">༄ᨒ𖠰 Pine</span>]] [[User talk:Pine|<span style="color:DeepSkyBlue">(<b style="color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em">✉</b>)</span>]]</span> 06:22, 18 October 2026 (UTC) == A new cosmological model == I would like to publish independent research that challenges the standard cosmological model. Does this fall within the scope of the "Original research" space? And what precautions should I take to ensure the page is not considered an inadmissible personal work, and is not open to modification by other users? [[User:Wb525|Wb525]] ([[User talk:Wb525|discuss]] • [[Special:Contributions/Wb525|contribs]]) 11:53, 20 October 2026 (UTC) : This sounds like it could be [[Wikiversity:Original research|original research]]. Some things to note: :* Sometimes what an author considers original research turns out to be original theory :* Some qualities that may help to ensure viability: objectivity, use of simple language, careful use of peer-reviewed citations, and minimal use of genAI with careful attribution/acknowledgement if AI is used per [[Wikiversity:Artificial intelligence]] :* Consider starting with a sub-page in user space e.g., [[User:Wb525/Subpage name]] :* As an open knowledge project, Wikiversity content is openly editable by others : -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 23:05, 20 October 2026 (UTC) == Interface administrators policy== This is a discussion of potential changes to [[Wikiversity:Interface administrators]] policy. Please refer to: "For now, Wikiversity does not have a need for permanent or long term interface administrators. Accordingly, interface administratorship may only be granted temporarily by bureaucrats (between 14 and 120 days based on discussion, at the discretion of that bureaucrat and perceived need)." --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 05:14, 23 October 2026 (UTC) === Temporary rights duration === I propose that temporary rights be granted for a duration of from one day to one year. (to conform with selections in [[Special:UserRights]] in the pull down menus.) --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 05:14, 23 October 2026 (UTC) ==== Discussion ==== *{{support}} ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 13:56, 23 October 2026 (UTC) * {{support}}. [[User:Codename Noreste|<span style="color: blue">Codename Noreste</span>]] /// [[User talk:Codename Noreste|⭐️✨]] 14:56, 23 October 2026 (UTC) * {{support}} as proposer.--[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 16:30, 23 October 2026 (UTC) * {{neutral}}; if we're changing the time period again, this time to align with the drop-down options, then I suggest that the policy require that nominators request one of the following time-frames supported by a corresponding plan of work: ** 1 day ** 1 week ** 1 month ** 3 months ** 6 months ** 1 year : -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 02:01, 25 October 2026 (UTC) === Temporary rights extention === I propose that we clarify that temporary rights can be extended. If the community approves of an extension, it can be granted more than once. The temporary rights can repeatedly be extended, for new temporary durations. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 05:14, 23 October 2026 (UTC) ==== Discussion ==== *{{support}} ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 13:56, 23 October 2026 (UTC) * {{support}} for more than one extension. [[User:Codename Noreste|<span style="color: blue">Codename Noreste</span>]] /// [[User talk:Codename Noreste|⭐️✨]] 14:56, 23 October 2026 (UTC) * {{support}} as proposer. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 16:30, 23 October 2026 (UTC) * {{Neutral}} because I'm not sure what the practical difference is between applying for an extension and applying for the right. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 02:04, 25 October 2026 (UTC) === Permanent IA === There has been a suggestion of permanent IA rights. Another suggestion is that there should be at least two members with such rights. Please express your preference. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 05:14, 23 October 2026 (UTC) ==== Discussion ==== *{{support}} ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 13:56, 23 October 2026 (UTC) * {{support}} for having at least two members because there shouldn't be only one. [[User:Codename Noreste|<span style="color: blue">Codename Noreste</span>]] /// [[User talk:Codename Noreste|⭐️✨]] 13:59, 23 October 2026 (UTC) * Conditional {{support}} with the understanding that there be a minimum of 2 (if only 1 is remaining, the other perm is set to temp until a 2nd becomes active) and perm can only be granted after having a temp. I also think a perm nom should have a requirement to demonstrate need - for example show activity logs so that the community can assess that IA is needed long term. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 17:55, 23 October 2026 (UTC) *: You mean that we may have at least 2 permanent IAs, but if one is removed, the other one's is set to temporary until another IA is appointed, makes sense? [[User:Codename Noreste|<span style="color: blue">Codename Noreste</span>]] /// [[User talk:Codename Noreste|⭐️✨]] 20:34, 23 October 2026 (UTC) *::Yes, I meant that if the second IA is no longer active, the remaining IA would be reset to 1 year temporary. That should give enough time to find a second perm IA. That 1 year could be extended, if it takes more time to find a second. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 04:15, 24 October 2026 (UTC) *:::Once the community approves a perm IA, there is no need to renominate. A crat can just reset the time to temp until the conditions are met, and reset to perm when we have two or more. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 04:25, 24 October 2026 (UTC) * {{Neutral}}. Perhaps if there are users who make effective use of IA rights across multiple temporary periods, permanent rights could be useful. My hunch is that is unlikely in the foreseeable future for en.wv to manifest two users who simultaneously maintain that level of engagement. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 02:09, 25 October 2026 (UTC) kxblrwinfuwuk48dfs64y9npcbk5ou8 2842516 2842511 2026-10-25T02:51:22Z Codename Noreste 2969951 /* Discussion */ reply to Jtneill ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]]) 2842516 wikitext text/x-wiki {{Wikiversity:Colloquium/Header}} <!-- MESSAGES GO BELOW --> == Paid part-time Administrative Officer position with WikiJournal == [[WikiJournal User Group|WikiJournal]] is an open-access, nonprofit scholarly publisher hosted on Wikiversity. It uses the wiki format to allow authors to write and develop scholarly articles directly online. Submitted articles then undergo formal academic peer review and, when accepted, are published as citable journal articles while remaining freely accessible. WikiJournal is seeking expressions of interest in a new part-time '''Administrative Officer''' position. The officer would support the project's day-to-day organizational work, including: * Setting up and administering a system for tracking contractor hours and issuing payments. * Coordinating the contracting, onboarding, timekeeping and payment of technical editors and other contractors. * Maintaining financial, contractor and administrative records. * Assisting with grant administration, budget tracking and reporting. * Following up on board action items, deadlines and recurring obligations. * Supporting recruitment and other operational tasks as needed. This is a remote, flexibly scheduled contractor role averaging approximately '''6 hours per week'''. Compensation is proposed at '''US$25 per hour''', with up to approximately '''320 hours (US$8,000) available over the year'''. Applicants may be based internationally, although appointment will depend on WikiJournal being able to establish a practical and legally appropriate contracting and payment arrangement in the applicant's country of residence. Relevant experience may include administrative organization, bookkeeping or financial administration, contractor or personnel coordination, nonprofit or grant administration, Wikimedia participation, academic publishing, and open-access work. Applicants are not expected to have experience in every area. For the complete role description and further details, see [[WikiJournal User Group/Administrative officer]]. Those interested should reply in the '''[[Talk:WikiJournal User Group#Call for interest: Administrative Officer|Call for interest: Administrative Officer]]''' discussion by '''30 September 2026''', briefly describing their relevant experience, any connection to WikiJournal or related communities, and their general availability. Expressions of interest received later may still be considered if the position has not been filled. Recommendations of potentially suitable candidates are also welcome. [[User:Mikael Häggström|Mikael Häggström]] ([[User talk:Mikael Häggström|discuss]] • [[Special:Contributions/Mikael Häggström|contribs]]) 20:00, 10 September 2026 (UTC) == Server switch - Your wiki will be read-only for a short time soon == <section begin="server-switch"/><div class="plainlinks"> [[:m:Special:MyLanguage/Tech/Server switch|Read this message in another language]] • [https://meta.wikimedia.org/w/index.php?title=Special:Translate&group=page-Tech%2FServer+switch&language=&action=page&filter= {{int:please-translate}}] The [[foundation:|Wikimedia Foundation]] will switch the traffic between its data centers. This will make sure that Wikipedia and the other Wikimedia wikis can stay online even after a disaster. All traffic will switch on '''{{#time:j xg|2026-09-23|en}}'''. The switch will start at '''[https://zonestamp.toolforge.org/{{#time:U|2026-09-23T14:00|en}} {{#time:H:i e|2026-09-23T14:00}}]'''. Unfortunately, because of some limitations in [[mw:Special:MyLanguage/Manual:What is MediaWiki?|MediaWiki]], all editing must stop while the switch is made. We apologize for this disruption, and we are working to minimize it in the future. A banner will be displayed on all wikis 30 minutes before this operation happens. This banner will remain visible until the end of the operation. You can contribute to the [https://meta.wikimedia.org/w/index.php?title=Special%3ATranslate&group=Centralnotice-tgroup-read_only_banner&task=view&language=&filter=&action=translate translation or proofreading] of this banner text. '''You will be able to read, but not edit, all wikis for a short period of time.''' * You will not be able to edit for up to an hour on {{#time:l j xg Y|2026-09-23|en}}. * If you try to edit or save during these times, you will see an error message. We hope that no edits will be lost during these minutes, but we can't guarantee it. If you see the error message, then please wait until everything is back to normal. Then you should be able to save your edit. But, we recommend that you make a copy of your changes first, just in case. ''Other effects'': * Background jobs will be slower and some may be dropped. Red links might not be updated as quickly as normal. If you create an article that is already linked somewhere else, the link will stay red longer than usual. Some long-running scripts will have to be stopped. * We expect the code deployments to happen as any other week. However, some case-by-case code freezes could punctually happen if the operation require them afterwards. * [[mw:Special:MyLanguage/GitLab|GitLab]] will be unavailable for about 90 minutes. This project may be postponed if necessary. You can [[wikitech:Switch_Datacenter|read the schedule at wikitech.wikimedia.org]]. Any changes will be announced in the schedule. '''Please share this information with your community.'''</div><section end="server-switch"/> [[m:user:Trizek (WMF)|Trizek (WMF)]] 12:58, 15 September 2026 (UTC) <!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Non-Technical_Village_Pumps_distribution_list&oldid=30513874 --> == Your feedback wanted: Remove ''(Language link added:) and similar'' edits from Special:RecentChanges == ''(Apologies for posting in English. You can help by translating it)'' Hello, I’m Danny from [[:en:w:Wikimedia_Deutschland|Wikimedia Deutschland’s]] [[m:Wikidata_For_Wikimedia_Projects|Wikidata For Wikimedia Projects]] team.<br/> We have been investigating and examining how Wikidata’s use in the [[Special:RecentChanges]] and [[Special:Watchlist]] pages can be made to be more useful, or less ‘noisy’ (fewer edits that have little value to the reader). We are currently exploring to make a change in the volume and the type of Wikidata edits that appear in these pages, and '''we need your feedback''' to help us ensure this is a positive change for the Wikis and won’t disrupt your workflows.<br/> Language links are also known as sitelinks or interlanguage links and describe when a Wikidata item is connected to a Wiki, adding it to the language dropdown tool for other connected Wikis. [[File:Types of Lang Link changes.png|x350px|frame|A variety of "Language Link" edits as seen in the Recent Changes feed]] The request for feedback is whether you support the removal of these language link edits from the Recent Changes and Watchlist feed, or would you oppose such a change and if so, why? Please share you thoughts on the dedicated '''[[m:Talk:Wikidata_For_Wikimedia_Projects/Clearer_Wikidata_Edit_Summaries/Hide_Language_Link_edits|metawiki Talk page]]'''. Further information about this change: [[m:Wikidata_For_Wikimedia_Projects/Clearer_Wikidata_Edit_Summaries/Hide_Language_Link_edits|m:Wikidata_For_Wikimedia_Projects/Hide_Language_Link_edits]] Thank you for any participation or feedback you give, -- [[m:User:Danny_Benjafield_(WMDE)|User:Danny Benjafield (WMDE)]] 12:23, 25 September 2026 (UTC) <!-- Message sent by User:Danny Benjafield (WMDE)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Danny_Benjafield_(WMDE)/MassMessage_test_list&oldid=31080985 --> == Wikidata Sitelink changes in Recent Changes and Watchlist == ''(Apologies for cross-posting in English, translating it will help spread the message)'' Editors on group 1 wikis, including Wikiquote, Wikibooks, Wikiversity, Wikivoyage, and Wikisource, will no longer see notifications on [[Special:RecentChanges|Recent Changes]] and [[Special:Watchlist|Watchlist]] when a link to ''another'' language version of a page is added, removed, or changed in its Wikidata item. This change will begin on '''October 5 at 07:00 UTC''' as a two-week trial to reduce high-volume, low-value notifications and make higher-value changes easier to identify. Editors who rely on these notifications should be aware of the change and can share feedback or concerns on the [[m:Wikidata For Wikimedia Projects/Clearer Wikidata Edit Summaries/Hide Language Link edits|project]] or [[m:Talk:Wikidata_For_Wikimedia_Projects/Clearer_Wikidata_Edit_Summaries/Hide_Language_Link_edits|talk page]].<br/>Please reach out if you have any questions, thank you! -- [[User:Danny_Benjafield_(WMDE)|Danny Benjafield (WMDE)]] ([[User_talk:Danny_Benjafield_(WMDE)|talk]]) sent via [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 12:42, 1 October 2026 (UTC) <!-- Message sent by User:Danny Benjafield (WMDE)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Danny_Benjafield_(WMDE)/MassMessage_test_list&oldid=31106967 --> == Demote [[Wikiversity:CheckUser policy]]? == I am proposing that we demote that policy to a standard page, mostly for two reasons: # Considering the few amount of custodians and the size of this wiki, I'd say that this project is considered too small to have any CUs, and relying on the stewards would be a more logical choice. # We haven't elected two CUs in a few years. [[User:Codename Noreste|<span style="color: blue">Codename Noreste</span>]] /// [[User talk:Codename Noreste|⭐️✨]] 20:53, 1 October 2026 (UTC) :Reasonable. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 17:09, 4 October 2026 (UTC) :I don't see much point in demoting it. Even if we are not currently (or recently) using the policy, it is there should a need ever arise. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 18:24, 23 October 2026 (UTC) == General Questions and Reference Questions == <p>Hey everyone,</p> <p> My apologies in advance if these questions are oblivious or their responses readily apparent, but I figured I'd ask just to be safe.</p> <p> '''Questions'''</p> <p> 1. Should pages have inline citations and a reference section? A [[Fundamentals of Neuroscience/What is Neuroscience|majority of pages]] I have explored have neither, while [[Neurobiology of emotion|others have a more traditional approach to references]].</br> 2. Are there widely-used templates for weasel-words, references, etc.? </p> '''CONTEXT''': I migrated from Wikipedia, so I'm not entirely sure what the norms/roc are here.</br> Thanks in advance,</br> [[User:SamLaxarox|SamLaxarox]] ([[User talk:SamLaxarox|discuss]] • [[Special:Contributions/SamLaxarox|contribs]]) 18:40, 12 October 2026 (UTC) :You're asking fine questions. I posted {{tl|welcome}} to your talk page which I think will help you. To answer your direct questions: :#This project spun out of [[:b:]] and both here and there, you can post original research. This does not mean that you can post just anything, so if you want to post an educational resource and you want to justify claims you make with inline citations, that's great. If you want to include a bibliography, that's also quite helpful for readers. Generally, the way that Wikiversity works in practice is that individual users make a resource and then tend to work on it solo and have broad latitude about what you can do and how it looks. :#You can take al look at [[:Category:Templates]] and the subcat [[:Category:Maintenance templates]]. At Wikipedia, it's very common to point out issues with tone or uncited claims. That's not really as common here. We [[:d:Q7780480|do not have an equivalent]] of [[:w:en:Template:weasel]], but we do have a [[Template:Citation needed|local version]] of [[:w:en:Template:Citation needed]]. :―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 20:31, 12 October 2026 (UTC) ::Thanks for the heads up on both accounts; I'm happy to join forces with all y'all! ::[[User:SamLaxarox|SamLaxarox]] ([[User talk:SamLaxarox|discuss]] • [[Special:Contributions/SamLaxarox|contribs]]) 23:39, 12 October 2026 (UTC) ==Agent editors== What about inviting agents to edit this wiki? And they would need a human being responsible for them. [[User:Michael Ten|Michael Ten]] ([[User talk:Michael Ten|discuss]] • [[Special:Contributions/Michael Ten|contribs]]) 06:31, 17 October 2026 (UTC) == October 2026 Wikimedia Café: legal issues on Commons, and WMF Board selection process == <div class="border-box" style="background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)"> <div class="box" style="float:left; padding-top: 10px; padding-right: 10px; padding-left: 10px; padding-bottom: 10px;">[[File:Wikimedia Café logo in plain SVG format.svg|60px|alt=The logo for the Wikimedia Café]]</div> Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities in October. In a change from the usual Café practice, this month's sessions will have different topics. Anyone who meets the Café attendance criteria is welcome to attend either or both sessions. Session A will happen on Saturday, 24 October 2026 at 15:00 UTC ([https://zonestamp.toolforge.org/1792854000 timestamp converter]), at a time friendly to the Americas, Africa, and Europe. This session will feature [https://meta.wikimedia.org/wiki/User:PBradley-WMF Wikimedia Foundation lawyer Phil Bradley-Schmieg]. We will discuss legal issues on Wikimedia Commons, including complexities of international law. (I am hoping that in a future month, a similar Café session will be offered at a time friendly to Asia and the Pacific.) Session B will happen on Sunday, 25 October 2026 at 03:00 UTC ([https://zonestamp.toolforge.org/1792897200 timestamp converter]), at a time friendly to Asia and the Pacific, we will discuss [https://meta.wikimedia.org/wiki/Wikimedia_Foundation_elections/2027/Proposed_eligibility_criteria proposed changes to how Wikimedia Foundation Board candidates are selected]. The latest proposals are much different from the earlier July 2026 proposals. The due date for feedback on current proposals is 27 October, and Wikimedians are welcome to participate on the [https://meta.wikimedia.org/wiki/Talk:Wikimedia_Foundation_elections/2027/Proposed_eligibility_criteria proposal talk page] regardless of Café attendance. Please see the [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Café] page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]! To subscribe or unsubscribe for Café notices on your talk page please [https://meta.wikimedia.org/wiki/Global_message_delivery/Targets/Wikimedia_Caf%C3%A9 go here]. <br /> [[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]</div> <span style="white-space:nowrap;">[[User:Pine|<span style="color:#01796f; text-shadow:#00BFFF 0 0 1.0em">༄ᨒ𖠰 Pine</span>]] [[User talk:Pine|<span style="color:DeepSkyBlue">(<b style="color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em">✉</b>)</span>]]</span> 06:22, 18 October 2026 (UTC) == A new cosmological model == I would like to publish independent research that challenges the standard cosmological model. Does this fall within the scope of the "Original research" space? And what precautions should I take to ensure the page is not considered an inadmissible personal work, and is not open to modification by other users? [[User:Wb525|Wb525]] ([[User talk:Wb525|discuss]] • [[Special:Contributions/Wb525|contribs]]) 11:53, 20 October 2026 (UTC) : This sounds like it could be [[Wikiversity:Original research|original research]]. Some things to note: :* Sometimes what an author considers original research turns out to be original theory :* Some qualities that may help to ensure viability: objectivity, use of simple language, careful use of peer-reviewed citations, and minimal use of genAI with careful attribution/acknowledgement if AI is used per [[Wikiversity:Artificial intelligence]] :* Consider starting with a sub-page in user space e.g., [[User:Wb525/Subpage name]] :* As an open knowledge project, Wikiversity content is openly editable by others : -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 23:05, 20 October 2026 (UTC) == Interface administrators policy== This is a discussion of potential changes to [[Wikiversity:Interface administrators]] policy. Please refer to: "For now, Wikiversity does not have a need for permanent or long term interface administrators. Accordingly, interface administratorship may only be granted temporarily by bureaucrats (between 14 and 120 days based on discussion, at the discretion of that bureaucrat and perceived need)." --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 05:14, 23 October 2026 (UTC) === Temporary rights duration === I propose that temporary rights be granted for a duration of from one day to one year. (to conform with selections in [[Special:UserRights]] in the pull down menus.) --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 05:14, 23 October 2026 (UTC) ==== Discussion ==== *{{support}} ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 13:56, 23 October 2026 (UTC) * {{support}}. [[User:Codename Noreste|<span style="color: blue">Codename Noreste</span>]] /// [[User talk:Codename Noreste|⭐️✨]] 14:56, 23 October 2026 (UTC) * {{support}} as proposer.--[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 16:30, 23 October 2026 (UTC) * {{neutral}}; if we're changing the time period again, this time to align with the drop-down options, then I suggest that the policy require that nominators request one of the following time-frames supported by a corresponding plan of work: ** 1 day ** 1 week ** 1 month ** 3 months ** 6 months ** 1 year : -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 02:01, 25 October 2026 (UTC) === Temporary rights extention === I propose that we clarify that temporary rights can be extended. If the community approves of an extension, it can be granted more than once. The temporary rights can repeatedly be extended, for new temporary durations. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 05:14, 23 October 2026 (UTC) ==== Discussion ==== *{{support}} ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 13:56, 23 October 2026 (UTC) * {{support}} for more than one extension. [[User:Codename Noreste|<span style="color: blue">Codename Noreste</span>]] /// [[User talk:Codename Noreste|⭐️✨]] 14:56, 23 October 2026 (UTC) * {{support}} as proposer. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 16:30, 23 October 2026 (UTC) * {{Neutral}} because I'm not sure what the practical difference is between applying for an extension and applying for the right. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 02:04, 25 October 2026 (UTC) *: On a nomination or IA request, a member may question what timeframe a candidate may request, or the candidate may state what timeframe they need for IA rights. [[User:Codename Noreste|<span style="color: blue">Codename Noreste</span>]] /// [[User talk:Codename Noreste|⭐️✨]] 02:51, 25 October 2026 (UTC) === Permanent IA === There has been a suggestion of permanent IA rights. Another suggestion is that there should be at least two members with such rights. Please express your preference. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 05:14, 23 October 2026 (UTC) ==== Discussion ==== *{{support}} ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 13:56, 23 October 2026 (UTC) * {{support}} for having at least two members because there shouldn't be only one. [[User:Codename Noreste|<span style="color: blue">Codename Noreste</span>]] /// [[User talk:Codename Noreste|⭐️✨]] 13:59, 23 October 2026 (UTC) * Conditional {{support}} with the understanding that there be a minimum of 2 (if only 1 is remaining, the other perm is set to temp until a 2nd becomes active) and perm can only be granted after having a temp. I also think a perm nom should have a requirement to demonstrate need - for example show activity logs so that the community can assess that IA is needed long term. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 17:55, 23 October 2026 (UTC) *: You mean that we may have at least 2 permanent IAs, but if one is removed, the other one's is set to temporary until another IA is appointed, makes sense? [[User:Codename Noreste|<span style="color: blue">Codename Noreste</span>]] /// [[User talk:Codename Noreste|⭐️✨]] 20:34, 23 October 2026 (UTC) *::Yes, I meant that if the second IA is no longer active, the remaining IA would be reset to 1 year temporary. That should give enough time to find a second perm IA. That 1 year could be extended, if it takes more time to find a second. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 04:15, 24 October 2026 (UTC) *:::Once the community approves a perm IA, there is no need to renominate. A crat can just reset the time to temp until the conditions are met, and reset to perm when we have two or more. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 04:25, 24 October 2026 (UTC) * {{Neutral}}. Perhaps if there are users who make effective use of IA rights across multiple temporary periods, permanent rights could be useful. My hunch is that is unlikely in the foreseeable future for en.wv to manifest two users who simultaneously maintain that level of engagement. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 02:09, 25 October 2026 (UTC) aakqnurhbdns6jb06ydavhcvunxu6yj 11 34400 2842485 1846161 2026-10-25T00:28:27Z ~2026-56788-08 3113246 /* welcome for anons? */ new section 2842485 wikitext text/x-wiki ==Old welcome== See also: [[Template talk:Welcome/old]]. --[[User:JWSchmidt|JWSchmidt]] 03:21, 12 May 2007 (UTC) *And also [[template:welcome/Unstable]].[[User:Hillgentleman|Hillgentleman]]|[[User talk:hillgentleman|Talk]] 11:43, 17 May 2007 (UTC) ::I don't see the point of a duplicate sandbox unless and until there are two divergent ideas of how to sandbox this template. {{tl|delete}} number 2. [[User:Cormaggio|Cormaggio]] ^{<small>[[User talk:Cormaggio|talk]]</small>} 13:46, 17 May 2007 (UTC) ==Simplifying and focusing== I like the initiative to get rid of the html for now, and focus on the links. Once we have an acceptable number of links, then we can perhaps work on a nice formatting (and I would again advocate keeping it basic, and avoid the mountain of code that currently gets added to people's talk pages. It firstly seems to me that linking to [[Wikiversity:Cite sources|Cite sources]] [[Wikiversity:Verifiability]], [[WV:NPOV|NPOV]] and [[Wikiversity:Scholarly ethics]] is slight overkill - perhaps linking to '''one''' of these would be enough, and then place this page (perhaps Scholarly ethics?) within a context of a few other key policies. I think "research guidelines" is unnecesary - unless the person shows an interest in research, and this can be added later. [[Wikiversity:Help desk]] is repeated twice (and replicated in [[Wikiversity:Questions]], and should really be linked from the Colloquium). I then think that the headings could be removed or reduced - for example, I've never liked the labelling of the Colloquium as "getting technical", as the Colloquium should be a place ''anyone'' can ask questions or make comments about Wikiversity - from the absolute newbie to the "veteran" (if anyone can be so named). The main thing this is missing is a way for people to understand learning in Wikiversity, and how to use Wikiversity for their own learning. [[Portal:Education/Wikiversity model]] is a good start (if a bit wordy) - perhaps we can make a user-friendly introduction to "learning by doing"? We also don't have a link to [[Wikiversity:Browse]] - this could be useful. What else could be useful? What does a newbie need to get up to speed with Wikiversity as quickly as they can? [[User:Cormaggio|Cormaggio]] ^{<small>[[User talk:Cormaggio|talk]]</small>} 22:14, 21 May 2007 (UTC) *There are different kinds of "newbies" and the [[Main Page]] should provide them with links to what they need. We should not be using the welcome template to provide what the Main Page should provide. I have been [[Main Page/Draft version 0.2|working towards]] a small set of "newbie tracks":<BR> #new to wiki<BR> #experienced with wiki but new to Wikiversity<BR> #browse track<BR> #track for teachers who want to use Wikiversity to host a classroom project<BR> #"heavy duty" content development track<BR> If we had a Main Page that provided newbies with what they need then the welcome template could really become a welcome template rather than a repair kit for the failings of the main page. --[[User:JWSchmidt|JWSchmidt]] 01:03, 22 May 2007 (UTC) :Good points. :-) A welcome template should not be making up for the Main Page's deficiencies, but have an emphasis on enculturation within Wikiversity, ie becoming part of the community and being motivated to explore and/or participate. However, I think it will still need to include some basic info on navigating and editing - which can be ignored if the user is already wiki-literate. [[User:Cormaggio|Cormaggio]] ^{<small>[[User talk:Cormaggio|talk]]</small>} 10:11, 22 May 2007 (UTC) *Is it assumed that a registered user knows how to browse and to obtain information? On wikipedia one can get away with entering almost any word into the search box and it would work. Not so here. I added "navigation".[[User:Hillgentleman|Hillgentleman]]|[[User talk:hillgentleman|Talk]] == What are the essentials? == I think the trick here is to go for something simple and direct, and not overwhelm the newcomer too much. Some ideas: * A welcome message should come across as sincere, warm, and friendly. * A welcome message should provide links for newcomers to help themselves easily, and have access to help easily * A welcome message should be nurturing - it should encourage participation in, and generation of content * A welcome message should be short - it is not supposed to express all the nuances that a Wiki should have * ... --[[User:HappyCamper|HappyCamper]] 03:13, 27 May 2007 (UTC) ::I heartily second ''all'' of that. :-) [[User:Cormaggio|Cormaggio]] ^{<small>[[User talk:Cormaggio|talk]]</small>} 09:00, 29 May 2007 (UTC) == Taking shape == OK, I think this new welcome template is taking shape. Comments? Suggestions? --[[User:HappyCamper|HappyCamper]] 02:04, 30 May 2007 (UTC) :It's looking good! I wonder if there should be a link to [[Wikiversity:Introduction]]? I also wonder why we are telling people they can "ask the support staff", when it's more of a distributed task (cf. [[Wikiversity:Questions]], as linked below). I also think there should be more clear emphasis or guidance on how to get involved - I'm trying to think of a good single link for that... (does it exist?) [[User:Cormaggio|Cormaggio]] ^{<small>[[User talk:Cormaggio|talk]]</small>} 19:42, 30 May 2007 (UTC) ::What if we just replaced that "ask the support staff line" with the "where to ask questions" links? Take a look...hmm...I just tried using the template. I think it's good, but something feels missing from it still. --[[User:HappyCamper|HappyCamper]] 00:53, 31 May 2007 (UTC) :::I've made one or two changes. I think what's missing is a single page which details how to '''get involved''' - this page should explian all of the ways that people can add to existing pages, upload their own work, and use Wikiversity to learn collaboratively. I made a start on the first parts of this at [[Wikiversity:Adding content]], but I think one single page (or perhaps tutorial) on [[Wikiversity:Getting involved|getting involved]] would be better. [[User:Cormaggio|Cormaggio]] ^{<small>[[User talk:Cormaggio|talk]]</small>} 20:59, 31 May 2007 (UTC) ::::I like that change re: policies - very, ''very'' nice! Other than that getting involved page, do you think this is ready for use? --[[User:HappyCamper|HappyCamper]] 00:00, 1 June 2007 (UTC) :::::I think it's pretty much ready, but could be significantly improved by developing the getting involved page, and of course some better main page design.. :-) [[User:Cormaggio|Cormaggio]] ^{<small>[[User talk:Cormaggio|talk]]</small>} 16:12, 2 June 2007 (UTC) ::Will this have the table formatting similar to {{tl|welcome}}? I like the simplification. Nice work! -- [[User:CQ|CQ]] 16:53, 2 June 2007 (UTC) :::I kinda prefer the simplified structure - it certainly cuts down on that sea of code that people are currently faced with when they go to edit their talk page after the welcome template has been added... [[User:Cormaggio|Cormaggio]] ^{<small>[[User talk:Cormaggio|talk]]</small>} 16:59, 2 June 2007 (UTC) ::::Let's experiment. I split the bullets into 2 columns. How does it look? If it looks ready for use, I think the next project to work on is the main page. Maybe we should point out the links to the left, and encourage people to click on them. --[[User:HappyCamper|HappyCamper]] 17:05, 2 June 2007 (UTC) :::::Personally, I reckon we should implement this as the default welcome message, and then tweak it and other related pages as we go. Thanks for all the work - especially HappyCamper! [[User:Cormaggio|Cormaggio]] ^{<small>[[User talk:Cormaggio|talk]]</small>} 21:43, 7 June 2007 (UTC) ::::::I'm using this template sporadically. I'd like to add colors to it, but perhaps simple is best. I think I might move on to the main page. --[[User:HappyCamper|HappyCamper]] 18:11, 8 June 2007 (UTC) ==Got one on my user discussion page== I got something like this put on my user discussion page and I'm here to tell you it is IMPRESSIVE. Boy did I feel welcome! Thanks so much.[[User:Harriska2|Harriska2]] 22:14, 4 June 2007 (UTC) :Me too. I've seen the old one on the pages of other users & I'm glad I got the new one instead—short & sweet ... and best of all, looks as if someone might just have written it. Nice work. [[User:Jimp|Jimp]] 02:19, 18 June 2007 (UTC) == template problem == The code for having this template include the message, "<nowiki>This welcome message was added by placing {{subst:welcome}} on the user page.</nowiki>" was creating a problem. It was generating "WARNING: template loop detected" and adding "on the user page. -->" to the talk page when used by substitution. I [http://en.wikiversity.org/w/index.php?title=Template%3AWelcome&diff=150369&oldid=149542 fixed the problem], but my fix is a hack. --[[User:JWSchmidt|JWSchmidt]] 18:47, 18 August 2007 (UTC) :I'm a little surprised this loop thing happens - anyway, I've reverted the changes. I think I might file a bug report. --[[User:HappyCamper|HappyCamper]] 19:32, 18 August 2007 (UTC) :*The template expansion does take place during substitution even if you put < ! -- ---> around it. The same is true for the < p r e> < / p r e> tags. If you really want to write <nowiki>{{subst:welcome}}</nowiki> on a page, you would need something like [[template:subst]] to get it correct, eg with the wikitext <code><nowiki>{{subst:subst}}welcome{{subst:rightbraces}}</nowiki></code>. Anyhow, I have removed the comment, because, if the user understands it, she doesn't really need much of the welcome message.[[User:Hillgentleman|Hillgentleman]]|[[User talk:hillgentleman|Talk]] 19:41, 20 September 2007 (UTC) ::This peculiarity or feature of the mediawiki has gone with the old preprocessor. [[User:Hillgentleman|Hillgentleman]]|[[User talk:hillgentleman|Talk]] 01:02, 3 May 2008 (UTC) == the basics== The basic wiki-mark-up's are <nowiki>== ==, [[]], '' '' and indentations *,:,#. </nowiki> I propose that the welcome template include at least one of each. Put the first line (greeting) in the section header. [[User:Hillgentleman|Hillgentleman]]|[[User talk:hillgentleman|Talk]] 00:46, 19 September 2007 (UTC) ==Smiley== I've got mixed feelings about the new box / smiley. Perhaps we could have a parameter to switch smiley on/off? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]]</small> 02:10, 5 May 2008 (UTC) :I find the smiley + box a change to the normal wiki-style (or I just got used too much to it) - brings new colorful life wiki-life. Well, what a newcomer thinks about it we should also find out. Let's make a survey :-) Why not also make a copy from the old one also for usage - and perhaps make it dependent on a random function and when using the template we also get a surprise each time which of it gets selected :-) :btw: what are the mixed feelings about ? ----[[User:Erkan_Yilmaz|Erkan Yilmaz]] <small>uses the [[Wikiversity:Chat|Wikiversity:Chat]] ([http://java.freenode.net//index.php?channel=wikiversity-en try])</small> 20:18, 5 May 2008 (UTC) :*I kept the version before McCormack at [[template:welcome/plain]]. :*Jtneill, Yes, it can be done. At the appropriate place, replace the relevant wikidcode for the smiley and the frame by: <kbd><nowiki>{{subst:#ifeq:{{{smiley|}}}|off|<!--nothing-->|the wikicode for the smiley and the frame|}}</nowiki></kbd>. :*Erkan, Yes, it can be done. Instead of the above, use something like <nowiki>{{subst:#ifeq:{{subst:#expr:{{SUBST:CURRENTTIMESTAMP}} mod 2}}|0|<!--nothing-->|wikicode for the smiley}}</nowiki> :*Each is good in its own way. Personally, I believe that the welcome message itself is often the ''first example'' for new-comers, wherefrom they learn their first set of wiki-syntax rules. Therefore I favour simplicity. It also partly explains why we had the three fundamental markups<kbd><nowiki>== ==, [[]], * </nowiki></kbd> appearing in the welcome message. [[User:Hillgentleman|Hillgentleman]]|[[User talk:hillgentleman|Talk]] 21:14, 5 May 2008 (UTC) :::Good arguments and tips. Perhaps in the new (and perhaps more complex appearing wiki syntax for a newcomer) welcome msg there could be made invisible comments which can help the newcomer when viewing the source code to adapt it ? ----[[User:Erkan_Yilmaz|Erkan Yilmaz]] <small>uses the [[Wikiversity:Chat|Wikiversity:Chat]] ([http://java.freenode.net//index.php?channel=wikiversity-en try])</small> 21:30, 5 May 2008 (UTC) ::::A little bit perhaps. The template is just a wikified welcome message; it can be a good first example (for a good wiki-style) but it shouldn't be a wikitext tutorial - that could be more confusing. [[User:Hillgentleman|Hillgentleman]]|[[User talk:hillgentleman|Talk]] 21:47, 5 May 2008 (UTC) ::::: Just reading [[User:Jtneill]]'s comment about the smiley. Yes - it is rather big and bold, isn't it? I increased the icon size until I was scared, and then added a bit more for good measure. Most newcomers never stay at WV for long, disappearing within a few hours for good. So my thinking was "seek attention", however much it scares you. But sure, lots of different versions to suit everyone's tastes. How about a version which changes colour every day? --[[User:McCormack|McCormack]] 03:41, 6 May 2008 (UTC) :::::: We could boost the shock value by having the Welcome messages signed randomly by one of the users in [[Wikiversity:Mascot contest|Mascot contest]]. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]]</small> 03:58, 6 May 2008 (UTC) ::::::: Eyes right. --[[User:McCormack|McCormack]] 04:06, 6 May 2008 (UTC) [[Image:Smiley green alien droling.svg|64px|thumb|right|McCormack's favourite.]] ::::::: Maybe if interested we can play with [[Template:Random mascot]]. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]]</small> 04:16, 6 May 2008 (UTC) ::::::::Nice, I wish I would have welcomed like this back then :-) ----[[User:Erkan_Yilmaz|Erkan Yilmaz]] <small>uses the [[Wikiversity:Chat|Wikiversity:Chat]] ([http://java.freenode.net//index.php?channel=wikiversity-en try])</small> 15:48, 6 May 2008 (UTC) ==mentioning the sandbox== Notice that the sandbox is linked from the very top of [[help:editing]], which was linked from the very first bullet (take a guided tour and learn to edit) in the welcoming message. The direct link to the sandbox may be useful, but it may not be useful enough (experienced editors know it already; newcomers may not know how to edit at all, and had better follow the advice and go to [[help:editing]]). [[User:Hillgentleman|Hillgentleman]]|[[User talk:hillgentleman|Talk]] 04:27, 6 May 2008 (UTC) : Feel free to remove the Sandbox link. I just got the feeling that some of the unwanted page creations are good faith edits by newcomers looking for somewhere to play. The Sandbox isn't really that user-friendly... So, what about a direct link to something like '''edit my user page'''? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]]</small> 04:33, 6 May 2008 (UTC) == Updates == I updated this a bit today. --[[User:McCormack|McCormack]] 15:43, 4 July 2008 (UTC) == Signature Icon == Please changing the signature icon with new version. Namely: [[File:Signature icon.png|link=|alt=]] replace with [[File:Insert-signature.png|link=|alt=]].--[[User:Mehdi|Mehdi]] 09:52, 8 September 2011 (UTC) :Thanks for catching that. --[[User:HappyCamper|HappyCamper]] 20:21, 19 November 2011 (UTC) ::Can you change the file as <nowiki>[[File:Insert-signature.png]]</nowiki> to <nowiki>[[File:Insert-signature.png|alt=|link=]]</nowiki> as it is in most Wikipedia templates. --[[User:Extra999|Extra999]] 05:55, 9 February 2012 (UTC) == welcome for anons? == I did not get a welcome message when I started editing here as an anon. I also don't see if there is a way to post a welcome template on new or anon usertalk pages without signing the message. I sometimes come (not often I admit) across a user I would like to welcome and provide them with some links to Help pages, but since anon accounts only stay around for 3 months, I don't think it's good practice to sign those welcome messages Is there a way to post such messages here? Thanks in advance, [[Special:Contributions/~2026-56788-08|~2026-56788-08]] ([[User talk:~2026-56788-08|talk]]) 00:28, 25 October 2026 (UTC) 2jzui9vj9twpkrcw6cgpnm90xu9dxzs 0 107808 2842446 2780011 2026-10-24T19:26:37Z CommonsDelinker 9184 Removing [[:c:File:Al_Akhawayn_Campus.jpg|Al_Akhawayn_Campus.jpg]], it has been deleted from Commons by [[:c:User:Jameslwoodward|Jameslwoodward]] because: per [[:c:Commons:Deletion requests/File:Al Akhawayn Campus.jpg|]]. 2842446 wikitext text/x-wiki Part of the [[Comparative law and justice]] Wikiversity Project {{Comparative law and justice project|region=Africa}} <!when you have set up your page, delete this line and replace it with four ~!> [[File:Morocco-flag.png|thumb|Morocco-flag]] [[Image:Animated-Flag-Morocco.gif|thumb|left|200px|alt=A|Moroccan Flag]] == Basic Information == '''Major Cities''' The capital of Morocco is Rabat. The government and key parliamentary offices are located in the capital. The largest city is its main port city of Casablanca.[[Image:CasablancaFAR.jpg|thumb|right|200px|alt=A|Casablanca]] [[Image:Rabat_Parliament.jpg|thumb|left|200px|alt=A|Parliament Building in Rabat]] '''Flag''' Red with a green pentacle (five-pointed, linear star) known as a variety of Solomon's six-pointed seal in the center of the flag. Red and green are traditional colors in Berber flags. The current Moroccan flag was made by the French governor-general Hubert Lyautey who replaced the David star in the Moroccan flag with a pentacle in 1915 and made it official<ref>http://www.theodora.com/wfbcurrent/morocco/morocco_government.html</ref>. '''National Anthem''' - Name: "Hymne Cherifien" (Hymn of the Sharif) - Lyrics/music: Ali Squalli Housainni/Leo Morgan - Note: music adopted 1956, lyrics adopted 1970<ref>http://www.theodora.com/wfbcurrent/morocco/morocco_government.html</ref> '''Geographical Information''' [[Image:Tizi'n'Toubkal.jpg|thumb|left|200px|alt=A|Jebel Toubkal Mountain Range]]Morocco is located in Northern Africa, bordering the Atlantic Ocean and the Mediterranean Sea, between Algeria and the Western Sahara desert. Morocco's landmass totals 446,550 sq. km. which is the 57 (slightly larger than California) largest country in the world. Morocco's northern coast and interior are mountainous with large bordering plateaus, valleys and rich coastal plains. the lowest point is Sebkha Tah (-55m) and its highest point is Jebel Toubkal (4,165m)<ref>https://www.cia.gov/library/publications/the-world-factbook/geos/mo.html</ref>.[[Image:Morocco & Western Sahara map.jpg|thumb|right|200px|alt=A|Morocco]] '''Climate''' The climate in Morocco is reliably dry, although small amounts of rain do fall between November and March. Temperature varies considerably by season and locale. While the southern and southeastern desert regions can reach extremely high temperatures during the hot summer months, the higher altitudes of the mountains are cool in summer evenings and freezing in winter<ref>http://www.geographia.com/morocco/</ref>. '''Resources''' Morocco has several key natural resources such as iron ore, lead, zinc, phosphates, manganese, fish, and salt. Large earthquakes and periodic droughts create natural hazards due to the mountainous terrain and the Sahara Desert<ref>https://www.cia.gov/library/publications/the-world-factbook/geos/mo.html</ref>. '''Population''' The population of Morocco (July 2010) is estimated at 31,627,428, 38 largest in the world. A 2010 estimate states that Morocco's population grows at an estimated 1.077% per year, good for 118 fastest in the world<ref>https://www.cia.gov/library/publications/the-world-factbook/geos/mo.html</ref>. The male population of 15,319,926 is outweighed by the female population of 15,875,248. The migration rate to Morocco is -3.88 migrant(s) per 1,000 population<ref>https://www.cia.gov/library/publications/the-world-factbook/geos/mo.html</ref>. '''Languages''' There are three main languages spoken in Morocco. The first, Arabic, is the official language spoken throughout the country. The second most popular language spoken is Berber. The third language spoken is French, which is used mainly for business, government, and diplomacy. French is popular with the government because Morocco was controlled by the French from about 1904 to 1956. '''Ethnicity And Religion''' Morocco is made up of several ethnic groups and religions. The ethnic group which makes up the largest population is Arab or Berber (99.1%) followed by a mixture of ethnicity (0.7%) and the smallest group is the Jewish consisting of a mere 0.2% of the population<ref>https://www.cia.gov/library/publications/the-world-factbook/geos/mo.html</ref>. As to religions, Islam has the largest following at 98.7% of the population. Christianity has a small following at 1.1%, followed by Judaism, with 0.2% of the total religious population in Morocco<ref>https://www.cia.gov/library/publications/the-world-factbook/geos/mo.html</ref>. ==Brief History== [[Image:Berber.jpg|thumb|200px|left|alt=A|Berber Descendant]]Morocco's history began with the Berbers, the aboriginal people who have inhabited the country since the end of the 2nd millennium BC Rome extended its rule over the area after defeating Carthage in 146 BC, and testimony to its presence still exists in the fine Roman ruins at Volubilis. As Rome fell into decline Morocco was invaded first by the Vandals and then, in the 7th century, by the Arabs. Although external Arab rule lasted little more than a century, the arrival of Islam proved to be a permanent addition to Moroccan culture. In the ensuing centuries a series of ruling dynasties came to power, including the Idrissids, the Almoravids, and the Almohads, but none seemed capable of long maintaining the critical support of the Berber leaders<ref>http://www.geographia.com/morocco/</ref>. By the 15th century Spain and Portugal began to intrude into Morocco, after having expelled the Moors from their own lands. Although Morocco successfully repulsed these invasions, the tide of European imperialism eventually proved too great. By the middle of the 19th century Morocco's strategic importance had become evident to all of the European powers, and they engaged in a protracted struggle for possession of the country. Finally, in 1911, France was formally acknowledged as protector of the greater part of the country, with Spain receiving a number of isolated locales. French rule came to an end in 1953, although its cultural influence on Morocco remains strongly in evidence. Today the country is ruled by King Mohammed VI. He appears to be leading Morocco toward both long-term stability and a greater degree of economic prosperity<ref>http://www.geographia.com/morocco/</ref>. ==Economic Development, Health, and Education== '''Economic Development''' Morocco's GDP is $153.8 billion (2010 est.), which is the 58 richest in the world. The GDP per capita is $4,900 (2010 est.), 147 richest in the world. Morocco's labor force is estimated to be around 11.63 million people<ref>https://www.cia.gov/library/publications/the-world-factbook/geos/mo.html</ref>. Labor Force-By Occupation Agriculture- 44.6% (barley, wheat, citrus fruits, grapes, vegetables, olives, livestock, wine) Industry- 19.8% (phosphate rock mining and processing, food processing, leather goods, textiles, construction, energy, tourism) Services- 35.5% The unemployment rate in Morocco is estimated at 9.8%, 109 in the world. 15% of the population is estimated to be below the poverty line (2007 est.)<ref>https://www.cia.gov/library/publications/the-world-factbook/geos/mo.html</ref>. Exports And Imports Morocco has an estimated $14.49 billion in exports in 2010. Materials Morocco exports are, clothing and textiles, electric components, inorganic chemicals, transistors, crude minerals, fertilizers (including phosphates), petroleum products, citrus fruits, vegetables, fish<ref>https://www.cia.gov/library/publications/the-world-factbook/geos/mo.html</ref>. Morocco has an estimated $34.19 billion in exports in 2010. Morocco imports much of their resources such as, crude petroleum, textile fabric, telecommunications equipment, wheat, gas and electricity, transistors, plastics<ref>https://www.cia.gov/library/publications/the-world-factbook/geos/mo.html</ref>. '''Health Care''' History In 1956, when Morocco became independent, there were only about 300 public health physicians and 400 private practitioners in the country. Since then, the government has made health care services more widely available and improved their quality. By 1992, health care was available to 70% of the population. Health education courses at schools and colleges, and programs to teach hygiene to children and parents, have also helped raise the quality of health. The current life expectancy is 66.5 years for men and 70.6 years for women<ref>http://gis.emro.who.int/HealthSystemObservatory/PDF/Morroco/Health%20system%20organization.pdf</ref>. In 1982, the Ministry of Public Health was formed. Since then, smallpox has been eliminated, typhus outbreaks are less frequent, and malaria and tuberculosis have been brought under control. The World Health Organization and UNICEF also support the government’s campaigns to reduce eye disorders and sexually transmitted diseases. Employers in industry and business are required to register their workers for benefits, but many workers are still not covered. Many other programs aiming at extending medical care to needy Moroccans are under wayDuring the early eighties, there was a decline in the overall state supremacy. With the world economic crisis, the State had to face a profound financial crisis leading to the adoption of restrictions and reforms. The coming of Alma-Ata declaration offered the opportunity to focus on the prevention of disease and the development of basic health care and health programs. Since then, prevention became the state priority while hospitals were left in the shadow of health policies. The Medical Doctors showed a growing interest in the liberal medicine. The development of the private sector increased in the urban area and in the most promising regions of the country, independently from the State<ref>http://gis.emro.who.int/HealthSystemObservatory/PDF/Morroco/Health%20system%20organization.pdf</ref>. During the early eighties, there was a decline in the overall state supremacy. With the world economic crisis, the State had to face a profound financial crisis leading to the adoption of restrictions and reforms. The coming of Alma-Ata declaration offered the opportunity to focus on the prevention of disease and the development of basic health care and health programs. Since then, prevention became the state priority while hospitals were left in the shadow of health policies. The Medical Doctors showed a growing interest in the liberal medicine. The development of the private sector increased in the urban area and in the most promising regions of the country, independently from the State<ref>http://gis.emro.who.int/HealthSystemObservatory/PDF/Morroco/Health%20system%20organization.pdf</ref>. Healthcare System The health system is organized according to a pyramidal hierarchy. Structures of primary healthcare (clinics, urban and rural health centers and local hospitals in rural districts for the public sector; medical offices and infirmaries for the private sector) represent the first resort for the patients. They provide preventive and promotional cares as well as ambulatory curative cares<ref>http://gis.emro.who.int/HealthSystemObservatory/PDF/Morroco/Health%20system%20organization.pdf</ref>. The second recourse corresponds to the provincial and prefectorial hospitals for the public sector and the specialized offices and clinics for private one. The third recourse includes regional hospital centers. Fourth recourse is the university hospital centers, one each in Rabat, Casablanca, Fez and Marrakech, where secondary healthcare requiring high-tech equipment and logistics is being provided<ref>http://gis.emro.who.int/HealthSystemObservatory/PDF/Morroco/Health%20system%20organization.pdf</ref>. The semi-public sector comprises of health care institutions managed by health insurance organization (CNOPS and CNSS) and other semi-public institutions (office chérifien of the phosphates; national office of the railways etc). They provide curative ambulatory and hospital cares<ref>http://gis.emro.who.int/HealthSystemObservatory/PDF/Morroco/Health%20system%20organization.pdf</ref>. Life Expectancy Total Population- 75.69 years Male- 72.63 years Female- 78.9 years Infant Morality Rate Total- 28.61 deaths/1,000 live births Male- 33.52 deaths/1,000 live births Female- 23.46 deaths/1,000 live births According to 2001 statistics, sexually transmitted infections (STIs) have become a real public health problem in Morocco, with 250,000 new cases reported per year. Women are particularly vulnerable to STIs and HIV/AIDS due to their lack of power over decisions regarding their reproductive health, such as the use of contraceptives for effective prevention and protection. While it is believed that HIV/AIDS is still relatively limited in Morocco, women are becoming as vulnerable as men. The subject of HIV/AIDS, however, is no longer taboo, and the Ministry of Health and health-specialized NGOs have organized several awareness programs. Nevertheless, the efforts of government and civil society to combat HIV/AIDS are often challenged by the activism of conservative political groups opposed to consciousness-raising and sexual education programs<ref>FreedomHouse.org.Morocco</ref>. '''Education''' [[Image:Referential-Outside.jpg|thumb|100px|left|alt=A|Referential High School in Dakhla Morocco]]Education in Morocco is free and compulsory through primary school (age 15). Nevertheless, many children--particularly girls in rural areas--do not attend school, and most of those who do drop out after elementary school. The country's literacy rate reveals sharp gaps in education, both in terms of gender and location; while country-wide literacy rates are estimated at 39.6% among women and 65.7% among men, the female literacy rate in rural areas is estimated only at 10%<ref>http://www.state.gov/r/pa/ei/bgn/5431.htm#</ref>. Morocco is home to 14 public universities. Mohammed V University in Rabat is one of the country’s most famous schools, with faculties of law, sciences, liberal arts, and medicine. Founded over 1,000 years ago, Karaouine University, in Fes, is the oldest center for Islamic studies in the Maghreb. Morocco’s most prestigious public English-language university, Al-Akhawayn, was founded in 1993 by King Hassan II and King Fahd of Saudi Arabia in Ifrane. Its curriculum is based on an American model<ref>http://www.state.gov/r/pa/ei/bgn/5431.htm#</ref>. ==Governance== Morocco's Constitution was adopted on September 13, 1996. The first link has the translated copy of the Constitution, while the second has the French copy: http://www.al-bab.com/maroc/gov/con96.htm (English). http://www.al-bab.com/maroc/gov/con96fr.htm (French). ===Structure=== '''Executive Branch''' - Chief of State: King Mohammed VI (since 30 July 1999)[[Image:King_Mohammed_VI_Morocco_23_April_2002.jpg|thumb|200px|left|alt=A|King Mohammed VI with President Bush, 2002]] - Head of Government: Prime Minister Abbas El Fassi (since 19 September 2007) - Cabinet: Council of Ministers appointed by the monarch - Elections: the monarchy is hereditary; prime minister appointed by the monarch following legislative elections<ref>http://www.theodora.com/wfbcurrent/morocco/morocco_government.html</ref> '''Legislative Branch''' - Bicameral Parliament consists of the Chamber of Counselors (or upper house) (270 seats; members elected indirectly by local councils, professional organizations, and labor syndicates to serve nine-year term<ref>http://www.theodora.com/wfbcurrent/morocco/morocco_government.html</ref>. - One-third of the members are elected every three years) and Chamber of Representatives (or lower house) (325 seats; 295 members elected by multi-seat constituencies and 30 from national lists of women. Members elected by popular vote to serve five-year terms)<ref>http://www.theodora.com/wfbcurrent/morocco/morocco_government.html</ref>. - Elections: Chamber of Counselors - last held on 3 October 2009 (next to be held in 2012); Chamber of Representatives - last held on 7 September 2007 (next to be held in 2012) election results: Chamber of Counselors - percent of vote by party - NA; seats by party - NA; Chamber of Representatives - percent of vote by party - NA; seats by party<ref>http://www.theodora.com/wfbcurrent/morocco/morocco_government.html</ref>. '''Judicial Branch''' - Supreme Court (judges are appointed on the recommendation of the Supreme Council of the Judiciary, presided over by the monarch)<ref>http://www.theodora.com/wfbcurrent/morocco/morocco_government.html</ref>. ===Elections=== The King is hereditary. The Prime Minister is appointed by the monarch. In the Chamber of Counselors (Majlis al-Mustacharin) 270 members are elected by indirect vote to serve 9-year terms. In the Chamber of Representatives (Majlis al-Nuwab), 295 members are elected through a closed-list proportional representation system to serve 5-year terms and 30 members are reserved for women to serve 5-year terms<ref>http://www.electionguide.org/country.php?ID=146</ref>. Members are elected indirectly by local councils, professional organizations, and labor syndicates. One-third of the members are renewed every three years. 295 members are elected in 95 multi-seat constituencies. The thirty reserved seats are from a national list of women<ref>http://www.electionguide.org/country.php?ID=146</ref>. '''Past Elections''' - Parliamentary - September 7, 2007 - Parliamentary - September 27, 2002 '''Future Elections''' - Parliamentary - September 2012 ===Courts=== Minister of Justice- '''Mr. Mohamed Taieb Naciri''' Morocco has a dual legal system consisting of secular courts based on French legal tradition, and courts based on Jewish and Islamic traditions<ref>http://www.nationsencyclopedia.com/Africa/Morocco-JUDICIAL-SYSTEM.html</ref>. The secular system includes communal and district courts, courts of first instance, appellate courts, and a Supreme Court. The Supreme Court is divided into five chambers: criminal, correctional (civil) appeals, social, administrative, and constitutional. The Special Court of Justice may try officials on charges raised by a two-thirds majority of the full Majlis. There is also a military court for cases involving military personnel and occasionally matters pertaining to state security. The Supreme Council of the Judiciary regulates the judiciary and is presided over by the king. Judges are appointed on the advice of the council. Judges in the secular system are university-trained lawyers. Since 1965 only Moroccans may be appointed as judges, and Arabic is the official language of the courts<ref>http://www.nationsencyclopedia.com/Africa/Morocco-JUDICIAL-SYSTEM.html</ref>. There are 27 Sadad courts, which are courts of first instance for Muslim and Jewish personal law. Criminal and civil cases are heard, and cases with penalties exceeding a certain monetary amount may be appealed to regional courts. The Sadad courts are divided into Shari'ah; Rabbinical; Civil, Commercial, and Administrative sections; and a criminal section<ref>http://www.nationsencyclopedia.com/Africa/Morocco-JUDICIAL-SYSTEM.html</ref>. The Morocco’s Constitution claims that “The judicial authority is independent from the legislative power and the executive power”. Magistrates are appointed by Dahir on the proposal of the High Council of the Magistracy<ref>http://www.llrx.com/features/morocco.htm#Judicial</ref>. Morocco has many courts that have different purposes and jurisdiction throughout the country. '''Jurisdictions of the First Degree''' '''First Instance Court''' They are competent for: - All civil affairs and relevant to the personal or inheritance - Commercial or social statute - In a first or last instance or in appeal<ref>http://www.llrx.com/features/morocco.htm#1st_degree</ref>. '''Trade Courts''' They rule the following: - Cases between tradesmen, involving commercial activities; - Disputes between associates of a trade country; - Cases related to trade effects; - Disputes related to business - Trade courts are competent to judge, on first and last instances, cases that do not exceed 9,000 Dirham (0.09$) in value and in the first instance all requests that exceed this value<ref>http://www.llrx.com/features/morocco.htm#1st_degree</ref>. - Communal and District judges are competent to adjudicate all personal estate actions brought against individuals who reside under their jurisdiction. The value of claims must be less than 1000 Dirham (0,09 $)<ref>http://www.llrx.com/features/morocco.htm#1st_degree</ref>. '''Jurisdictions of the Second Degree: The Courts of Appeals''' They try criminal cases, appeals against judgments passed by Tribunals of Original Jurisdiction and appeals against rulings made by the latter's presiding judges<ref>http://www.llrx.com/features/morocco.htm#1st_degree</ref>. '''The Supreme Court''' - Appeals for cassation of sentences without appeal decided by anyone of the kingdom's courts. - Appeals for cancellation of the Prime Minister's decisions. - Jurisdiction disputes arising among courts above which there is no high court other than the Supreme Court. - Suits for bias filed against magistrates and courts with the exception of the Supreme Court. - Proceedings aimed at judge disqualifying because of likelihood of bias. - Disqualifying for reasons of public security or for the sake of a good administration of justice<ref>http://www.llrx.com/features/morocco.htm#1st_degree</ref>. '''Administrative Courts''' They are competent to make initial rulings - On claims for cancellation of acts filed against administrative authorities; - Disputes related to administrative contracts; - Claims for compensation of prejudice caused by public entities' acts or activities; - To set up the consistency of administrative acts with legal provisions<ref>http://www.llrx.com/features/morocco.htm#1st_degree</ref>. '''Other Jurisdictions''' '''The Special Court of Justice''' They are competent for cases (corruption, embezzlement, etc) in which magistrates or Government officials are involved<ref>http://www.llrx.com/features/morocco.htm#1st_degree</ref>. '''The High Court''' It is competent for offences or crimes committed by government members during the discharge of their functions. They may be indicted by the two Houses of Parliament and referred to the High Court of Justice for trial<ref>http://www.llrx.com/features/morocco.htm#1st_degree</ref>. '''The Standing Tribunal of the Armed Forces''' It is competent for cases about: - Unauthorized carrying of firearms; - Offenses committed by soldiers<ref>http://www.llrx.com/features/morocco.htm#1st_degree</ref>. '''The Audit Court''' The Audit Court is responsible for conducting overall supervision of the implementation of the budget. It’s ensure the sound conduct of receipt and expenditure operations and evaluate the management of agencies placed under its control by law. When necessary, The Audit Court takes action against violation of the rules governing such operations. The Audit Court provides assistance to Parliament and the government in its fields of competence<ref>http://www.llrx.com/features/morocco.htm#1st_degree</ref>. ===Punishment=== The median prison population rate for Western African countries is 37 as opposed to 267 in Southern African countries. Table 1 shows the Prison Population for the countries in Africa<ref>Walmsley, Roy.World Prison Population.Pg. 2.http://www.kcl.ac.uk/depsta/law/research/icps/downloads/world-prison-pop-seventh.pdf</ref>. Morocco has a prison population of 54,542 dating back to 2004. The prison population rate is 175 per 100,000 of the national population. The death sentence in Morocco is still in existence. In the year 2010, Morocco imposed 4 death sentences. However, the authorities of Morocco have refrained from carrying out those death sentences<ref>Amnesty International.Death Sentences And Executions.2010.Pg. 26.http://www.amnesty.org/en/library/asset/ACT50/001/2011/en/ea1b6b25-a62a-4074-927d-ba51e88df2e9/act500012011en.pdf</ref>. Morocco is considered an abolitionist country, where they retain the death penalty for ordinary crimes such as murder but can be considered abolitionist in practice in that they have not executed anyone during the past 10 years and are believed to have a policy or established practice of not carrying out executions<ref>Amnesty International.Death Sentences And Executions.2010.Pg. 45.http://www.amnesty.org/en/library/asset/ACT50/001/2011/en/ea1b6b25-a62a-4074-927d-ba51e88df2e9/act500012011en.pdf</ref>. The criminal code provides for severe punishments for rape and other forms of sexual violence; however, due to the difficulties in establishing proof of rape and the taboos surrounding this issue, women tend not to report these crimes<ref>FreedomHouse.org.Morocco</ref>. '''Prisons''' [[Image:Tamekinprison.jpg|thumb|200px|right|alt=A|Prisoner in Ait Melloul Prison]]The country of Morocco has 4 state prisons. The prisons are overcrowded with unhealthy conditions. These unhealthy conditions include, minimal hygiene, medical care is normally non-existent and disease is rife<ref>BBC News Africa.Morocco Prisons.David Bamford.2001.http://news.bbc.co.uk/2/hi/africa/1318740.stm</ref>. Other problems of these prisons include overcrowding.T hey say the country's prisons house anything up to 80,000 detainees in a system designed for less than half that number. What makes the overcrowding worse is that, while some prisons are half-empty, in others the only place left to sleep for some prisoners are the toilets<ref>BBC News Africa.Morocco Prisons.David Bamford.2001.http://news.bbc.co.uk/2/hi/africa/1318740.stm</ref>. Morocco has several prisons that house men facing terrorist charges. Two of those prisons are Zaki prison in Saleh and Oukasha prison in the port city of Casablanca. '''Children's Rights''' Of the 31.5 million inhabitants of Morocco, 37% are under the age of 18. The population of minors in Morocco is very large, therefore the rights for these children are very important. Since the ratification of the Convention on the Rights of the Child in 1993, Morocco has demonstrated keen interest in addressing problems related to the promotion and the protection of children’s rights<ref>Making Children's Rights Work in North Africa: Country Profiles on Algeria, Egypt, Libya, Morocco, Tunisia.2007.Pg. 109.http://www.juvenilejusticepanel.org/resource/items/I/B/IBCRCountryProfilesNorthAfricaAugust07_EN.pdf</ref>. ===Legal Personnel=== Morocco is a constitutional, democratic and social monarchy. The King is the Supreme Representative of the Nation and the Symbol of the unity thereof. He is the guarantor of the perpetuation and the continuity of the State. As Defender of the Faith, he ensures the respect for the Constitution. He is the Protector of the rights and liberties of the citizens, social groups and organizations<ref>http://www.nyulawglobal.org/Globalex/Morocco.htm</ref>. ===Law Enforcement=== '''History''' Morocco’s police force was established by the French colonial authorities and remains based on the French model. When Morocco’s long struggle for independence from France (and Spain) ended in 1956, however, the existing police force had virtually ceased to exist. The new government quickly created a police force by reconstituting what elements it could of the colonial police, cobbling them together with former guerilla units and using French advisors.<ref>Nasr,Hesham,Jill Crystal,Nathan Brown.2004.Criminal Justice And Prosecution In The Criminal World.Pg.10http://en.wikiversity.org/w/index.php?title=Morocco&action=edit&section=10</ref>. In May a government decree created the new force: the Surete Nationale, under Interior (and later Defense) Minister Muhammad Oufkir. He built this force into a centralized, powerful, independent, and often ruthless force. During the 1960s opposition to the government grew and in 1965 the government declared a state of emergency which lasted until 1970. After riots in Casablanca in 1965, the Surete was also reorganized as an autonomous organization, directly answerable to the king.33 After coup attempts in the early 1970s, the king strengthened the paramilitary Gendarmerie to balance the army’s power. Morocco’s police today are still based on the French model. Morocco has a number of different police organizations, with overlapping responsibilities, nearly all paramilitary in structure<ref>Nasr,Hesham,Jill Crystal,Nathan Brown.2004.Criminal Justice And Prosecution In The Criminal World.Pg.10http://en.wikiversity.org/w/index.php?title=Morocco&action=edit&section=10</ref>. '''Branches''' The Moroccan police, La Direction Générale de la Sûreté Nationale is the authoritative and main State police body of Morocco. The Sûreté Nationale is tasked with upholding the law and public order. It works alongside the Gendarmerie Royale<ref>http://www.worldlingo.com/ma/enwiki/en/Law_enforcement_in_Morocco</ref>.[[Image:Police_officer_of_Morocco.jpg|thumb|200px|left|alt=A|La Direction Générale de la Sûreté Nationale]] - La Sûreté Nationale (National Police Force) - La Gendarmerie Royale (Royal Military Police Force) '''Police''' The Groupes urbains de sécurité (GUS) or Urban Society Groups, is a defunct Moroccan special police unit which dealt with urban matters basing on "rapid intervention approach<ref>http://www.worldlingo.com/ma/enwiki/en/Law_enforcement_in_Morocco</ref>. GUS was established by the Direction Générale de la Sûreté Nationale (DGSN) (the main National police body) in October 17, 2005. The initiative lies within the scope of a general reform whose principle was "the Police force, near to you for your safety". As of October 2006, GUS comprised between 4,000 and 5,000 agents<ref>http://www.worldlingo.com/ma/enwiki/en/Groupes_urbains_de_s%C3%A9curit%C3%A9</ref>. The GUS are considered first-aid workers and all members speak English. They were attached to arrondissements. Because of its nature as an auxiliary police, GUS, for example did not have the right to confiscate driver licenses or to file official reports. GUS were equipped with Peugeot Partner vans and Honda motorcycles. The body was disbanded exactly one year later; in October 16, 2006. Before its disbanding, GUS comprised 6 groups. The action came as a result of the reforms which started to be executed within the national police and army bodies. The members of the GUS are to be redeployed to other police departments such as criminal investigation departments and judiciary police<ref>http://www.worldlingo.com/ma/enwiki/en/Groupes_urbains_de_s%C3%A9curit%C3%A9</ref>. While many Moroccans regarded the presence of GUS as a relief, many others considered it as a step back to the rule of the Makhzen (Elite). The disbanding came after many criticisms about excesses or abuses of power were noted. Some irresponsible actions of certain members of this police turned over the public opinion which became discreditory. GUS were also accused of corruption especially. In many cases, civil offenders used to pay a bribe (between 10 and 20 dirhams (Moroccan Currency. This led to the appearance of the popular nickname; "10 drahem"<ref>http://www.worldlingo.com/ma/enwiki/en/Groupes_urbains_de_s%C3%A9curit%C3%A9</ref>. '''Military''' [[Image:Moroccan military Toyota Land Cruiser.JPEG|thumb|200px|left|alt=A|Moroccan Military Jeep]]The Moroccan Royal Gendarmerie was founded in 1957 by late King Mohammed V<ref>http://www.worldlingo.com/ma/enwiki/en/Royal_Moroccan_Gendarmerie#History</ref>. The legislation which founded the Royal Moroccan Gendarmerie describes it as a public force designed to guarantee public security and public order and the implementation of laws<ref>http://www.worldlingo.com/ma/enwiki/en/Royal_Moroccan_Gendarmerie#History</ref>. This legislation text attaches the Gendarmerie to the Royal Moroccan Army, then constituting a military force in its structure, administration and command forms. It consists of officers, NCOs and privates<ref>http://www.worldlingo.com/ma/enwiki/en/Royal_Moroccan_Gendarmerie#History</ref>. Although the Moroccan Royal Gendermerie is a military force it has also been tasked with policing duties throughout the country. The duties of The Royal Moroccan Gendarmerie are numerous and various in kind. However, the legislative body has given great importance to policing duties. The seventh Article of the founding text indicates that "The service aim of the Gendarmerie is to ensure especially administrative, judicial and military policing activities directly and to help to the competent authorizations with the envisaged laws"<ref>http://www.worldlingo.com/ma/enwiki/en/Royal_Moroccan_Gendarmerie#History</ref>. According to the legislative body, the three tasks the Royal Gendarmerie is tasked with are as follows: 1. To the National Defence Administration especially on the subjects of organisation equipment and military judicial policing ; To the Ministry of Justice for the implementation of judicial policing; and, To the Ministry of the Interior on the subject of Common Administration Policing<ref>http://www.worldlingo.com/ma/enwiki/en/Royal_Moroccan_Gendarmerie#History</ref>. 2. The duties of the Royal Gendarmerie, in this field, have been determined by the code of Penal Trial Procedures of 1959. It performs all these duties under the administration of the royal prosecutor and control of the Court of Appeal. Duties consist of determination of the crime before the prosecution gathering the evidence, detection of perpetrators and bringing them before the authorized courts<ref>http://www.worldlingo.com/ma/enwiki/en/Royal_Moroccan_Gendarmerie#History</ref>. 3. And executing the orders of the judges after the prosecution. The personnel of the Royal Gendarmerie participates in the implementation Judicial Policing. The Royal Gendarmerie who acts with the characteristics of being a judicial oriented police officer according to the Court of Penal Proceedings<ref>http://www.worldlingo.com/ma/enwiki/en/Royal_Moroccan_Gendarmerie#History</ref>. In order to perform various duties, the Royal Gendarmerie is organized as follows : - Headquarter which conducts the classic duties of the organization; - 22 Regional Gendarmeries; 64 Companies of Gendarmerie and 322 posts of Gendarmerie where they are deployed to divided small security areas<ref>http://www.worldlingo.com/ma/enwiki/en/Royal_Moroccan_Gendarmerie#History</ref>. Today the Moroccan Royal Gendarmerie is commanded by a Lieutenant General. They are notorious for taking bribes to allow for the delivery of contraband, including gasoline smuggled in from Algeria and Hashish smuggled out through Tangiers<ref>http://www.worldlingo.com/ma/enwiki/en/Royal_Moroccan_Gendarmerie#History</ref>. '''Corruption''' Morocco's Law Enforcement rates a 3.4 on the corruption list according to Transparency International<ref>http://www.transparency.org/policy_research/surveys_indices/cpi/2010/results</ref>. Morocco has substantial problems with illegal migration, human smuggling, transnational terrorists, marijuana (hashish) production and trafficking, and commercial smuggling owing to its long and poorly controlled border, extensive coastline, and proximity to Europe. Smuggling of commercial products and trafficking in persons could ultimately provide revenue sources to terrorists<ref>http://web.archive.org/web/20081213093716/http://www.state.gov/p/inl/rls/fs/113050.htm</ref>. Due to these problems in Morocco, it is very difficult for the law enforcement agencies in the country to effectively combat crime. Morocco also consistently ranks among the world’s largest producers and exporters of cannabis. The value of this illegal trade is estimated at $13 billion per year<ref>http://web.archive.org/web/20081213093716/http://www.state.gov/p/inl/rls/fs/113050.htm</ref>. However, the United States has set goals for assisting Moroccan Law Enforcement and they have funded and set up programs. '''United States Law Enforcement Goals''' - To further counterterrorism and security cooperation with Morocco and continue to promote the long-standing US-Morocco relationship<ref>http://web.archive.org/web/20100528044618/http://www.state.gov/p/inl/rls/fs/141676.htm</ref>. - To support Morocco’s efforts to promote and sustain effective judicial reform<ref>http://web.archive.org/web/20100528044618/http://www.state.gov/p/inl/rls/fs/141676.htm</ref>. '''United States Programs''' - U.S. funds are used to provide training, technical assistance, and equipment to improve border control detection techniques and security at Morocco’s main air and sea ports<ref>http://web.archive.org/web/20081213093716/http://www.state.gov/p/inl/rls/fs/113050.htm</ref>. Based on the different military and law enforcement organizations in Morocco, the type of law enforcement taxonomy, according to Philip Reichel, would be Multiple Coordinated<ref>Reichel, Philip L.2008.Comparative Criminal Justice Systems.Chapter 6, Pg. 194."An International Perspective on Policing"</ref>.The reason for Morocco to fall into this category is because it has many types of organizations of law enforcement that work together in dealing with crime, protection, security and other important duties necessary to the safety of the country. ===Crime Rates and Public Opinion=== ===Legal System=== Morocco's legal system is a mixture of several from around the world. Morocco's legal system is a combination of both Muslim Law and Civil Law<ref>http://www.juriglobe.ca/eng/sys-juri/class-poli/sys-mixtes.php)</ref>. Civil Law originates from Continental Europe and it consists of an actual written code. It is a rational (based on reason) code that is universal (applies to everyone). ===Crime=== Crime in Morocco is a serious concern, particularly in the major cities and tourist areas. Aggressive panhandling, pick-pocketing, purse-snatching, theft from occupied vehicles stopped in traffic and harassment of women are the most frequently reported crimes. Criminals have used weapons, primarily knives, during some street robberies and burglaries. These have occurred at any time of day and night, not only in isolated places or areas less frequented by visitors, but in crowded areas as well<ref>http://travel.state.gov/travel/cis_pa_tw/cis/cis_975.html#crime</ref>. Residential break-ins also occur and have on occasion turned violent, but most criminals look for opportunities based on stealth rather than confrontation<ref>http://travel.state.gov/travel/cis_pa_tw/cis/cis_975.html#crime</ref>. Fraud in Morocco may involve a wide range of situations from financial fraud to relationship fraud for the purpose of obtaining a visa<ref>http://travel.state.gov/travel/cis_pa_tw/cis/cis_975.html#crime</ref>. '''Drugs''' The most occurring crime in Morocco is the use of drugs. This includes a large amount of drugs being trafficked. The main illicit drug that is being trafficked through Morocco is cannabis (form of marijuana).[[Image:DEA_mar_loose.jpg|thumb|200px|left|alt=A|Cannabis]] Over 65% of the cannabis seized in Europe was of Moroccan origin<ref>https://www.unodc.org/pdf/egypt/country_profile_morocco.pdf</ref>. As evident from the data here, marijuana or cannabis is the most common drug used and trafficked throughout Morocco and it finds its way into Europe and other parts of Africa. Other drugs such as heroine and morphine are common, but at a much lower rate. Only 3.971 kilograms of heroine was seized by Moroccan officials in 2001<ref>https://www.unodc.org/pdf/egypt/country_profile_morocco.pdf</ref>. '''Crime Rates''' At the end of the 2010 year Morocco had accounted for 283,702 total crimes. The most typical crimes in Morocco are assaults and drug offenses. The most common form of crime in Morocco is assaults numbering at 46,341 in 2010<ref>http://www.nationmaster.com/country/mo-morocco/cri-crime</ref>. In 2010 there were 199 manslaughter's and 143 murders, with only 1 with a firearm. Theft is another important issue in Morocco with the number of car thefts at 1,523. Crime is a very large problem in Morocco especially with a relatively small police force numbering at 41,977 officers<ref>http://www.nationmaster.com/red/country/mo-morocco/cri-crime&all=1</ref>. The homicide rate in Morocco however, is much smaller. In 2004 according to an international survey conducted by the United Nations reports that there were less than 10 homicides per 100,000 inhabitants<ref>http://www.unodc.org/documents/data-and-analysis/Crime-statistics/International_Statistics_on_Crime_and_Justice.pdf page 12</ref>. ==Rights== Citizenship of Morocco was dated on September 5, 1958. The citizenship of Morocco is governed by the code of Moroccan Nationality, which is the identifying term for citizens. There are multiple requirements for a person to be considered a citizen of Morocco. 1. By Birth- Birth within the territory of Morocco does not automatically confer citizenship. 2. By Descent- Child of a Moroccan father, regardless of the child’s country of birth. Child of a Moroccan mother and an unknown or stateless father, regardless of the child's country of birth. 3. Marriage- A foreign woman who marries a Moroccan citizen can become a Moroccan citizen by declaration after two years residency and marriage. 4. By Naturalization- Moroccan citizenship may be acquired upon fulfillment of the following conditions: Person has continually resided in the country for five years and is of full age. Citizenship must be approved by Cabinet decree<ref>Citizenship Laws of the World.2001.Pg. 138.http://www.opm.gov/EXTRA/INVESTIGATE/is-01.PDF</ref>. Morocco does recognize dual citizenship. However, there are exceptions. Moroccan law recognizes dual citizenship, but permission must be granted by the government before a second citizenship is acquired. Dual citizenship by default is not recognized<ref>Citizenship Laws of the World.2001.Pg. 138.http://www.opm.gov/EXTRA/INVESTIGATE/is-01.PDF</ref>. As many countries, a person can lose their citizenship. There are two ways in which a person can lose their citizenship. 1. Voluntary- Voluntary renunciation of Moroccan citizenship is permitted by law. Renunciation is not automatic and must be approved by the Ministry of Justice. Contact the Embassy for details and required paperwork. 2. Involuntary- The following are grounds for involuntary loss of Moroccan citizenship (unless the government has previously permitted the activity): - Person voluntarily acquires a foreign citizenship. - Moroccan woman marries a foreign national and acquires the husband's citizenship. - Person serves in the military or public employment of a foreign state and refuses the Moroccan government's demand that they resign<ref>Citizenship Laws of the World.2001.Pg. 138.http://www.opm.gov/EXTRA/INVESTIGATE/is-01.PDF</ref>. ===Family Law=== The following laws regarding marriage in Morocco are described in the Moudawana(Moroccan Family Law). '''Marriage''' [[Image:Jüdische Hochzeit in Marokko-1024.jpg|thumb|200px|left|alt=A|Jewish Wedding in Morocco in 1024]]Marriage is a legal contract by which a man and a woman mutually consent to unite in a common and enduring conjugal life. Its purpose is fidelity, virtue and the creation of a stable family, under the supervision of both spouses according to the provisions of this Moudawana<ref>The Moroccan Family Code.Book One of Marriage.Article 4.2004.http://www.hrea.org/moudawana.html#book1</ref>. The following articles of the Moudawana break down the different smaller provisions of marriage in Morocco for those who have established residency in the country, not those living abroad: Article 10 1. Marriage is legally concluded by an offer expressed by one of the parties and acceptance by the other, in any accepted expression from which the meaning of marriage is inferred verbally or conventionally. 2. Persons unable to speak may legally write their consent if they are literate, and if they cannot write, any clear sign understood by the other party and the witnesses is legally sufficient<ref>The Moroccan Family Code.Book One of Marriage.Article 10.2004.http://www.hrea.org/moudawana.html#book1</ref>. Article 11 To be valid, the offer and acceptance must be: 1. expressed verbally whenever possible; otherwise, they must be written; or else expressed by means of an unequivocal sign; 2. congruent and concurrent; 3. irrevocable and not restricted by a condition or a suspensive or nullifying deadline<ref>The Moroccan Family Code.Book One of Marriage.Article 11.2004.http://www.hrea.org/moudawana.html#book1</ref>. Article 12 Any marriage contract concluded under duress or by fraud is subject to the provisions of Articles 63 and 66 below<ref>The Moroccan Family Code.Book One of Marriage.Article 12.2004.http://www.hrea.org/moudawana.html#book1</ref>. Articles 63 and 66 explain that if a marriage contract is concluded under duress or by fraud, the violating party may face penal sanctions. Article 13 The conditions required to contract marriage are: 1- The legal capacity of both spouses to marry; 2- No intention or agreement to cancel the dowry; 3- A marital tutor, if required; 4- The hearing and notarized statement by two adouls (public notaries) of the offer and acceptance pronounced by the two spouses; 5- The absence of any legal impediments<ref>The Moroccan Family Code.Book One of Marriage.Article 13.2004.http://www.hrea.org/moudawana.html#book1</ref>. '''Engagement''' Engagement is the reciprocal promise of marriage between a man and a woman. Engagement takes place through the expression by the two parties of a reciprocal promise to marry by any accepted means, including the reading of the Sura of Al Fatiha from the Holy Koran and the customary exchange of presents<ref>The Moroccan Family Code.The Book of Marriage.Engagements.Article 5.2004.http://www.hrea.org/moudawana.html#book1</ref>. The two parties are considered to be in an engagement period until the marriage is officially registered. Each party has the right to break off the engagement<ref>The Moroccan Family Code.The Book of Marriage.Engagements.Article 6.2004.http://www.hrea.org/moudawana.html#book1</ref>. No compensation is due for breaking off the engagement. However, if one of the parties causes any harm to the other, the injured party may ask for compensation from the other<ref>The Moroccan Family Code.The Book of Marriage.Engagements.Article 7.2004.http://www.hrea.org/moudawana.html#book1</ref>. Either party may recover the presents he or she offered to the other provided that the breaking off of the engagement was not his or her decision. Presents are returned in the state in which they were initially offered or reimbursed according to their value, depending on the circumstances<ref>The Moroccan Family Code.The Book of Marriage.Engagements.Article 8.2004.http://www.hrea.org/moudawana.html#book1</ref>. If the engaged man pays the dowry or part of it and the engagement is broken off, or if one of the parties dies during the engagement period, he or his inheritors recovers what the latter initially offered in its original state, something comparable, or something of equivalent value. If the engaged woman refuses to reimburse the money with which she bought furniture, the person responsible for breaking off the engagement assumes the loss between the value of the furniture and the amount of money initially paid<ref>The Moroccan Family Code.The Book of Marriage.Engagements Article 9.2004.http://www.hrea.org/moudawana.html#book1</ref>. '''Divorce''' Article 94 If either or both spouses ask the court to settle a dispute that risks to breakdown their marriage, the court must make all efforts to reconcile them according to the provisions of preceding Article 82<ref>The Moroccan Family Code.Title Four:Divorce.Article 94.2004.http://www.hrea.org/moudawana.html#book1</ref>. Article 95 The two arbitrators or two persons that can assume this role shall investigate the causes of the dispute between the spouses and do their best to resolve the conflict. If the two arbitrators reconcile the two spouses, they shall write a report, make three copies of it, signed by them as well as the spouses, and submit the three copies to the court, which remits a copy to each spouse and files one in the record, and certifies it<ref>The Moroccan Family Code.Title Four:Divorce.Article 95.2004.http://www.hrea.org/moudawana.html#book1</ref>. Article 96 When the two arbitrators disagree on the content of the report or in the attribution of responsibility, or do not submit the report within the requisite deadline, the court may conduct an additional investigation by all means it judges appropriate<ref>The Moroccan Family Code.Title Four:Divorce.Article 96.2004.http://www.hrea.org/moudawana.html#book1</ref>. Article 97 In the event reconciliation is impossible to reach, and the conflict between the spouses persists, the court shall make written mention of this in an official report of the proceedings, and grant the divorce as well as fix the vested rights to be paid according to preceding Articles 83, 84 and 85, taking into account each spouse's responsibility for the cause of the separation when considering measures it will order the responsible party to take in favour of the other spouse. The irreconcilable differences suit shall be settled within a deadline not to exceed six months from the date the petition was filed<ref>The Moroccan Family Code.Title Four:Divorce.Article 97.2004.http://www.hrea.org/moudawana.html#book1</ref>. '''Inheritance''' According to the Maliki Muslim laws as interpreted in Morocco, sisters inherit less than half of their brothers' share of inheritance. If there is no male heir, the daughter still does not get the full inheritance, and part of her parents' estate goes to her uncles and aunts. The only changes regarding inheritance in the 2004 revision of the CSP concern the rights of the children of a deceased mother to inherit from the maternal grandparents in the same way as children of a deceased father. Laws that forbid a non-Muslim Moroccan wife from inheriting from her Muslim husband remain intact. Discrimination against women in matters of inheritance is exacerbated by cases in which women, particularly rural women, are excluded from receiving their inheritance altogether. The government has not addressed this issue, and the access of rural women to their rightly deserved land and credit continues to be restricted.<ref>http://www.freedomhouse.org/template.cfm?page=178.Economic Rights and Equal Opportunity</ref>. '''Rights of Juveniles''' ===Social Inequality=== ===Human Rights=== '''Voting''' [[Image:Woman_in_Morocco.jpg|thumb|200px|left|alt=A|Veiled Moroccan Woman]]In December of 2002, King Mohammed VI announced that the voting age of citizens will be lowered from 20 to 18<ref>http://news.bbc.co.uk/2/hi/africa/2565373.stm</ref>. September's parliamentary elections saw a record low turnout of just 52% and the rise of the Islamist Justice and Development Party (PJD). The PJD gained 42 seats to become the third largest party in parliament but it was left out of the new coalition government<ref>http://news.bbc.co.uk/2/hi/africa/2565373.stm</ref>. Women have the right to take part in elections, which in Morocco are based on a multiparty system. However, all women are not freely able to exercise their right to vote and participate in fair elections, as some candidates exploit the poverty and illiteracy of the poorest women to buy their votes. Government authorities are also known to intervene and direct the voting process, especially in rural and semi-urban areas; nevertheless, these practices, which were widespread in the past, are on the decline<ref>FreedomHouse.org</ref>. '''Women's Rights''' According to Morocco's 2004 landmark family law, a hotly disputed reform to empower women, cast off their vulnerability and set a trend in Muslim North Africa, was passed and backed by King Mohamed VI<ref>http://www.chinapost.com.tw/life/discover/2009/03/09/199351/Womens-rights.htm</ref>. The 2004 reforms gave women the right to divorce their husbands and to own assets and property obtained during marriage, and clamped down sharply on a man's ability to repudiate his wife. However, there are still areas that women wish to improve in, the marriage of minors and the practice of polygamy, which are still allowed under a judge's discretion in this country of 34.3 million where an estimated 31 percent of the population is under 14<ref>http://www.chinapost.com.tw/life/discover/2009/03/09/199351/Womens-rights.htm</ref>. Today, 10 percent of all marriages in Morocco involve minors. Women's custody rights under the new family laws guarantee that a Moroccan mother no longer automatically loses her custody rights if she remarries or moves to a town other than the town where her husband resides. However, a mother may still lose custody of her children over the age of seven if she remarries and her husband requests custody; she may obtain legal guardianship of her minor children only in cases in which the father is deceased or legally incompetent<ref>FreedomHouse.org.Morocco</ref>. According to the Maliki Muslim laws as interpreted in Morocco, sisters inherit less than half of their brothers' share of inheritance. If there is no male heir, the daughter still does not get the full inheritance, and part of her parents' estate goes to her uncles and aunts<ref>FreedomHouse.org.Morocco</ref>. Moroccan women have made significant progress in the field of labor, and the overall quality of women's employment has improved. More women are in wage-earning positions (24.7 percent of women report being at the level of paid employment), the number of women in managerial positions has increased, and a larger proportion of women in the workforce have an advanced degree or diploma. However, while women wage-earners in urban areas are significantly better educated than men, women still tend to work in occupations that lack status, such as family helpers, home workers, or cleaning staff. The social protection system, associated with the formal sector, excludes most women workers in these types of jobs<ref>FreedomHouse.org.Morocco</ref>. ===Works Cited=== <references /> ekwsfvh08vhxv8q95ye8bs94q9qofc5 0 152431 2842467 2764862 2026-10-24T23:43:42Z Jtneill 10242 /* Written expression */ ### use past tense when describing research studies, although implications of findings could be in the present tense 2842467 wikitext text/x-wiki {{title|General feedback about<br>book chapters}} {{RoundBoxTop|theme=3}} * General feedback about the [[Motivation and emotion/Book/2025|2025 student-authored book chapters]] * Detailed feedback about each individual chapter is available on the respective talk pages {{RoundBoxBottom}} __TOC__ ==Overall== # The overall quality of chapters was reasonably good, but there was a wide range # The main issues were: ## Unacknowledged and/or poor use of genAI ## Insufficient review and synthesis of the best research about the topic ## Lack of social contributions <!-- # The best chapters will be [https://twitter.com/jtneill/status/1554447127639519232 tweeted in this thread] --> ==Overview== # The best overviews consisted of: ## A scenario/case study in a feature box with a relevant image ## A simple, evocative description of the psychological problem or phenomenon ## Clear and relevant focus questions in a feature box; the focus questions are closely aligned to the top-level headings. ==Theory== # Most chapters used a well selected range of relevant theories # Weaker chapters tended to use more general, historical theories; stronger chapters tended to identify more specific, recent theories # Theories were generally well explained, but theoretical depth and relevance was a key differentiator between chapters # All chapters could be improved by building more strongly on related chapters (i.e., by embedding links from the first mention of key words to related chapters) # More practical examples would have been useful to demonstrate how theories work in practice ==Research== # The best chapters provided comprehensive and critical synthesis of the best research # Most chapters review some relevant research, but could often the research could have been more relevant and/or indepth # Place more emphasis on major reviews such as meta-analyses # Sometimes there was insufficient citation of academic peer-reviewed sources to support claims ==Integration== # The best chapters had excellent integration between theory and research # Many chapters emphasised theory more than research — strive for balance ==Conclusion== # Useful summaries of key points were provided # The best chapters emphasised explicit take-home messages for each focus question ==Style== ===Written expression=== # Overall, the quality of written expression was reasonably good, but varied widely. # The most common problems were: ## grammar and spelling ### poor grammar — there's many ways to get help to develop professional-level written expression (e.g., Studiosity, genAI) ### use 3rd person, rather than 1st person, perspective[https://www.grammarly.com/blog/first-second-and-third-person/] ### use past tense when describing research studies, although implications of findings could be in the present tense ### [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should] should be used (they are part of APA style) ### use of [https://grammar.yourdictionary.com/punctuation/apostrophe-rules.html ownership apostrophes] ### Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling]. ## insufficient citation — scientific communication needs evidence to support claims ## clarity of written expression ### "People" is often a better term than "individuals" ### Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the content/key point to be communicated, with the citation included along the way or, more typically, in parentheses at the end of the sentence. # APA style: ## Citations with three or more authors should use the first author's surname followed by et al. and the year ## Reference formatting areas for improvement: ### capitalisation ### italicisation ### use of hyperlinked dois ===Learning features=== # Embedded interwiki links to Wikipedia articles help to connect reader to further information about key concepts # Embedded interwiki links to related Wikiversity [[Motivation and emotion/Book|book chapters]] were underutilised — network the chapter to related chapters # Figures were reasonably well used, with several students uploading their own images (thankyou!) # Tables were less commonly used, but usually very useful # Feature boxes were generally well used # Quizzes were generally well used # Scenarios/case studies were reasonably well used, but often more examples would have been helpful ==Social contributions== # There were many valuable improvements made to past and current chapters # The amount and quality of contributions varied widely — some students made no contributions whilst some excelled and were awarded bonus marks # Unless there were ''direct links'' to evidence, no marks were provided == See also == * [[Motivation and emotion/Assessment/Chapter/Feedback/2021|2021 feedback]] * [[Motivation and emotion/Assessment/Chapter/Feedback/2022|2022 feedback]] * [[Motivation and emotion/Assessment/Chapter/Feedback/2023|2023 feedback]] * [[Motivation and emotion/Assessment/Chapter/Feedback/2024|2024 feedback]] * [[Motivation and emotion/Assessment/Chapter|Book chapter - Guidelines]]<!-- * [[Motivation and emotion/Assessment/Multimedia/Feedback|Multimedia presentation feedback]] --> [[Category:Motivation and emotion/Assessment/Chapter]] oz3e6d1d90h7poajlh6srrd1esvryac 1 171302 2842461 2842427 2026-10-24T22:01:19Z James500 297601 /* */ Add 2842461 wikitext text/x-wiki Terms that sometimes appear in the titles of bibliographies, guides to literature/reference, etc: Bibliography, catalogue, guide, list, index, in print, books, literature, sources, works, material, titles, reference. [[User:James500|James500]] ([[User talk:James500|discuss]] • [[Special:Contributions/James500|contribs]]) 13:54, 14 November 2020 (UTC) For future use: *Non-fiction, art, fine arts, music, painting, sculpture, government, politics, philosophy, theology, religion, agriculture, forestry, fisheries, mining, manufacturing, technology, computer science, engineering, astronomy, physics, chemistry, biology, botany, natural history, prehistory, classics, anthropology, library science, librarianship, information studies, places, exploration, transport, Earth science, paleontology, official publications, universities, museums *Reprography, lithography, engraving *Bookbinding, early printed books, incunabula, manuscripts, papyri, papyrology *Languages, philology, ancient languages, medieval languages, dead languages, extinct languages *Writing, scripts, decipherment, alphabets, hieratic, demotic, hieroglyphic, figurative, ideographic, pictographic, cuneiform, inscriptions, stelae [stela], clay tablets, ostraka [ostrakon], papyrus, Minoan scripts, Linear A, Linear B *Indo-European languages; Italo-Celtic; Celtic languages *Latin, Greek, middle english, old english, anglo-norman [anglo-french], old french, old norse, old welsh, middle welsh, egyptian, hittite, luwian, lycian, akkadian, assyro-babylonian, elamite, amorite *Special libraries, national libraries, public libraries, university libraries, college libraries, school libraries, museum libraries *Archaeological theory, methodology, geology, stratigraphy etc. Field archaeology, behavioural archaeology *Tide tables *George Watson (ed). The Cambridge Bibliography of English Literature. Cambridge University Press. 1966. Volume 5. [https://books.google.co.uk/books?id=ebM6AAAAIAAJ] *[[s:Page:Cornwall (Mitton).djvu/230|"Some Books on Cornwall"]] *[[s:Author:Richard Colt Hoare]] *Yorkshire literature *Catalogues régionaux des incunables des bibliothèques publiques de France. [https://books.google.co.uk/books?id=K38VcZrjrj0C&pg=PA1#v=onepage&q&f=false] (regional catalogues of incunabula in the public libraries of France) * [[User:James500|James500]] ([[User talk:James500|discuss]] • [[Special:Contributions/James500|contribs]]) 16:24, 24 November 2019 (UTC) For future use: *Palmer's Index to The Times **editions:2Faf22c-dSwC *Routledge Handbook of Soviet and Russian Military Studies France: *Les Dictionnaires départementaux.  [https://books.google.co.uk/books?id=FZEwAQAAMAAJ] (series includes departmental biographical dictionaries) Creuse: *Guide des archives de la Creuse. 1972 [https://books.google.co.uk/books?id=eEsCAAAAMAAJ] *Joanne. Géographie du département de la Creuse. 1895 [https://books.google.co.uk/books?id=izeVfoPbIc8C] *Jamain. Le département de la Creuse: ses origines et sa pérennité [https://books.google.co.uk/books?id=kY-KMwcsJMEC&pg=PP1#v=onepage&q&f=false] *Annuaire du Département de la Creuse [https://books.google.co.uk/books?id=7wpBAAAAcAAJ&pg=PA1#v=onepage&q&f=false] *Journal du département de la Creuse (1807-1819) Bibliography: *[[w:Fredson Bowers|Bowers, Fredson]]. Principles of Bibliographical Description. Princeton University Press. 1949. Russell & Russell, New York. 1962. St Paul's Bibliographies (No 15). 1986. [https://books.google.co.uk/books?id=hA7hAAAAMAAJ] Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&pg=PA525#v=onepage&q&f=false] *Bowers, Fredson. Bibliography and Textual Criticism. Clarendon Press, Oxford. 1964. [https://books.google.co.uk/books?id=zOFhNcQsihAC] Series: *Studies in Bibliography. Formerly "Papers in Bibliography". (Bibliographical Society of the University of Virginia). (1948 onwards) [https://books.google.co.uk/books?id=HGEUAAAAIAAJ] Commentary: "A History of Studies in Bibliography: The First Fifty Volumes" in The First Fifty Years, p 125 [https://books.google.co.uk/books?id=ZukVAQAAIAAJ] **Tanselle. Selected Studies in Bibliography. (Bibliographical Society of the University of Virginia) [https://books.google.co.uk/books?id=OnktAQAAIAAJ] Commentary: [https://books.google.co.uk/books?id=Utcb32E7rsMC&pg=PA526#v=onepage&q&f=false] Library associations: *IFLA. Functional Requirements for Bibliographic Records: Final Report. Saur. 1998. [https://books.google.co.uk/books?id=YW8hAAAAQBAJ&pg=PP4#v=onepage&q&f=false] *Patrick Le Boeuf (ed). Functional Requirements for Bibliographic Records (FRBR): Hype or Cure-All? Haworth Press. 2005. Routledge. 2013 [https://books.google.co.uk/books?id=Ka7hAQAAQBAJ&pg=PP1#v=onepage&q&f=false] *Guidelines for Bibliographic Description of Reproductions. American Library Association. 1995. [https://books.google.co.uk/books?id=9zG6MGudKv8C&lpg=PP1&pg=PP1#v=onepage&q&f=false] Serials: *Serials Review Nostgia *Nostalgia cycles: [https://slate.com/culture/2012/04/the-golden-forty-year-rule-and-other-nostalgia-cycles-could-trends-possibly-return-every-40-years-20-years-and-12-15-years.html] [https://www.vice.com/en/article/how-the-20-year-trend-cycle-collapsed/] [https://www.npr.org/2022/03/01/1081115609/from-tumblrcore-to-2014core-the-nostalgia-loop-is-getting-smaller-and-faster] [https://www.theguardian.com/music/2023/apr/20/lets-go-round-again-the-ridiculous-rise-of-fifth-anniversary-vinyl-reissues] [https://www.independent.co.uk/life-style/fashion/gen-z-fashion-trends-tumblr-aesthetic-b2042247.html] *Stuck in the Seventies: 113 Things from the 1970s that Screwed Up the Twentysomething Generation. 1991. 1995. [https://books.google.co.uk/books?id=bZ03g5ylsaEC]. Review: [https://books.google.co.uk/books?id=6V4cAQAAMAAJ]. *1969: [https://books.google.co.uk/books?id=yuwjAAAAIBAJ&pg=PA25#v=onepage&q&f=false]. *Decade nostalgia [https://www.thetimes.com/uk/politics/article/rolling-back-the-years-qg8fs6sh06d] *[https://books.google.co.uk/books?id=COhVAAAAIBAJ&pg=PA20#v=onepage&q&f=false] [https://books.google.co.uk/books?id=Xf4rAAAAIBAJ&pg=PA3#v=onepage&q&f=false] [https://books.google.co.uk/books?id=nFtWAAAAIBAJ&pg=PA31#v=onepage&q&f=false] [https://books.google.co.uk/books?id=HqkfAAAAIBAJ&pg=PA27&article_id=4890,1282350#v=onepage&q&f=false] [https://books.google.co.uk/books?id=_WZGAAAAIBAJ&pg=PA29#v=onepage&q&f=false][https://books.google.co.uk/books?id=bzpHAAAAIBAJ&pg=PA27#v=onepage&q&f=false][https://books.google.co.uk/books?id=s1MgAAAAIBAJ&pg=PA7#v=onepage&q&f=false] *1970s: [https://books.google.co.uk/books?id=YT5QAAAAIBAJ&pg=PA8#v=onepage&q&f=false] *1990s Tamagotchi [https://www.nssmag.com/en/lifestyle/38204/tamagotchi-y2k-nostalgia] *[https://www.bbc.co.uk/programmes/m002g6k5 90s Nostalgia]. BBC. Other *[[w:Haruko Ichikawa|Haruko Ichikawa]] (市河晴子 or 市川晴子) (1896-1943) **Japanese Lady in Europe. 1937. [https://books.google.co.uk/books?id=xi42AAAAMAAJ] (Japanese: 欧米の隅々) **Japanese Lady in America. First published 1938. [https://books.google.com/books?id=KQY7AQAAIAAJ] (Japanese: 米国の旅日本の旅) Search terms: *"press in japan" *"japanese press" *"press of japan" *"journalism of japan" *"journalism in japan" *"journalists in japan" *"japanese journalism" Generations *[https://www.vox.com/2018/8/15/17686668/millennials-explained Stop calling teenagers millennials]. Vox. 15 August 2018. Planet Earth *[[w:Encyclopedia of Earth|Encyclopedia of Earth]] Crime *The Concise Encyclopedia of Crime and Criminals [https://books.google.co.uk/books?id=uGTQAAAAIAAJ] Showa era *懐かしの昭和・平成流行事典: 2001-1945. [[w:ja:新人物往来社|Shin-Jinbutsuoraisha]]. 2002. [https://books.google.co.uk/books?id=QCYyAQAAIAAJ] *少年ブーム～昭和レトロの流行もの. 串間努著. 2003. [https://books.google.co.uk/books?id=sK0nAQAAIAAJ] Fading of the Showa era *[https://www.theguardian.com/world/gallery/2025/dec/02/fading-japan-showa-era-in-pictures The fading of Japan’s Shōwa era – in pictures]. The Guardian. 2 December 2025. *[https://soranews24.com/2026/04/02/fading-tokyo-searching-for-signs-of-the-showa-era-as-local-neighborhoods-evolvephotos/ Fading Tokyo – Searching for signs of the Showa era as local neighborhoods evolve]. SoraNews24. 2 April 2026. Shakespeare *Demmon (compiler). Catalogue of the Shakespearian Books and Pamphlets in the Joseph Crosby Library. 1885. [https://books.google.co.uk/books?id=GSZLAQAAMAAJ&pg=PA1#v=onepage&q&f=false]. *Shakespeare the Man and the Book. Part 2: Occasional Papers. 1881. [https://books.google.co.uk/books?id=gfUUAAAAYAAJ&pg=PP11#v=onepage&q&f=false] *Best films: [https://www.empireonline.com/movies/features/best-movies-of-all-time-us/] Survivability of past atmosphere *[https://news.climate.columbia.edu/2022/09/20/you-asked-dinosaurs-survived-when-co2-was-extremely-high-why-cant-humans/ You Asked: Dinosaurs Survived When CO2 Was Extremely High. Why Can’t Humans?] Columbia Climate School. 20 September 2022. *[https://www.sciencefocus.com/planet-earth/how-far-back-in-time-would-we-be-able-to-go-and-still-breathe-our-planets-air How far back in time would we be able to go and still breathe our planet's air?] BBC Science Focus. *[https://gizmodo.com/the-one-basic-fact-about-history-that-time-travelers-al-1652649190 The One Basic Fact About History That Time Travelers Always Forget]. io9. Gizmodo. 31 October 2014. [[User:James500|James500]] ([[User talk:James500|discuss]] • [[Special:Contributions/James500|contribs]]) 10:55, 26 October 2020 (UTC) ovq1aqlythyp79n3yq7ngh13pnn02fj 0 235228 2842458 2388724 2026-10-24T21:06:47Z The Citer 3110681 Added phylogeny! 2842458 wikitext text/x-wiki [[Image:Limnognathia maerski.PNG|thumb|300px|right|Illustration of ''Limnognathia maerski'', a lozenge animal.]] *Rhombozoa, also called Dicyemida *Name Meaning: Lozenge animal *English Common Name: Lozenge animals *Major distinguishing characteristics: Single axial cell surrounded by ciliated cells *Approximate number of species described: 75 ==Natural History== ==Taxonomy== This phylum contains 3 families: *Conocyemidae *Dicyemidae *Kantharellidae ==Phylogeny== {{clade |label1=Rhombozoa |1={{clade |1=''Kantharella antarctica'' |label2=Dicyemida |2={{clade |label1=Conocyemidae |1={{clade |1=''Conocyema'' |2=''Microcyema''}} |label2=Dicyemidae |2={{clade |1=''Dodecadicyema loligoi |2=''Pleodicyema delamarei |6=''Dicyema |5=''Dicyemennea |label3=''Dicyemodeca |3={{clade |1=''Dicyemodeca anthinocephalum |2=''Dicyemodeca deca |3=''Dicyemodeca dogieli }} |label4=''Pseudicyema |4={{clade |1=''Pseudicyema cappacephalum |2=''Pseudicyema nakaoi |3=''Pseudicyema truncatum}} }} }} }} }} ==Anatomy== ==The Fossil Record== ==Quiz== * [[/Rhombozoa Quiz/]] ==References and Further Reading== *[https://en.wikipedia.org/wiki/Dicyemida Rhombozoa] at Wikipedia *[http://eol.org/pages/8764/overview Rhombozoa] at Encyclopedia of Life {{Animalia}} [[Category:Animals]] [[Category:Biology]] [[Category:Taxonomy]] [[Category:Zoology]] au30qfsfcl78thuq3hmcwm1yo171by8 0 247473 2842438 2696443 2026-10-24T14:28:37Z Z. Patterson 1936818 /* Key Terms */ Copy edit 2842438 wikitext text/x-wiki <noinclude>{{{{NAMESPACE}}:{{BASEPAGENAME}}/Sidebar}}</noinclude> This lesson introduces the JavaScript programming language and environments. __TOC__ == Objectives and Skills == Objectives and skills for this lesson include:<ref>[https://www.microsoft.com/en-us/learning/exam-98-382.aspx Microsoft: Exam 98-382 Introduction to Programming Using JavaScript]</ref> * Apply JavaScript best practices ** Comments; indentations (code formatting); naming conventions; noscript; constants; reserved keywords; debugger keyword; setting breakpoints; console.log * Evaluate the use of inline and external scripts ** When to use, how to use, and what happens when both are used == Readings == # [[Wikibooks: JavaScript/Introduction]] # [[Wikibooks: JavaScript/First program]] # [[Wikibooks: JavaScript/Placing the code]] # [https://www.codecademy.com/resources/docs/html/file-paths CodeCademy: File Paths] == Multimedia == # [https://www.youtube.com/watch?v=xpZLS6R91rQ YouTube: JavaScript Tutorial for Beginners - 01 - Introduction] # [https://www.youtube.com/watch?v=tkw8QXIcmU4 YouTube: JavaScript Tutorial for Beginners - 02 - Statements] # [https://www.youtube.com/watch?v=r--dRuIiV7A YouTube: JavaScript Tutorial for Beginners - 05 - Using an external file] # [https://www.youtube.com/watch?v=efN50MBOSvc YouTube: JavaScript - Output] # [https://www.youtube.com/watch?v=yUyJ1gcaraM YouTube: Beginner JavaScript Tutorial - 2 - Comments and Statements] # [https://www.youtube.com/watch?v=q9jAFZjPFHo YouTube: JavaScript - How to use the console in a browser] # [https://www.youtube.com/watch?v=CCZ2pHpJe2U YouTube: noscript Tag - JavaScript Programming] # [https://www.youtube.com/watch?v=fP2IUZXF0sk YouTube: Using Alert & Console.log in JavaScript] # [https://youtu.be/RLCpvCTiDRo YouTube: JavaScript for Beginners — Console Logging] # [https://www.youtube.com/watch?v=HzWf-EeE3uI YouTube: Clean Code: Formatting and Comments] # [https://www.youtube.com/watch?v=SFZXXUY_MHg YouTube: Javascript The innerHTML property] # [https://www.youtube.com/watch?v=EZQany2bhLY YouTube: javascript: part1, console.log, getElementByID & innerHTML] # [https://www.youtube.com/watch?v=GoEBQCp2Nzg YouTube: JavaScript Loading Strategies (async and defer)] # [https://youtu.be/JlgLDfINXvY?si=sHzcSqo6u-wXPkGp Youtube: JavaScript DOM Tutorial #4 - The Query Selector] # [https://youtu.be/VJ8_-mEF3ss?si=6vtrnkXtCFMtLmm8 Youtube: The innerText vs. textContent properties in JavaScript] == Examples == To test the following examples, copy the code and paste it in the body of an HTML file and view the file in a browser, or use a JavaScript test environment such as [https://www.w3schools.com/js/tryit.asp?filename=tryjs_intro_inner_html W3Schools: Tryit Editor]. === document.write() === <code>document.write()</code> writes a string of text to a document stream. <script> document.write("Hello World"); </script> === window.alert() === <code>window.alert()</code> displays an alert box with an optional message and an OK button. <script> document.alert("Hello World"); </script> === <code>innerHTML and InnerText</code> === The <code>innerHTML</code> and <code>innerText</code> returns or sets the inner HTML and text of an element, respectively. <nowiki><p id="id-name"> This text will be replaced</p></nowiki> <script> document.getElementById("id-name").innerHTML = "Hello World"; </script> === console.log() === The <code>window.alert()</code> outputs a message or a value to the console. <script> console.log("Hello World") </script> * [https://www.w3schools.com/js/js_intro.asp W3Schools: JavaScript Introduction] * [https://www.w3schools.com/js/js_comments.asp W3Schools: JavaScript Comments] * [https://www.w3schools.com/js/js_output.asp W3Schools: JavaScript Output] * [https://www.w3schools.com/js/js_whereto.asp W3Schools: JavaScript Where To] * [https://www.w3schools.com/tags/tag_noscript.asp W3Schools: HTML noscript tag] * [https://www.w3schools.com/jsref/tryit.asp?filename=tryjsref_node_textcontent_innerhtml_innertext W3Schools: Difference between innerText and innerHTML] *[https://www.w3schools.com/jsref/met_win_alert.asp W3Schools: Window Alert()] *[https://www.w3schools.com/jsref/met_doc_write.asp W3Schools: DOM write() Method] *[https://www.w3schools.com/jsref/met_document_getelementbyid.asp W3Schools: GetElementById()] *[https://www.w3schools.com/jsref/prop_html_innerhtml.asp W3Schools: HTML DOM Element innerHTML] * [[/Example Code/]] == Activities == # Create a web page that uses JavaScript to display <code>Hello <name>!</code>, where <code><name></code> is your name. Test JavaScript output in a variety of ways: #* Use <code>document.write()</code> #* Use <code>window.alert()</code> (which is the same as <code>alert()</code>). <code>window.alert()</code> is best practice. #* Use <code>document.getElementById().innerText</code> or <code>document.getElementById().innerHTML to change the text of a paragraph</code> #* Use <code>console.log()</code> #**Use your browser's built-in [[Wikipedia:Web development tools|web development tools]] to view console output. #**To open the console panel in Chrome press Ctrl+Shift+J (or Command+Option+J on Mac).<ref>[https://developers.google.com/web/tools/chrome-devtools Chrome DevTools]/</ref> #**To open the console panel in Firefox press Ctrl+Shift+J (or Command+Shift+J on a Mac).<ref>[https://developer.mozilla.org/en-US/docs/Tools/Browser_Console MDNː Browser Console]</ref> #**To open the console panel in Internet Explorer press F12 and then Ctrl+2.<ref>[https://msdn.microsoft.com/en-us/ie/dn322041(v=vs.94) MSDNː Using keyboard shortcuts]</ref> #**To open the console panel in Safari press Option-Command-I in the Develop Menu and click on the Console tab.<ref>[https://support.apple.com/guide/safari-developer/safari-developer-tools-overview-dev073038698/mac Safari Developer Helpː Developer Tools overview]</ref> # Using the Hello program above, test JavaScript code placement in a variety of ways: #* Use <code><script></code> in <code><head></code> for the <code>document.write()</code> #* Use <code><script></code> in <code><body></code> for the <code>window.alert()</code> #* Use <code><script src="..."></code> in <code><head></code> or <code><body></code> for the <code>document.getElementById().innerText</code> or <code>document.getElementById().innerHTML</code> and <code>console.log()</code> #** <code><script src="..."></code> is how you link to an external JavaScript page, the name of which should end in .js, and should have a closing <code></script></code> tag # Add comments to the external JavaScript src code above, describing the program. # Add a <code><noscript></code> section to the Hello HTML page above, displaying an appropriate message to users who have JavaScript disabled. Disable JavaScript in your browser to test the noscript section, then re-enable JavaScript. # Research best practices for where to place JavaScript code and how to format and structure it (style guides). # Research the difference between <code><script async ...></code> and <code><script defer ...></code>. == Lesson Summary == * JavaScript, often abbreviated as JS, is a high-level, interpreted programming language that conforms to the ECMAScript specification.<ref>[[Wikipedia: JavaScript]]</ref> * JavaScript comments are created using either <code>//</code> for single-line comments or <code>/* ... */</code> for block comments.<ref>[https://developer.mozilla.org/en-US/docs/Web/JavaScript/Guide/Grammar_and_types#Basics MDN: JavaScript Guide]</ref> * The HTML <code><noscript></code> element defines a section of HTML to be inserted if a script type on the page is unsupported or if scripting is currently turned off in the browser.<ref>[https://developer.mozilla.org/en-US/docs/Web/HTML/Element/noscript MDN: noscript]</ref> * The HTML <code><script></code> element is used to embed or reference executable code, and is typically used to embed or refer to JavaScript code.<ref>[https://developer.mozilla.org/en-US/docs/Web/HTML/Element/script MDN: script]</ref> * <code><script src="..."></code> specifies the URI of an external script and is used as an alternative to embedding a script directly within a document.<ref>[https://developer.mozilla.org/en-US/docs/Web/HTML/Element/script MDN: script]</ref> * JavaScript scripts required during the loading of the page are added to the document head section.<ref>[https://developer.mozilla.org/en-US/docs/Learn/HTML/Howto/Author_fast-loading_HTML_pages MDN: Author Fast-Loading HTML Pages]</ref> * JavaScript scripts not required during the loading of the page are added at the end of the document body section.<ref>[https://developer.mozilla.org/en-US/docs/Learn/HTML/Howto/Author_fast-loading_HTML_pages MDN: Author Fast-Loading HTML Pages]</ref> * JavaScript's best practice is to use external JavaScript files and load them at the end of the document body section.<ref>[https://developer.mozilla.org/en-US/docs/Learn/HTML/Howto/Use_JavaScript_within_a_webpage MDN: Use JavaScript Within a Webpage]</ref> Alternatively, external JavaScript files may be added at the end of the document head section and loaded asynchronously using the <code>async</code> or <code>defer</code> attributes.<ref>[https://developer.mozilla.org/en-US/docs/Web/HTML/Element/script MDN: script]</ref> * The <code>async</code> attribute indicates that the browser should, if possible, load the script asynchronously. The script is executed as soon as it is loaded.<ref>[https://developer.mozilla.org/en-US/docs/Web/HTML/Element/script MDN: script]</ref> * The <code>defer</code> attribute indicates that the browser should, if possible, load the script asynchronously. The script is executed after the document has been parsed.<ref>[https://developer.mozilla.org/en-US/docs/Web/HTML/Element/script MDN: script]</ref> * The <code>Document.write()</code> method writes a string of text to a document stream.<ref>[https://developer.mozilla.org/en-US/docs/Web/API/Document/write MDN: Document.write]</ref> * The <code>Window.alert()</code> method displays an alert dialog with the optional specified content and an OK button.<ref>[https://developer.mozilla.org/en-US/docs/Web/API/Window/alert MDN: Window.alert]</ref> * The <code>document.getElementById()</code> method returns an Element object representing the element whose id property matches the specified string.<ref>[https://developer.mozilla.org/en-US/docs/Web/API/Document/getElementById MDN: Document.getElementById]</ref> * The Element <code>innerHTML</code> property gets or sets the HTML or XML markup contained within the element.<ref>[https://developer.mozilla.org/en-US/docs/Web/API/Element/innerHTML MDN: innerHTML]</ref> * The HTML Element <code>innerText</code> property represents the text content of a node and its descendants.<ref>[https://developer.mozilla.org/en-US/docs/Web/API/HTMLElement/innerText MDN: innerText]</ref> * The <code>console.log()</code> method outputs a message to the web console.<ref>[https://developer.mozilla.org/en-US/docs/Web/API/Console/log MDN: console.log]</ref> == Key Terms == ;Console :Is an object which provides access to the browser debugging console. The console object provides us with several different methods, like :log(), error(), warn(), clear(),time(), etc. To open the developer console window for Chrome, use the keyboard shortcut Ctrl - Shift - J on Windows or Cmd + Option + J on a Mac. To open it for Firefox, use the shortcut Ctrl + Shift + K on Windows, or Cmd + Option + K on a Mac.<ref>{{Cite web|url= https://www.geeksforgeeks.org/console-in-javascript/#:~:text=In%20javascript%2C%20the%20console%20is,error()</ref> ;JavaScript :The proprietary name of a high-level, object-oriented scripting language used especially to create interactive applications running over the internet.<ref>{{Cite web|url=https://www.dictionary.com/browse/javascript|title=Definition of JavaScript {{!}} Dictionary.com|website=www.dictionary.com|language=en|access-date=2021-01-22}}</ref> ;comments :Used in programming languages, it allows the programmer to explain the code in a more understandable way to others. In JavaScript, there are two types of comments that can be created which includes single-line comments and block comments.<ref>{{Cite web|url=https://www.w3schools.com/js/js_comments.asp|title=JavaScript Comments|website=www.w3schools.com|access-date=2020-08-26}}</ref> ;ECMAScript :A scripting-language specification, standardized by Ecma International, which was created to standardize JavaScript and foster multiple independent implementations.<ref>[[Wikipedia: EMCAScript]]</ref> ;external script :A script file that is attached to the HTML using the <script> element. Example - &lt;script src='myscript.js'&gt;.{{source}} ;Nonscript Tag :The <noscript> tag defines an alternate content to be displayed to users that have disabled scripts in their browser or have a browser that doesn't support script. The <noscript> element can be used in both <head> and <body>.<ref>{{Cite web|url= https://developer.mozilla.org/en-US/docs/Web/HTML/Element/noscript</ref> ;statement :A command that performs an action. An example being, alert("string"), where alert prints the string within the parenthesis.{{source}} == Review Questions == * What are three different ways to implement JavaScript into an HTML file? * Is it mandatory to end a statement with a semicolon in JavaScript? * What can you do with <code>document.getElementById()</code> in JavaScript? * How do you create a windowed message with JavaScript? * What is a common use for comments during code testing? * How do you embed JavaScript code in an HTML file? * What is <code><noscript></code> used for? *Why is linking to external scripts considered the best practice for script placement? *Where is the best place to put <script></script> for faster loading webpages? *What is the difference between defer and async? *What is the purpose of the <script> tag in an HTML document? == See Also == * [[Wikipedia: JavaScript]] * [[Wikipedia: Input/output]] * [https://grasshopper.codes/ Grasshopper: JavaScript Coding App for Beginners] * [https://google.github.io/styleguide/jsguide.html Google: JavaScript Style Guide] * [https://developers.google.com/speed/docs/insights/v5/get-started Google: Get Started with the PageSpeed Insights API] * [https://javascript.info/script-async-defer JavaScript Info: Scripts: async, defer] * [https://www.codecademy.com/learn/introduction-to-javascript Codecademy: Introduction to Javascript] * [https://www.learn-js.org/ Javascript Learning Resource] * [https://javascript.info/ The Modern Javascript Tutorial] * [https://learnjavascript.online/ Learn Javascript Online] * [https://www.geeksforgeeks.org/understanding-basic-javascript-codes/?ref=lbp Geeks for Geeks: Understanding basic JavaScript codes] *[https://www.w3schools.com/js/default.asp W3Schools: JavaScript] *[https://blog.webdevsimplified.com/2019-12/javascript-loading-attributes-explained/ Async Vs Defer - JavaScript Loading Explanation] == References == {{reflist}} <noinclude>{{subpage navbar}}</noinclude> {{CourseCat}} 4lgixeztsur0xf07skrcvoo7c6xau16 0 247474 2842439 2697506 2026-10-24T14:33:42Z Z. Patterson 1936818 /* Lesson Summary */ Spelling and grammar correction 2842439 wikitext text/x-wiki <noinclude>{{{{NAMESPACE}}:{{BASEPAGENAME}}/Sidebar}}</noinclude> This lesson introduces JavaScript variables, data types, expressions, and operators. __TOC__ == Objectives and Skills == Objectives and skills for this lesson include:<ref>[https://www.microsoft.com/en-us/learning/exam-98-382.aspx Microsoft: Exam 98-382 Introduction to Programming Using JavaScript]</ref> * Apply JavaScript best practices ** Comments; indentations; naming conventions; noscript; constants; reserved keywords; debugger keyword; setting breakpoints; console.log * Declare and use variables of primitive data types ** Number; Boolean; String; Null; Undefined; typeof operator; type checking functions; use strict; converting between data types; formatting numbers; string operations; single quote vs double quote (nesting); initialization * Complete or debug code that uses assignment and arithmetic operators ** Assignment; increment; decrement; addition; subtraction; division; multiplication; modulus; compound assignment operators. == Readings == # [[Wikibooks: JavaScript/Variables and types]] # [[Wikibooks: JavaScript/Numbers]] # [[Wikibooks: JavaScript/Strings]] # [[Wikibooks: JavaScript/Operators]] == Multimedia == *'''Variables''' # [https://www.youtube.com/watch?v=IhKWfiNCoSY YouTube: JavaScript Tutorial for Beginners - 03 - Variables] # [https://www.youtube.com/watch?v=UR71iavwMkE YouTube: JavaScript Tutorial for Beginners - 04 - Variables Part 2] # [https://www.youtube.com/watch?v=M1xuykxResM YouTube: JavaScript Programming Tutorial 6 - Variables and Expressions] # [https://www.youtube.com/watch?v=dzEieWaOJE0 YouTube: JavaScript Let, Const & Var] # [https://www.youtube.com/watch?v=9aGIAL16DL4 YouTube: 7: How to Create Variables in JavaScript | JavaScript Tutorial | Learn JavaScript] # [https://www.youtube.com/watch?v=y9PscU1uMrA YouTube: What is a Variable? | JavaScript in less-than 3 minutes] # [https://youtu.be/YptrAq5O-jQ?si=lP0bXs3X1qooWqCK YouTube: Variables & Data Types | Javascript | Tutorial 6] *'''Data Types and Operators''' # [https://www.youtube.com/watch?v=edlFjlzxkSI YouTube: Javascript Variables & Data Types] # [https://www.youtube.com/watch?v=hWdTfi-68Bo YouTube: JavaScript Tutorial for Beginners - 08 - Operators] # [https://youtube.com/watch?v=opz_-qeDi2E YouTube: A Basic Calculator] # [https://www.youtube.com/watch?v=VRz0nbax0uI YouTube: 20 String Methods in 7 Minutes] *'''Input''' # [https://www.youtube.com/watch?v=15oNAHVuwIE YouTube: Getting User Input | Javascript | Tutorial 9] # [https://www.youtube.com/watch?v=JeXqaKeJSRI&t=253s YouTube: How to accept JavaScript USER INPUT in 5 minute] *'''Debugging''' # [https://www.youtube.com/watch?v=AX7uybwukkk YouTube: Debugging JavaScript (Google Chrome and Visual Studio Code)] # [https://www.youtube.com/watch?v=bwUNifZ4WrY YouTube: Firefox JavaScript Debugger] # [https://www.youtube.com/watch?v=H0XScE08hy8 YouTube: Debugging JavaScript - Chrome DevTools 101] *'''Type Coercion''' # [https://www.youtube.com/watch?v=0yKFDXB3gNw&t=243s YouTube: JavaScript Type Coercion - Javascript Tutorial #4] # [https://academind.com/tutorials/javascript-type-coercion JavaScript & Type Coercion] *'''Expressions vs. Statements''' # [https://www.youtube.com/watch?v=WVyCrI1cHi8 Expressions vs. Statements in JS / Intro to JavaScript ES6 programming, lesson 14] == Examples == * [https://www.w3schools.com/js/js_syntax.asp W3Schools: JavaScript Syntax] * [https://www.w3schools.com/js/js_variables.asp W3Schools: JavaScript Variables] * [https://www.w3schools.com/js/js_operators.asp W3Schools: JavaScript Operators] * [https://www.w3schools.com/js/js_statements.asp W3Schools: JavaScript Statements] * [https://www.w3schools.com/js/js_arithmetic.asp W3Schools: JavaScript Arithmetic] * [https://www.w3schools.com/js/js_assignment.asp W3Schools: JavaScript Assignment] * [https://www.w3schools.com/js/js_popup.asp W3Schools: JavaScript Popup Boxes] * [https://guide.freecodecamp.org/javascript/popup-boxes/ Freecodecamp: JavaScript Popup Boxes with Examples] * [https://www.w3schools.com/js/js_debugging.asp W3Schools: JavaScript Debugging] * [https://www.w3schools.com/js/js_datatypes.asp W3Schools: JavaScript Data Types] * [https://www.w3schools.com/js/js_numbers.asp W3Schools: JavaScript Numbers] * [https://www.w3schools.com/js/js_number_methods.asp W3Schools: JavaScript Number Methods] * [https://www.w3schools.com/js/js_strings.asp W3Schools: JavaScript Strings] * [https://www.w3schools.com/js/js_string_methods.asp W3Schools: JavaScript String Methods] * [https://www.w3schools.com/js/js_strict.asp W3Schools: JavaScript Strict Mode] * [https://www.w3schools.com/js/js_type_conversion.asp W3Schools: JavaScript Type Conversion] *[https://www.w3schools.com/js/js_datatypes.asp W3Schools: Null vs Undefined] *[https://codeburst.io/javascript-null-vs-undefined-20f955215a2 JavaScript — Null vs. Undefined] * [[/Example Code/]] * [https://developer.mozilla.org/en-US/docs/Web/JavaScript/Guide/Expressions_and_Operators MDN Web Docs: Expressions and Operators] * [https://www.geeksforgeeks.org/javascript-expressions-complete-reference/ GeeksforGeeks: Javascript Expressions] * [https://www.tutorialrepublic.com/javascript-tutorial/javascript-data-types.php Data types: Data Types in JavaScript] == Activities == Complete two of the following activities using external JavaScript code. Apply JavaScript best practices, including comments, naming conventions, and constants. In at least one program, prompt the user for input and use console.log() for output. In another program, prompt the user for input and use document.getElementById().innerText or document.getElementById().innerHTML for output. Use the strict directive. Create test data to validate the accuracy of each program. Add comments at the top of the program and include references to any resources used. === Arithmetic === # Create a program that uses variables for hours and rate per hour and then calculate and displays weekly, monthly, and annual gross pay (hours * rate). Base monthly and annual calculations on 12 months per year and 52 weeks per year.<ref>[http://www.pythonlearn.com/html-270/book003.html PythonLearn: Variables, expressions, and statements]</ref> # Create a program that uses variables for years, and then calculate and displays an approximate age in months, days, hours, and seconds. For example, a person 1-year-old is 12 months old, 365 days old, etc. # Review [http://www.mathsisfun.com/measure/us-standard-length.html MathsIsFun: US Standard Lengths] and [https://www.mathsisfun.com/metric-imperial-conversion-charts.html MathsIsFun: Metric-US/Imperial Conversions]. Create a program that uses variables for a distance in miles, and then calculate and displays the distance in yards, feet, and inches, or calculate and display the distance in kilometers, meters, and centimeters. # Review [http://www.mathsisfun.com/area.html MathsIsFun: Area of Plane Shapes]. Create a program that uses variables for the dimensions of different shapes and then calculate and displays the area of the shapes. Do not include shape choices. That will come later. For now, just include multiple shape calculations in sequence. # Create a program that calculates the area of a room to determine the amount of floor covering required. The room is rectangular with the dimensions measured in feet with decimal fractions. The output needs to be in square yards. There are 3 linear feet to a yard.<ref>[[Wikibooks: Programming Fundamentals/Practice: Data and Operators]]</ref> # Review [http://www.mathsisfun.com/operation-order-bodmas.html MathsIsFun: Order of Operations]. Create a program that demonstrates the order of operations. Include parentheses, exponents, multiplication, division, addition, and subtraction in your program. Use variables for the calculations and label the output. For example, part of the program might display:<br>&nbsp;&nbsp;&nbsp;&nbsp;<code>1 + 2 * 3 = 7</code><br>&nbsp;&nbsp;&nbsp;&nbsp;<code>(1 + 2) * 3 = 9</code><br>&nbsp;&nbsp;&nbsp;&nbsp;<code>...</code> === Data Types === # Create a program that initializes different variables with integer, floating point, string, null, and undefined values. Demonstrate various operations and converting between data types. For example, user input is always a string, but adding string values of "1" + "1" is typically "11", whereas, adding numeric values of 1 + 1 is 2. Use <code>typeof</code> to label each variable and the resulting output. # Create a program that demonstrates string operations and string formatting. Define string constants or string literals that include apostrophes ('). Define string constants or string literals that include "quotes". Convert between string and numeric data types using Number() and .toFixed() with two decimal places. Concatenate string literals and variables to display output. === Debugging === # Use the <code>debugger</code> statement to create a breakpoint in one of the programs above. Experiment with single-stepping through the code and viewing script and global values. Add one or more variables to the watch window. # Add a breakpoint to the program somewhere after the <code>debugger</code> statement. After the program stops with the debugger, resume script execution and observe the script stopping at the breakpoint. == Lesson Summary == * By convention, JavaScript variable names are written in camelCase.<ref name=":2">https://google.github.io/styleguide/jsguide.html</ref> * In JavaScript, I explored variable declarations with <code>let</code> and <code>const</code>, string properties and methods such as <code>length</code> and <code>toUpperCase()</code>, primitive data types like <code>number</code>, <code>string</code>, <code>boolean</code>, <code>null</code>, and <code>undefined</code>, and various operators for string concatenation, arithmetic, assignment, increments, comparisons, and logical evaluations. * JavaScript Class, interface, record, and typedef names are often written in Title Case.<ref name=":2" /> * JavaScript constant names are written in UPPER_CASE.<ref>https://google.github.io/styleguide/jsguide.html</ref> * JavaScript variable and constant names must be unique. These unique names are called identifiers.<ref name="W3Schoolsː JavaScript Variables">[https://www.w3schools.com/js/js_variables.asp W3Schoolsː JavaScript Variables]</ref> * Reserved words (for example, continue, debugger, break, etc.) can't serve as variable names.<ref name = "W3Schoolsː JavaScript Variables" /> * Names can begin only with a letter, an underscore or a dollar sign. Names are allowed to contain numbers, but a number can't be the first character.<ref name = "W3Schoolsː JavaScript Variables" /> * In JavaScript, names are case sensitive.<ref name = "W3Schoolsː JavaScript Variables" /> *Comments, which are annotations or explanations of written code<ref>{{Cite journal|date=2021-01-08|title=Comment (computer programming)|url=https://en.wikipedia.org/w/index.php?title=Comment_(computer_programming)&oldid=999054165|journal=Wikipedia|language=en}}</ref>, are done in JavaScript using // for a single line, and /* ... */ for multiple lines. * JavaScript has 7 data types: Number, String, Boolean, Null, Undefined, Function, and Object. Null and Undefined data types can't contain values.<ref name = "W3Schoolsː JavaScript Data Types">[https://www.w3schools.com/js/js_type_conversion.asp W3Schoolsː JavaScript Data Types]</ref> * There are 6 types of objects in JavaScriptː Object, Date, Array, String, Number and Boolean.<ref name = "W3Schoolsː JavaScript Data Types"/> * JavaScript has only 1 type of number.<ref name="W3Schoolsː JavaScript Numbers">[https://www.w3schools.com/js/js_numbers.asp W3Schoolsː JavaScript Numbers]</ref> * With some other languages, like Java and C, there is a special “character” type. JavaScript consists of only one type, string.<ref>{{Cite web|url=https://javascript.info/types|title=Data types|website=javascript.info|language=en|access-date=2019-09-05}}</ref> * The <code>typeof</code> operator is used to find the data type of the variable.<ref name = "W3Schoolsː JavaScript Data Types"/> * JavaScript has built-in functions to work with strings and numbers<ref name="MDN Web Docs: Data Types and Data Structures">[https://developer.mozilla.org/en-US/docs/Web/JavaScript/Data_structures MDN Web Docs: Data Types and Data Structures]</ref> * Strings are written using single or double quotation marks. Quotation marks can be used inside the string if they don't match the ones outside of it.<ref name = "W3Schoolsː JavaScript Data Types"/> Quotation marks used inside can match the outside ones only if they are preceded by the backslash escape character.<ref>[https://www.w3schools.com/js/js_strings.asp W3Schoolsː JavaScript Strings]</ref> * You can convert between data types in JavaScript.<ref name = "W3Schoolsː JavaScript Data Types">[https://www.w3schools.com/js/js_type_conversion.asp W3Schoolsː JavaScript Data Types]</ref> * "use strict" is a literal expression that requires / enforces variable declaration. It allows for cleaner code to be written.<ref>[https://www.w3schools.com/js/js_strict.asp W3Schoolsː JavaScript Use Strict]</ref> == Key Terms == ;array :An array is a type of variable that stores a collection of values. The position of each value inside the array is that value's index. Indexes in JavaScript always start at 0.<ref>{{Cite web|url=https://www.w3schools.com/js/js_arrays.asp|title=JavaScript Arrays|website=www.w3schools.com|language=en-US|access-date=2024-03-08}}</ref> ;assignment :Assigning a value to a variable or constant with the "=" operator.{{source}} ;Boolean :Data type which has only two values, True and False.<ref name = "JavaScript.Infoː Data Types">[https://javascript.info/types JavaScript.Infoː Data Types]</ref> ;camel case :Method of joining multiple words into one variable name where the first word is not capitalized and every subsequent first letter of following words are capitalized. (Ex. CamelCase)<ref name=":1">[https://docs.microsoft.com/en-us/previous-versions/dotnet/netframework-1.1/x2dbyw72(v=vs.71)?redirectedfrom=MSDN Microsoft: Capitalization Styles]</ref> ;Title case or pascal case :Method of joining multiple words into one variable name where the first letter of every word is capitalized.<ref name=":1" /> ;concatenation :Method of joining two strings, data, or other text together in programming. In JavaScript, strings are concatenated using the '+' operator. <ref>{{Cite web|url=https://www.computerhope.com/jargon/c/concaten.htm|title=What is Concatenate?|website=www.computerhope.com|language=en|access-date=2020-09-04}}</ref> ;console.log :A function in JavaScript which is used to print any kind of variables defined before in it or to just print any message that needs to be displayed to the user. ;constant :A keyword introduced from ES2015, It is a container for storing a permanent data value that cannot be changed later in the program.<ref name=":0">[https://www.w3schools.com/js/js_const.asp W3Schools: JavaScript const]</ref> ;Equality :An operator referencing when two values are equal, "==", this is distinct from the assignment ("=") operator referenced above.<ref name=":0" /> ;let :A keyword introduced from ES2015, It is used to declare a variable in JavaScript. it is used as block scoped variable while "var" is used as a function scope variable <ref>{{Cite web|url=https://www.w3schools.com/js/js_let.asp|title=JavaScript Let|website=www.w3schools.com|access-date=2020-09-01}}</ref> ;null :A special value which represents "nothing." It is not a reference to a null pointer.<ref name = "JavaScript.Infoː Data Types"/> ;number :Data type in JavaScript which represents both integers and floating-point value.<ref name = "JavaScript.Infoː Data Types"/> ;object :A variable that can contain many values.<ref>[https://www.w3schools.com/js/js_objects.asp W3Schoolsː JavaScript Objects]</ref> ;order of operations :A set of rules for for evaluating mathematical expressions. While different programming languages have different rules for precedence, the order of operations start from greatest precedence with parentheses, exponents, multiplication/division, and addition/subtraction.<ref>{{Cite web|url=https://en.wikibooks.org/wiki/Programming_Fundamentals/Order_of_Operations|title=Programming Fundamentals/Order of Operations - Wikibooks, open books for an open world|website=en.wikibooks.org|access-date=2020-09-04}}</ref> '''scope''' The current context of the code, refers to the accessibility of functions and variables. ;string :Can consist of one character or many. A string must be surrounded by quotation marks.<ref name="JavaScript.Infoː Data Types" /> ;type coercion :The automatic or implicit conversion of values from one data type to another (such as strings to numbers).<ref>[https://developer.mozilla.org/en-US/docs/Glossary/Type_coercion MDNː Type coercion]</ref> ;undefined :A variable which has not been assigned a value. If a variable is simply declared, but not assigned anything, it becomes undefined.<ref name="JavaScript.Infoː Data Types" /> ;variable :A container for storing a data values. Variables are used to hold values in expressions. <ref name="W3Schoolsː JavaScript Variables" /> == See Also == * [[Wikipedia: Variable (computer science)]] * [[Wikipedia: Constant (computer programming)]] * [[Wikipedia: Data type]] * [[Wikipedia: Expression (computer science)]] * [[Wikipedia: Statement (computer science)]] * [[Wikipedia: Assignment (computer science)]] * [[Wikipedia: Order of operations]] * [[Wikipedia: Self-documenting code]] * [https://www.robinwieruch.de/javascript-naming-conventions Robin Wieruch: Javascript Naming Convention] * [https://www.freecodecamp.org/news/var-let-and-const-whats-the-difference/ Var, Let, and Const – What's the Difference?] *[https://www.w3schools.com/js/js_datatypes.asp The typeof Operator] * [https://developers.google.com/web/tools/chrome-devtools/javascript Get Started with Debugging JavaScript in Chrome DevTools] == References == {{reflist}} <noinclude>{{subpage navbar}}</noinclude> {{CourseCat}} 8ijaoczmfbpixvxv7dtame8990jgojv 0 254749 2842517 2722084 2026-10-25T02:56:46Z ~2026-56923-92 3113251 /* Problems with discounting */ Fixed a typo where the word "sued" was used instead of "used". 2842517 wikitext text/x-wiki == Capital Expenditure Decisions == Capital expenditure is a cash purchase of assets expected to generate future cash flows that extend beyond a one-year period. It is different from a normal operating expenditure, which is expected to result in cash flow within a one-year period. Examples of capital expenditures include land, buildings, machinery, business vehicles or intangible assets (such as a licence or patent). Capital expenditure is important as it has a long-term impact on the future of a firm; determining what products will be produced, markets that will be entered, the location of plants and facilities, and the type of technology that the firm will use. Therefore, capital expenditures need to be carefully thought through and analysed as they are costly decisions and are usually more costly to change. (James R. McGuigan, R. Charles Moyer, Frederick H. deB. Harris 2013) <ref>https://learn.uq.edu.au/bbcswebdav/pid-4885320-dt-content-rid-21494565_1/courses/ECON3430S_6960_61095/Investment.pdf</ref> === Capital Budgeting Process === Capital budgeting is the process a firm uses to plan and evaluate capital expenditures. Capital budgeting techniques can be used, in addition to firm expansion and asset replacement decisions, to also analyse research and development expenses, investing in education and training for employees, lease-vs-buy decisions, and potential mergers and acquisitions. (James R. McGuigan, R. Charles Moyer, Frederick H. deB. Harris 2013) <ref>https://learn.uq.edu.au/bbcswebdav/pid-4885320-dt-content-rid-21494565_1/courses/ECON3430S_6960_61095/Investment.pdf</ref> The process of selecting capital investment projects consists of the following important steps: '''1. Generate alternative capital investment project proposals.''' New capital investment ideas can originate from multiple revenues, inside and outside the company. Proposals can be initiated at different levels in the firm, from contract workers to the board of directors. Often, medium and large-sized firms consist of departments whose job involves studying and finding capital expenditure projects. These departments include cost accounting, industrial engineering, marketing research, research and development, and corporate planning. (James R. McGuigan, R. Charles Moyer, Frederick H. deB. Harris 2013) <ref>https://learn.uq.edu.au/bbcswebdav/pid-4885320-dt-content-rid-21494565_1/courses/ECON3430S_6960_61095/Investment.pdf</ref> '''2. Estimate cash flows for the project proposals.''' Research has shown that a relationship exists between the presentation of financial information and user decisions. Whilst earnings are very important, investors perceptions of cash flow information has increased significantly and is useful in the assessment of financial decisions. When estimating the cash flows, it is useful to follow certain basic guidelines. Firstly, cash flows should be measured on an incremental premise. This means that the difference between the cash flows streams to the firm with and without acceptance of the investment project, should be represented in the cash-flow stream for that project. Secondly, use the firm’s marginal tax rate to measure cash flows on an after-tax basis. Thirdly, all indirect effects of the firm throughout the firm should be incorporated in the calculations for cash-flow. If a department is considering a capital investment that will change other department’s revenues or costs, these external effects should then be included in the cash-flow estimates. Fourth, when evaluating the project, sunk costs should not be considered. Sunk costs are expenditures that have been made and cannot be avoided; hence, they should not be considered when deciding whether to accept or reject a project. Lastly, opportunity costs should be used to measure the value of the resources used in the project. Opportunity cost is the value of a resource’s next best alternative use. (James R. McGuigan, R. Charles Moyer, Frederick H. deB. Harris 2013) <ref>https://learn.uq.edu.au/bbcswebdav/pid-4885320-dt-content-rid-21494565_1/courses/ECON3430S_6960_61095/Investment.pdf</ref> '''3. Evaluate and choose investment projects to implement.''' After identifying the capital expenditure project and the cash flows have been estimated, a decision needs to be made whether to accept or reject the project. Accepting the project would mean a cash-flow stream to the firm – series of either cash inflows or outflows in future years. A project will typically see an initial outflow (first-year investment), followed by a series of cash inflows (returns) over a number of succeeding years.(James R. McGuigan, R. Charles Moyer, Frederick H. deB. Harris 2013) <ref>https://learn.uq.edu.au/bbcswebdav/pid-4885320-dt-content-rid-21494565_1/courses/ECON3430S_6960_61095/Investment.pdf</ref> '''4. Review''' Finally, after the investment project has been implemented a review must be completed to ensure the accuracy of assumptions. Any inaccurate assumptions should be modified and noted for similar future investment projects. (James R. McGuigan, R. Charles Moyer, Frederick H. deB. Harris 2013) <ref>https://learn.uq.edu.au/bbcswebdav/pid-4885320-dt-content-rid-21494565_1/courses/ECON3430S_6960_61095/Investment.pdf</ref> === Estimating Cash Flows === When estimating Cash Flows some basic guidelines are helpful: 1. Cash flows should be measured on an incremental basis (It should represent the difference between the cash-flow streams to the firm with and without acceptance of the investment project) 2. Cash flows should be measured on an after-tax basis, using the firm’s marginal tax rate. 3. All indirect effects of the project throughout the firm should be included in the cash-flow calculations. 4. Sunk costs should not be considered when evaluating the project (Sunk costs = Costs that has already been incurred and cannot be recovered) 5. The value of resources used in the project should be measured in terms of their opportunity costs. In a typical investment project, an initial investment is made in year 0, generating a series of yearly net cash flows over the life of the project (n). The net investment (NINV) of a project is defined as the initial net cash outlay in year 0, including the acquisition cost of any new assets plus the installation and shipping costs and tax effects. The incremental, after-tax net cash flows (NCFs) of a particular investment project is equal to cash inflows minus cash outflows. These could be defined as the difference in net income after tax (ΔNIAT) with and without the project plus the difference in depreciation (ΔD): NCF = Δ NIAT + ΔD ΔNIAT is equal to the difference in net income before tax (ΔNIBT) times (1 – T), where T is the corporate marginal income tax rate: ΔNIAT = ΔNIBT (1 – T) ΔNIBT is defined as the difference in revenues (ΔR) minus the difference in operating costs (ΔC) and depreciation (ΔD): ΔNIBT = ΔR – ΔC – ΔD Substituting some equations, the net cash flow could be yield as follows: NCF = (ΔR – ΔC – ΔD) (1 – T) + ΔD == Investment Projects Evaluation == In capital budgeting, there are various approaches that can be used to evaluate a project. In order to determine whether or not to invest in a project, qualitative methods namely Internal Rate of Return (IRR) and Net Present Value (NPV), can be used. Other relevant investment criteria include the payback period and the average accounting return. The payback period determines the point in time when the firm will have generated enough cash to cover the cost of investment, and the average accounting return measures return using book values as opposed to market values. As long as the cash inflows and outflows are known or accurately estimated over a number of years, the value or return of the project can be determined. An investment project is usually incurs a negative cash flow in the first year, known as the cost of investment. Then, during the following years, the investment typically generates positive incomes known as returns. ===Net Present Value (NPV)=== Net present value (NPV) can be defined as a tool to calculate the present value of the cash inflows (returns) and outflows (costs) of a project over it's working life, with a required rate of return<ref>Mcguigan, J., Moyer, C. & Harris, F., 2013. Managerial Economics: Applications, Strategies and Tactics, Andover: Cengage Learning.</ref>. A positive NPV of a project denotes that the stream of revenues generated from the project exceeds the cost of the project while also being discounted at the required rate of return. This then infers that a project with a positive NPV is profitable and should be accepted. If a project has a predicted negative NPV then the project will not be profitable and should not be accepted by the firm. The NPV is the main deciding factor when a firm is considering whether or not to accept a project and is a more important consideration than the IRR. The calculation for NPV is seen below. <math> NPV = \displaystyle\sum_{t=1}^{n} \frac{NCF_{t}}{(1+k)^t} - NINV</math> where; NCF, is the net cash flow at time t, NINV, is the net investment, k, is the required rate of return, n, is the expected working life of the project, ====Time inconsistent preferences==== When given a choice between getting $10 right now or $12 in a week, many people chose to take $10, even though the amount they could get if they waited would be higher. This can be explained because, for most people, the $2 difference is not worth waiting a week for. However, when given the choice between $10 in a week and $12 in 2 weeks, people tend to change their mind and go for the second option. In both cases, individuals chose between $10 or $12 a week later. The choice should, therefore, be the same. Conversely, is that the first scenario lets individuals chose between a reward in the present or one in the future. The second one only allows for future rewards. The shift in people's choice shows that preferences can be inconsistent through time. One way to explain this is through the concept of present bias. Present bias is the assumption that people are more impatient if benefits or costs are immediate. Procrastination and self-control problems can be explained using time-inconsistent preferences. For example, when you decide on a Monday that you will go to the gym on Friday, you have made this decision because you know that the benefits outweigh the costs. However, once you get to the end of the week, your decision may change because you actually have to face the costs of going to the gym. This time-inconsistent behaviour cannot be explained using an exponential model, because exponential discounting does not account for present bias. However, the use of hyperbolic discounting functions can help shine a light on this phenomenon. This bias can be accounted for via the use of a discount factor which makes it possible to explain why an individual can change their opinion between time periods. ===== Benefits ===== ''1. Quantified cash flow'' - NPV will only consider the visible money value flow of the project. It can help the decision-makers to understand how much actual money they can earn. ''2. Considering time value of money '' - The advantage of using the NPV method over IRR is that NPV can handle multiple discount rates as a year's cash flow can be discounted separately from the others. NPV will be adjusted by the discount rate to affect the future cash flow to the value of today. It can help the decision-makers know about the whole project’s income as of today's value. The NPV is primarily used to determine whether an investment such as a project, merger or acquisition will add value to a company. Positive net values will ensure shareholder satisfaction, whereas negative values are not feasible. ===== Weakness ===== ''1. Difficult to determine the discount rate(cost of capital)'' - The cost of capital is a vital component in the NPV. Wrongly estimate the discount rate to lead the NPV error and not reliable. ''2. Does not available to make the decision without comparison to another project.'' ===Internal Rate of Return (IRR)=== IRR is a method of project evaluation which calculates the interest rate where the NPV of all project’s cash flows is equal to zero. In other-words, given the initial investment, the IRR is the discount rate/interest rate needed for the project to breakeven. It is an annualised rate that ranks prospective projects. A project is more attractive when its IRR is higher than the other comparable options. When the market’s opportunity cost (the project’s cost of capital) is lower than the estimated average return (IRR) on a project, at a similar maturity period and risk, the project should be undertaken. In contrast, when the cost of capital is higher than the IRR, the project should be turned down. However, IRR may provide misleading investment decisions as the IRR rule may disagree with the NPV rule due to certain situations such as delayed investments (positive cash flows precede negative cash flows), non-existent IRR, multiple IRR and comparing projects with different scale, timing and risk. in this case (when the rules conflict), the NPV decision rule should be followed. The calculate the IRR the following equation is used:<ref name="name">Mcguigan, J., Moyer, C. & Harris, F., 2013. Managerial Economics : Applications, Strategies and Tactics, Andover: Cengage Learning. </ref><br> <math> NPV = \displaystyle\sum_{t=1}^{n} \frac{NCF_{t}}{(1+r)^t} - NINV</math> <br> Where;<br> NCF, is the net cash flow at time t, <br> NINV, is the net investment<br> n, is the life of the investment<br> r, is the IRR ===== Benefits ===== 1. Easy to understand IRR is defined as an easy method to calculate and provides a simple way to compare the value of the various projects under consideration. IRR provides for the small business owner which capital projects will provide the greatest potential cash flow. It can also be used for budgetary purposes, such as providing the potential value of buying new equipment or cost savings, rather than repairing old equipment. （There is no need of the pre-determination of cost of capital or cut off rate.） 2. Considering the time value of money Time value of money is the basic concept of finance. Money has a time value, which is based on people's desire to obtain money now rather than in the future. Therefore, when money is used for saving or investing, it should earn interest. In this way, the future value of money should be greater than the present value 3. IRR provides for maximizing profitability. IRR is able to correlate the lifetime return of a project with the total amount of investment, indicating the maximum return that can be made on the project. To compare IRR with the industry benchmark return on investment then can determine whether the project is worth the investment. ===== Weakness ===== 1. As discount rates change substantially over time, IRR does not account for this, therefore this is not an adequate method for long term projects where discount rates are volatile. Example: A one year T-bill returned between 1% and 15% in the last 20 years, evidently the discount rate is varying. 2. IRR only can show one constant '''Cost of borrowing''' of the whole project IRR can not present the changing of the cost of borrowing during the cash inflow and outflow and it also does not represent the rate of return. Therfore, IRR calculations are ineffective when a project has a mixture of both positive and negative cash flows. This could be the case if a firm has to redesign the brand every few years in order to remain competitive and trendy in the evolving market. 3. For the IRR to be a valid method to evaluate a project it must be compared to a discount rate, therefore if this is not known, then the IRR bears no value. Hence, if the IRR is above the discount rate, the project is deemed practical and if it is below the discount rate, then the project is not feasible. 4. Ignoring the risk of the investment project. IRR assumes the NPV is equal to 0 to calculate the cost of borrowing. But NPV set the cost of capital which considered the risk of the investment project. In contrast, IRR has ignored the risk to reflect the projects. 5. IRR focuses on profitability, not on the early recovery of capital expenditure. Also, IRR sometimes prefers projects that take longer to recoup capital expenditure. With uncertainty in the future, following IRR may not recoup all capital expenditure. ==== Problems with discounting ==== # The first problem is the determining of the discount rate, to be used either to calculate the NPV or to be compared to the IRR. Defining this as the opportunity cost of capital gives some insight but is of limited value since the opportunity cost is essentially subjective, depending both on the circumstance of the investor and the investors view of these circumstances. # The opportunity cost of capital used for investment is dependent on both the method of financing that investment and the degree of risk associated with that investment. # The treatment of uncertainty is another concern; if the alternative to investing in risky productive assets is investing in risk-free financial assets, the discount rate for productive assets will exceed the interest rate on financial assets by a margin determined by the attitude towards the risk of the investor. <ref>Hill, S. (2016). ''Managerial Economics''. London: Macmillan Education, Limited, pp.301-324.</ref> === Conclusion: NPV vs IRR <ref name="name">Mcguigan, J., Moyer, C. & Harris, F., 2013. Managerial Economics : Applications, Strategies and Tactics, Andover: Cengage Learning. </ref> === The above discussion provides a detailed analysis on NPV and IRR. The table presented below provides a brief analysis defining the key differences between the two. <br> {| class="wikitable" |- ! !! Net Present Value Method (NPV) !! Internal Rate of Return Method (IRR) |- | Decision Rule || It the projects NPV is zero or positive accept the project i.e. if the NPV of cash flows is greater than the required net investment || If the IRR is greater than or equal to the cost of capital of the firm accept the project |- | Benefits || Takes into account the timing of the cash flows <br> An objective is specified<br> Criteria regarding whether to accept or reject a project || The meaning of IRR is simple to interpret<br> Takes into account the timing of the cash flows<br> An objective is specified<br> Criteria regarding whether to accept or reject a project |- | Weaknesses || The meaning of NPV is somewhat difficult to interpret || The decision outcome can sometimes disagree with the NPV <br>Problems with multiple rates of return |} ==Estimating the Cost of debt== The cost of debt faced by a firm is the rate of interest that it pays on loans and bonds. Riskier firms tend to have a higher cost of debt due to a higher premium to compensate for the additional risk by the debtholder. The following formula is used to calculate the cost of debt<ref name="name">Mcguigan, J., Moyer, C. & Harris, F., 2013. Managerial Economics : Applications, Strategies and Tactics, Andover: Cengage Learning. </ref> <br> k<small>i</small>=k<small>d</small>(1-T)<br> With;<br> k<small>i</small>, is the after tax cost of debt<br> k<small>d</small>, is the pre tax cost of debt<br> T, is the firms marginal tax rate <br><br> An example of how a firm raises debt capital is shown below. Assume Disney sells $100 million worth of 7% corporate bonds at par value. If the corporate tax rate is equal to 30%, then the after-tax cost of debt is equivalent to:<br> = 7*(1-0.3) = 4.9% The cost of debt is usually influenced by a firm’s credit rating. Credit ratings range between AAA and D. For instance, again using Disney from the example above, Standard and Poor’s currently assigns it a credit rating of A. Any rating above BBB- is classed as an “investment grade”. <ref>{{Cite web|url=https://www.marketwatch.com/story/sp-downgrades-disney-by-one-notch-to-a-ahead-of-fox-deal-2019-03-12|title=S&P downgrades Disney by one notch to A ahead of Fox deal|last=Linnane|first=Ciara|website=MarketWatch|language=en-US|access-date=2019-10-23}}</ref> ==Estimating the Cost of capital== The cost of funds supplied to a firm are known as the cost of capital. The cost of capital can be comprised of both debt and equity components, and forms the minimum rate of return that must be earned on new investments. The debt component (or the cost of debt capital) is the rate of return required by investors, whereas the equity component (or the cost of internal equity capital), is the rate required by the firm's stock investors. Firms raise equity capital through retained profits or the sale of shares, while debt capital is raised through loans or the issue of debt instruments such as bonds. The proportion of debt and equity form the company's target capital structure, and are used to calculate the weighted cost of capital figure. This figure is used to determine a discount rate to use when evaluating proposed capital expenditure projects. Both the net-present-value and the internal-rate-of-return methods result in identical decisions to either accept or reject individual projects. NPV is greater than 0 if and only if the IRR is greater than the required rate of return k. ''For example:'' Two projects: X and Y are mutually exclusive, requiring both same net investment but different cash flows over the years. If: '''IRR:''' Project X > Project Y --> Project X is preferred '''NPV:''' Project Y > Project X --> Project Y is preferred In this case, it is necessary to determine which of the two criteria is the correct one to use. The outcome depends on what assumptions the decision maker chooses to make about the implied reinvestment rate for the net cash flows generated from each project. The NPV method assumes that cash flows are reinvested at the firm’s cost of capital, while the IRR method assumes that the cash flows are reinvested at the computed IRR. In general, the cost of capital is considered to be a more realistic reinvestment rate than the computed IRR because it is the rate the next marginal investment project can be assumed to earn. '''Cost of Equity Capital''' The cost of equity capital, to the firm, is the equilibrium rate of return required by the firm's common stock investors. Firms can raise equity capital in two ways: 1. Internally: through retained earnings 2. Externally: through the sale of new common stock. The cost of internal equity to the firm is less than the cost of new common stock because the sale of new stock requires the payment of flotation costs. Flotation cost is the total cost incurred by a firm offering its securities to the general public. The concept of the cost of equity can be developed using several different methodologies, including the dividend valuation model. The cost of external equity is greater than the cost of internal equity because of two reasons. First, associated flotation costs with new shares are generally so high that realistically, it cannot be ignored. Secondly, the selling price of new shares to the public need to be lower than the market price of the stock before the new issue is announced, if not the shares may not sell. Before being announced, the current market price of a stock usually represents the supply and demand equilibrium. If supply increase, ceteris paribus, the new equilibrium price will be lower. (James R. McGuigan, R. Charles Moyer, Frederick H. deB. Harris 2013) <ref>McGuigan, J, Moyer, C, & Harris, F 2013, Managerial Economics : Applications, Strategies and Tactics, Cengage Learning, Andover. Available from: ProQuest Ebook Central. [21 October 2019].</ref> <br /> ====== Cost of Equity Capital and Capital Asset Pricing Model ====== The cost of equity capital for a particular firm is often estimated by reference to the '''Capital Asset Pricing Model''' (CAPM). The CAPM is a useful framework for substantiating cost of equity assumptions used in valuations that may otherwise be arbitrary (e.g. 10% or 15%). The model is expressed as follows: '''Cost of equity capital''' = risk-free rate + equity beta * (market risk premium) The '''risk-free rate''' is the return on an investment that presents (virtually) no risk of default. Government bonds are typically used as a proxy for a risk-free asset. The '''equity beta''' measures the risk of a firm relative to the market. For example, a beta above 1 indicates that the firm faces greater risk than that of the market and are more exposed to economic conditions (e.g. technology stocks, tourism-related companies). Meanwhile, a beta below 1 indicates that the firm is less risky than the market as a whole (e.g. water and energy utilities, telecommunication companies etc.). Beta can be estimated by looking at other similar companies that are listed on the stock market. The '''market risk premium''' (MRP) is the return (in excess of the risk-free rate) required by investors in the market as a whole. The cost of equity capital reflects the ''forward-looking'' rate of return required by investors, so past performance is not necessarily an indicator of future performance. However, extensive research has been undertaken into estimating historical returns. Jorda et al. (2019) estimate real rates of return for equity for a broad sample of countries, including Australia. For the post-1950 sample, the real rate of return on equity was 7.53%; for the post-1980 sample the real rate of return on equity was 8.70%. This compared to real rates of return on equity for the United States of 8.89% and 9.31% for the post-1950 and post-1980 periods, respectively.<ref>''Jorda et al. (2019). The rate of return on everything, 1870-2015, The Quarterly Journal of Economics, pp.122 5-1298''</ref> '''Dividend Valuation''' is a measurement of estimating the cost of equity capital of a firm. A firm's value, such as the wealth of the shareholders, is equivalent to the firm's future dividend payment's present value, discounted at the required rate of return of the shareholders. '''Weighted Cost of Capital''' Firms calculate their cost of capital to determine a discount rate to use when evaluating proposed capital expenditure projects. As the purpose of a capital expenditure analysis is to determine which proposed project a firm should actually undertake, it is logical that the capital whose cost is measured and compared with the expected benefits from these projects should be the next capital the firm raises. Typically, companies estimate the cost of each capital component as the cost they expect to have to pay on these funds during the coming year. Additionally, a firm normally does not specify the proportions of debt and equity financing for each individual project. Instead, each project is presumed to be financed with the same proportion of debt and equity contained in the company's target capital structure. Therefore, the appropriate cost of capital figure to be used in capital budgeting is not only based on the next capital to be raised but also weighted by the proportions of the capital component in the firm's long range target capital structure. This figure is called the weighted cost of capital. A formula for the weighted cost of capital is shown by: WACC=(E/V)*Ke+(D/V)*Kd Where E: Equity fraction of capital structure D: Debt fraction of capital structure V: Firm value where E + D = V Ke: Cost of equity capital Kd: Cost of debt capital Eg. Telstra has a target capital structure of 40% equity and 60% debt. A new project is planned to be financed with $50 million of retained earnings and $80 million of long-term debt. The cost of equity is 10% and the cost of debt is 8%. The corporate tax rate is 30%. Applying the WACC formula gives: WACC = 0.4 * 10% + 0.6 * 8% * (1-0.3) = 7.36% ==Cost-Benefit Analysis (CBA)== Cost-Benefit Analysis (CBA) is a tool used to identify, measure and compare the implications of a project from a social benefit and cost perspective. CBA can also be used to assess on-going and existing projects, and whether suggested improvements or alternatives are economically justifiable. Finally, CBA is useful as a policy intervention to guide firms and decision-makers in making more informed decisions by accounting for externalities <ref>http://www.railassociation.ir/Download/Article/Books/Benefit-Cost%20Analysis-%20Financial%20and%20Economic%20Appraisal%20using%20Spreadsheets.pdf</ref>. Shadow-pricing is often used within CBA to account for externalities which are traditionally not awarded a monetary value. By accounting for shadow pricing CBA is able to provide a greater level of detail to decision-makers which may not be accounted for in a financial analysis. Equivalent annual cost (EAC), net present value (NPV) and internal rate of return (IRR) are critical tools used in conjunction with a CBA for decision-makers to determine whether an investment is viable. === Steps in Cost-Benefit Analysis === The general principles of cost-benefit analysis may be summarised by answering the following set of questions: 1. What is the objective function to be maximised? 2. What are the constraints placed on the analysis? 3. What costs and benefits are to be included, and how may the be valued? 4. What investment evaluation criterion should be used? 5. What is the appropriate discount rate? ===Constraints in Cost-Benefit Analysis=== Although a project may seem attractive if its benefits exceed the costs, it does not necessarily mean it will be implemented. This is because of the following constraints as defined by James McGuigan in 2014 (McGuigan et al, 2013): '''1. Physical Constraints''' Physical constraints refer to the limitations imposed by the current state of technology and production inefficiencies. An example of this could be skin cancer treatment. As there is currently no cure for skin cancer, an emphasis must instead be placed on early detection and treatment. '''2. Legal Constraints''' Legal constraints refer to laws pertaining to the project under consideration. Examples of legal constraints could be the necessity of a government permit to build a hospital, or the illegality of marijuana preventing someone from farming it. '''3. Administrative Constraints''' Although a program may be conceived as extremely attractive, effective implementation of such a program requires individuals with a mix of the required knowledge and administrative skills to carry it out. Due to administrative constraints, even some of the best conceived projects may fail if the individuals involved lack the proper skills to implement it effectively. '''4. Distributional Constraints''' Distributional constraints refers to the discrepancies of gains and losses distributed to all parties involved with a project. When the distributional impacts of a project are a concern, the constraint imposed on the project will be making sure the benefits-less-costs for a particular group (usually the group who benefits the least from the project) reaches a desirable level. '''5. Political Constraints''' Political constraints are imposed when the feasibility of a project may be at risk due to the inefficiency of the political process. This can be caused by a ruling government being unfriendly to a given business or by the processes themselves within a given government. '''6. Financial or Budget Constraints''' Although a project may seem desirable if its benefits exceed costs, it may not be feasible if it is not actually affordable. Therefore the budget/financial constraint imposed requires maximising benefits of a project with a fixed budget. '''7. Social or Religious Constraints''' Different societal and religious beliefs may impose constraints which will restrict the potential feasibility of a project. An example of this would be attempting to build a residential development on land reserved for wildlife protection. ==== Cost-Effectiveness Analysis ==== Cost-effectiveness analysis is a tool used by public decision makers to assist in the allocation of resources when benefits, in monetary terms, are not simply quantified but the costs, in monetary terms, can be quantified. When governments are determining the feasibility of a potential investment such as programs that result in health, safety and well-being benefits, it is difficult to measure the benefits of a project if they are not monetarily based. In this situation, it is much more useful to use cost-effectiveness analysis to determine the expected payoff of the project. There are two main approaches to determining the feasibility of a project with non-monetary benefits: #'''''Least-Cost Approach''''' - This method focuses on achieving a certain level of output/result using the least amount of money. For example, a government may want to reduce crime by 10% and wants to figure out the best policy to achieve this. While it is difficult to determine the monetary benefit from the reduction in crime, the government can evaluate the best option by comparing the expected costs of different policies; #'''''Objective-Level Approach -''''' The other way to conduct a cost-effective analysis is to use an objective-level approach. This method compares the cost of achieving different levels of the same objective. For example, if the government wants to reduce emissions given a certain budget, it can estimate the cost of reducing different levels of emissions. This technique can illustrate the relationships (linear/exponential) between emission reduction and the cost of the endeavour. From: McGuigan, J, Moyer, C, & Harris, F 2013, Managerial Economics : Applications, Strategies and Tactics, Cengage Learning, Andover. Available from: ProQuest Ebook Central. [16 October 2019]. ==Categorisation of Costs and Benefits within a CBA== In the private sector, firms are traditionally governed by the notion that revenue must be equal to or greater than expense in the long-term for the firm to be profitable and survive, this is known as profit-and-loss accounting. Cost-benefit analysis is based on the same principle with two key differences, firstly, the analysis is implemented from the point of view of society, not a firms bottom line. Cost-benefit analysis’s ask the question whether society as a whole will benefit or not benefit, from the adoption of a specified project. Secondly, the benefits and costs measured are not as static as revenues and expenses, benefits and costs do not have to be directly quantifiable and can also be based on assumptions rather then pure facts. Because of this it is possible to evaluate benefits and costs into three forms, direct, indirect and intangible. It is also important to note that because the recommendations of cost-benefit analysis’s can be based off assumptions, sensitivity analyses of key assumptions must always accompany CBA’s.<ref>McGuigan, J., Moyer, R. and Harris, F. (2017). Managerial economics. 14th ed. Australia: Cengage learning, pp.665-666.</ref> ===Direct Costs=== Direct or primary costs are generally quite easy to measure and report. They are costs that are necessary for the project to be undertaken and are directly tied to the production of goods or services. They may include capital costs, such as land, buildings and materials incurred throughout a project, ongoing operating and maintenance costs or employee wages. For example, direct costs involved in the construction of a dam would include: materials, equipment, labourers and operators’ wages and machinery. ===Direct Benefits=== Direct or primary benefits can be more difficult to recognise then direct costs as they are not as easy to quantify. Direct benefits are the value of goods and services produced by a project compared to the benefits before the project was built. Using the example from above, if a dam was constructed and the water used for crop irrigation, the direct benefit of the project would be the value produced by the irrigated crops minus the expenses incurred producing the crops (e.g. equipment, labour, seeds). ===Indirect Costs and Benefits (Real and Pecuniary)=== Any project invariably creates indirect or secondary costs and benefits, and indirect effects may be categorized further into real or pecuniary effects. Indirect costs and benefits are effects caused by the project, which are not the express purpose of the project. Real, indirect costs and benefits are effects which, despite being indirect cause actual, quantifiable impacts that were not present before the project. For example, a real indirect benefit is the reduced amount spent in the healthcare system on supporting smoking relating to illnesses due to a campaign aimed at reducing smoking. Similarly, a real indirect cost would be damaging to an endangered species as a result of a new housing development. Pecuniary costs and benefits should not be included in a cost-benefit analysis. They are costs or benefits which arrive due to a project; however, rather than being created due to a new project they have merely been distributed from another project or location. For example, upgrading an existing highway may increase the traffic and business for shops along that highway as motorists opt to take the upgraded route. However, the increased business is only at the expense of businesses along the alternative highways. Therefore, the benefits have not been created but only distributed from another location. ===Intangible Costs and Benefits=== The last type of costs and benefits are intangibles, these are categorised as impacts resulting from a project which are close to impossible to quantify. Therefore, the cost/benefit is not represented by a single dollar value, instead intangibles are quantified by the extra increase or decrease in known quantities which is caused by incrementally changing a related tangible cost/benefit. An example of an intangible cost would be a if a company’s product caused a consumer to get sick and the consumer sued the company. Expenses the company pays, such as settling the lawsuit represent tangible costs, however the damage to the company’s reputation represents an intangible cost. <ref>McGuigan, J., Moyer, R. and Harris, F. (2017). Managerial economics. 14th ed. Australia: Cengage learning, pp.666.</ref> ==The Appropriate Rate of Discount== For comparison purposes, the costs as well as the benefits are to be discounted to a mutual time for any given program (AKA the time value of money). As many individuals desire current consumption as the measurement, rather than future consumption, discount rates are usually configured to fit that preference. The percentage rate of return that the goods being used in the project would otherwise give in the market is the correct discount rate for the evaluation of a project. The appropriate choice of discount rate for the evaluation of public investments is crucial to conclude to any type of cost-benefit analysis. A particular chosen discount rate influences whether or not the project will be accepted or rejected. For example, if a justified 5% discount rate is selected, then a higher rate of 15% may indicate a gross misallocation of resources. High discount rates favour projects who received benefits shortly after the initial investment, however, a low rate favours an investment opportunity that has a high project lifespan such as a “brick and mortar” type of business. In addition, differentiating between the public and private sector can be determined by the chosen discount rate. The appropriate discount rate will indicate which resources belong in which sector. If resources can earn 10% in the public sector, then they should not be transferred into the private sector to invest their resources. In the case of a government project, the right discount rate will reflect directly as the percentage rate of return where the resources utilized would otherwise provide in the private sector. (McGuigan, 2013) <ref>McGuigan, J. M. (2013). Managerial Economics : Applications, Strategies and Tactics, Cengage Learning, Andover. In J. M. McGuigan, Managerial Economics : Applications, Strategies and Tactics, Cengage Learning, Andover. (p. 666).</ref> While the discount rate is a useful tool, it should not be considered the be all and end all of project viability. ==Present Value of Perpetuity== Financially speaking, a perpetuity is an annuity, aka, a monetary instrument which supplies a never-ending stream of cash flows. It is calculated to find a company’s future cash flow’s present value, when discounted back to a certain rate. Although set payments are used, the value of the perpetuity can differ over time as the discount used may change. The formula used to determine the present value of a perpetuity includes dividing cash flows by some specific discount rate. An example of a perpetuity flow being used in finance is within British issued bonds (consols). When an individual purchases a bond through the British Government, they are then entitled to receive continuous (never-ending) interest payments. Though slightly illogical, there is a finite present value in countless cash flows. As the value of money changes overtime, the infinite payments made will decrease in value. '''PV of Perpetuity''' = Σδ^t x d_t = d/r for a constant dividend rate. '''where''' ''PV'' = Present value ''d'' = Dividend per period ''r'' = Discount rate ''δ'' = 1/(1+r) = Discount factor ==Discounting functions== The ‘Discount function’ is a function used in economic models to understand the different weighted values given to rewards set at different periods of time. Usually in finance, this process is most commonly modelled using exponential discounting. However, studies have shown that individuals can deviate from their initial preferences from the continuous discount rate given in exponential discounting, and therefore hyperbolic discounting can be a more appropriate model. '''Exponential discounting''' , which is a specific system of the discounting function, decreases value by a fixed given percentage over any period of time. This means that the preference of any two rewards does not change, regardless of the time leading up to obtaining these rewards. Therefore, the rate of discounting is not affected and does not change with time such as hypobolic discounting is.<ref>Gomes, O., Ferreira-Lopes, A. & Sequeira, T., 2014. Exponential discounting bias. Journal of Economics, 113(1), pp.31–57.</ref> '''Hyperbolic discounting''' on the other hand, shows the change in preference over the period of time. Originally, a larger reward with a later obtainment may be preferred over a more immediate smaller reward, however, as the smaller reward lures closer, the individual may then decide to choose the closer/smaller reward as the rate of discounting changes with time to reward. '''Time inconsistent preferences''' refers to how the different decision-making preferences of an individual become inconsistent over time as the different ‘selves’ of an individual make different decisions. For example, one day a person may be given the option to receive a $50 gift card at the end of the week, or a $100 at the end of the month. Initially, the person may pick the $100 gift card at the end of the month as they value the higher payment at that time, however, as time gets closer to the end of the week, the individual may change their mind and take the more immediate reward. Therefore, time-inconsistent preferences cannot be explained by exponential discounting, rather hyperbolic discounting can be used to understand this occurrence. == Save More Tomorrow == Recently, individuals have been placed in control of savings decisions affecting their retirement. These type of savings decisions are in reality highly complex as they involve distinct levels of self-control and forward planning. Hence, these types of decisions can be assisted through guided plans and consistency of saving. '''For example:''' In the US multiple companies have gone from a defined-benefit system to a defined-contribution system. This allows individual employees to make decisions about their retirement funds, however many people are short-sighted and do not make these contributions. ===SMarT (Thaler & Benartzi, 2004)=== Thaler and Benartzi (2004) created a program known as Save More Tomorrow (SMarT) utilising behavioural economics in order to assist people in saving for the future. The initiative was formed to address and overcome many of the following behavioural biases that individuals have a tendency to pertain: '''1. Loss Aversion''' Loss Aversion (associated with Prospect Theory) refers to an individual's irrational preferences, emphasising the high amount of psychological pain associated with losing something compared to the proportionally smaller amount of happiness received by gaining an equivalent amount - losses hurt roughly twice as much as gains yield pleasure. For example, if an individual receives a £300 bottle of wine, they will gain a small amount of happiness (utility). However, if that individual bought a £300 bottle of wine and then dropped it, they would place a heavier weight on the loss compared to the gain. Empirically speaking, it has been proven in studies that people weight losses almost twice as heavily as they do equivalent gains. Loss aversion tends to affect savings as households get used to a certain level of disposable income, and any reductions in this value are seen as a loss.<ref name=":0">Thaler, R. and Benartzi, S. (2004). Save More Tomorrow™: Using Behavioral Economics to Increase Employee Saving. ''Journal of Political Economy'', [online] 112(S1), pp.S164-S187. Available at: <nowiki>https://www-jstor-org.ezproxy.library.uq.edu.au/stable/pdf/10.1086/380085.pdf?refreqid=excelsior%3A05c30c9ada6ed9fd74986dd0adfcc4bc</nowiki>.</ref> '''2. Hyperbolic Discounting''' Hyperbolic Discounting is defined as people's myopic preferences to place a higher value on short term gains compared to larger rewards received at a later date. This is a widely accepted irrational preference that is taken into account in all forms of future Cost-Benefit Analysis. Discount Factors (or Discount Rates) are applied to future gains in order to estimate their reduced present value, with a 7% per annum rate being the standard normally applied <ref>Harrison, M, 2010, Valuing the future: the social discount rate in cost-benefit analysis, Visiting Researcher Paper, Productivity Commission, Canberra</ref>. For example, consumers may have a preference of receiving 40 dollars today compared to 50 dollars next week. '''3. Procrastination''' Defined as the tendency to put off and delay something that needs to be completed, usually unpleasant tasks. From a behavioural economics standpoint, it can be viewed as a result of irrational consumers time-inconsistent preferences, a notion closely associated with both loss aversion and hyperbolic discounting. Procrastination can be considered a direct result of an individuals tendency to place greater weight on current satisfaction than on future satisfaction. The most common examples experienced include quitting smoking, eating healthier, beginning an exercise regime or saving more money for the future. As all of these examples display higher future gains than what present satisfaction can account for. Individuals procrastinate because they (falsely) think that whatever they will be doing later will not be as important as what they are doing now. The more naive the individual, the more pronounced the tendency to procrastinate<ref name=":0" />. There is considerable evidence showing that people display time inconsistent behaviour specifically weighing current and near term consumption especially heavily. '''4. Inertia''' Procrastination produces a strong tendency toward inertia. Inertia is defined as ones tendency to remain unchanged or to do nothing, and is closely associated with maintaining the status-quo bias. For example, when a decision cannot be made and you are left within inaction. '''5. Status-quo Bias''' The tendency to favour decisions that maintain the status quo (i.e., the existing state of affairs). Those affected by this bias choose not to divert from established behaviours unless there is compelling incentive to change. For example, people often leave their money in low-yield savings accounts. This status quo bias leads people to maintain their financial situation as it currently is, rather than taking a the initiative to improve their financial outlook. Some firms may promote free for trial is another example taking advantage of this bias. ====Four Main Processes of the SMarT Program==== '''1. Approaching Employees''' - A considerable time prior to scheduled pay increases, employees were approached about increasing their savings contribution rates. Separation of dates between schedules pay increases and approaching employees was used to account for hyperbolic discounting. Where the gap between should be as long as feasible. '''2. Increase in Contribution''' - For employees who decided to join, their contribution according to the plan was increased at the beginning of their fist pay check after a pay rise was done. This mitigates the perceived loss aversion of a cut in take home pay.<ref name=":0" /> '''3. Continued increase in Contribution''' - The contribution rates then would continue to be raised according to a scheduled raise put in place. This would occur until the contribution reached a pre-set maximum. Here, status quo bias and inertia work toward keeping people in the SMarT plan '''4. Opt Out''' - During the implementation of the initiative employees were told that they were allowed to opt out of the contribution schedules at any time. This was also to help insecurities about joining the program. ===SMarT Findings=== Overall, the study found that individuals who implemented the SMarT program were able to increase savings the most, over time. This was opposed to other individuals who followed financial consultants advice or followed their own intuitive saving methods. Individuals who did not acknowledge the complex behavioural biases in making such savings decisions ultimately recorded lower contributions as time progressed. It is distinct that a prescriptive savings program aids individual in making savings to their retirement fund. Helping individuals to consistently save and decrease the behavioural biases that cloud peoples thinking. ==Type of Research== {| class="wikitable" |- ! Comparison !! Basic Research !! Applied Research |- | Meaning || Refers to the study that is aimed at expending the existing base of knowledge || Designed to solve the specific and practical problems or answer certain questions |- | Nature || Theoretical || Practical |- | Application || General || Limited |- | Concerns || Developing existing knowledge and predictions || Developing technique skills |- | Goal || To add some knowledge to the existing study || To find out solutions for the current problems |- |} === Basic Research === Basic research is designed to drive curiosity and public interest via a scientific question<ref>http://www.sjsu.edu/people/fred.prochaska/courses/ScWk170/s0/Basic-vs.-Applied-Research.pdf</ref>. Conventionally conducted within universities the basic research aims to expand knowledge and understanding. An example might be examining how coffee/caffeine consumption impacts students' study efforts. Discoveries founded from pure basic research were not intended to hold commercial value. Researchers believe all branches of research depend on some level of basic research to drive knowledge and understanding. In principle, basic research is essential in assisting the development of applied research. To summarise, basic research is not applicable in the real world. The philosophy used within basic research is predominately theoretical in nature. It uses advanced scientific knowledge of a natural phenomenon. The benefit provided by using a basic research approach is increased flexibility to control over variables. However, results may be subject to biases and human error. === Applied Research === Applied research is used to solve practical problems via developing new products and processes. Applications of applied research vary greatly and include business optimisations and improving the human condition<ref>https://www.ucentralasia.org/Content/Downloads/UCA-IPPA-OP4-Beginners%20Guide%20for%20Applied%20Research%20Process-Eng.pdf</ref>. A motivation for firms to conduct applied research is the possibility to profit from product developments and patents. It is used to answer any particular and specific questions so as to solve any problems that firms/ society/ business organisations may have. This is possible as it has real world application and is present in most industries, for example, the medical industry. There are a few advantages of applied research; these include resolving issues that arise in our everyday lives, developing innovative technology that can possibly reduce business costs, and even curing illnesses when used in the medical industry. Some disadvantages of applied research also include results being only limited to the problem that was researched on. It can also be rather inflexible with tight deadlines to get results causing inefficiencies and a decrease in quality of work produced. ==Reasons to Invest in Research== Investing in research and development (R&D) is one of the many strategic decisions made by firms that consume current resources to generate future returns. An obvious posited difference in risk-taking is evident from who’s perspective is considered. Shareholders, i.e. Owners, implore participation in projects with a positive NPV. Whereas CEO’s and managers ration projects from their current appetite for risk. Hence management decisions might conform to the majority and sacrifice profitable projects (Rashad, 2014). <ref>https://onlinelibrary-wiley-com.ezproxy.library.uq.edu.au/doi/full/10.1111/abac.12029</ref> From a holistic standpoint, firms seek to ensure their survival in a continuously changing market. Research is required to retain or increase their current market share and profit margins. In the long run, it is imperative for firms to invest in research to maintain a competitive edge. Research can allow for innovations to occur, which can give firms an advantage should they choose not to share their findings through patents, copyright protection, trade secrets, etc. The patentability of an innovation is not guaranteed, however, as it depends on its subject matter, utility, novelty and non-obviousness. It is worth highlighting, that another crucial reason for investing in research, is to for companies to be more aware of their competitive landscape. In this way, organizations will be able to optimize their decision making. As a result, they can be more efficient on their internal operations and also be one step ahead from its competitors. ==Why Invest in Research== ===For Firms=== '''1. Increase future revenue''' R&D efforts may result in patents, trademarks and copyrights which can be used to create new or improved products and help the business achieve long-term profitability. To ensure profitability, the firm must first undertake a cost-benefit analysis to find the net present value of any research endeavour, if the net present value is positive, firms can expect increased future profits. Many long-term investors will seek out companies with aggressive R&D efforts, as this can be an indicator of increased revenue in the future. '''2. Ensure future survival''' Through R&D, firms are able to gain a competitive advantage by developing competencies that its competitors are unable to easily replicate. These competitive advantages help ensure the firm can fight off attackers and maintain market share in their industry. Examples of R&D based competitive advantages are innovations which provide increased productivity or the development of a new product. '''3. Increase market Share''' New and innovative products created through R&D and protected by patents or copyrights can create a competitive edge which firms are able to exploit in order to increase their market share. This is beneficial for firms as increased market share can build their market power, build a strong consumer base and help create barriers to entry for new competitors. '''4. Buyouts''' Investing in research and development can allow a firm to come across a good innovation or idea which then either be monetized, or in the case of many smaller companies which don't have the capital to actually proceed with monetizing their research findings. The good idea can be sold to companies with this ability for large sums of money. This is especially common with companies finding new software features or easily licenseable products. '''General reasons''' When their is incentive to innovate, and usually this incentive is monetary related, firms will always be participating in research and development. Potentially designing and creating new products in any market or optimizing and enhancing previous technology or processes used within a firm all have extreme monetary value and always have a positive effect on a firm's future. ===For Universities=== There is no exact answer for this question. However, the following are common factors considered by most universities:<br> 1. Reputation of the university (market share)<br> 2. Future student admission (survival)<br> 3. Having adequate academics for innovation as well as understanding innovations done by other scholars from other universities. (Ability for future development) 4. Increase in knowledge and ideas that lift innovation and productivity in the firms 5. Increase the knowledge and idea generation of university graduates 6. Create proximal benefits with ideas and knowledge that spill over due to a mix of formal and informal interactions between firms and universities 7. Ensure growth of direct commercial activity, increasing income generated directly by universities 8. Generate partnership interest from companies that partake in similar research and development directions. (Gain internship or long-term reputation surplus) 9. Attracting more research grants. For instance, UQ ranks at the top in the research of the mining industry. A high reputation helps UQ to attract more research funds for the mining industry from public or private companies. ==Innovation== Innovation describes firms producing novel products or services that give them an edge on competitors. For example, Apple, a highly innovative firm, developing the iPhone, which completely revolutionized what a phone could be. Information goods are non-rival and, in-principal, nonexcludable. An innovative company or innovative entrepreneur may generate higher profits and return. ===Reasons to Encourage Innovation=== Innovation gives firms an edge on competitors and ensures products and services remain unique. Consistent innovation is an efficient way to prevent competitors from copying designs and ideas and draws the consumers attention to market leaders. Reasons for innovation include:<br> 1. Patents: Awards monopoly rights for 17-20 years(ie:medicine) * Allows publication of research findings that lead to further innovations. * Patent rights(for certain applications) can be transferred. 2. Copyright protection: Pertains to books, brand names, and the media. 3. Companies also have their own trade secrets that give them an edge for future innovation. This information needs to be kept secret to prevent other companies negating the advantage, this could be recipes or formulas. An example of such would be: * plant breeders'right: it is the right granted to breeders who are developing new plants. It allows exclusive sales of varieties and allows farmers to reuse seeds. 4. Prizes: may be awarded to innovators who find a technical solution to a problem. '''Advantages of innovation''' First-mover advantage: it is the profit gain by first opening a market niche or to explore new technology.It can gain sustainable competitive advantage by establishing themselves before other competitors of the industry enter to the market. 1. learning curve: firm may gain advantage in selling more in early periods than its competitors and achieve lower costs than its competitors. 2.reputation: firms with innovation may gain advantage in raising reputation and competitiveness, which can further strengthen market position. It has potential to leave a lasting impression on the customer, which lead to brand recognition and brand loyalty. ===Patentability=== A patent is a form of intellectual property granted by a government authority or licence conferring a right or title for a set period of time that gives its owner the legal right to exclude others from selling, making, importing or using an invention An innovation is patentable if it meets the relevant legal conditions to be granted a patent. However, not all innovations are patentable: '''The threshold requirement for patents''' '''1. Patentable subject matter''' It is important to know if your invention is patentable or not. For example: ''Patentable'' * New products like toys, tools, medical devices, pharmaceutical drugs * New process, like a manufacturing process or an industrial method * Composition of matter, a compound created through two or more elements * Improvement, requires novel or improved element in a known invention ''None-patentable'' * Non-functioning products * Mathematical algorithms or models * Informative presentations, such as plans or schemes for performing mental task * Immoral inventions and software/business methods that are not technical thus non patentable * Principles or theories '''2. Utility, usefullness''' In this case, not only is it necessary to demonstrate that the subject matter of the invention is patentable, but it is also required to demonstrate that the claimed invention is “useful” for some purpose. '''3. Novelty, New''' You cannot patent something that is already publicly known, as it would be unfair to confer the economic benefits of a patent for the discovery of something that is already publicly known. '''4. Non-obviousness''' If the invention is 'obvious' to a skilled person, it is not patentable. EU patent law: The following shall not be regarded as inventions: # discoveries, scientific theories and mathematical methods;(ie.mrth solutions) # aesthetic creations;(ie.beauty, taste or preferences that involve subjective judgement) # schemes, rules and methods for performing mental acts, playing # games or doing business, and programs for computers; # presentations of information. ===R&D Spillovers=== * Since not all innovations are patentable, or worth patenting, industries that are agglomerated have high R&D Spillovers mainly via personnel movements between firms. * Firms often use non-disclosure agreements or non-compete clauses in employment contracts to reduce such spillovers. - Leads to monopsony power in labour markets <br> === Shortcomings === Enforceability: Even if a confidentiality agreement is signed, the company cannot ensure that the information will not be disclosed. ===Importance of market power=== Innovation can drive market power and increase competitive intensity. competitive intensity - refers to a company's potential profitability. A higher competitive intensity indicates that the company is more competitive and able to transfer more values. It is presented as an inverted U shape with the amount of innovation.<br> <br> As Sharp and Currie (2008) state that, more and more researches indicate that open and competitive as drivers of innovation and productivity is overwhelming. Innovation is used to increase with growth in competition up to a point and then decline. <ref name="csls"> {{cite web | author = Andrew Sharpe, Ian Currie | title = COMPETITIVE INTENSITY AS DRIVER OF INNOVATION AND PRODUCTIVITY GROWTH | date = 2008-03-15 | url = http://www.csls.ca/reports/csls2008-3.pdf | accessdate = 22 October 2019 }} </ref> ===Innovation in Entrepreneurship=== In William's research has given some insight of innovation in entrepreneurship(innovative entrepreneur). It represents truly new ideas and explore a new market niche. William has stated that innovative company can be regarded as "entering the white space", which means creating new products to new customers. It is the most risky investment as company need to adjust supply accordingly to sustain(transformation). If the company invest in innovation and succeed, a high return and profit can be secured. When the entrepreneurs are creative in their business ideas and making innovation, their business are succeed. Under the global environment, innovation and creativity acts as a platform for the organisations to compete and sustain in the marketplace. （Rangarajan & Lakshmi, 2013) Enterprise innovations include: Thinking innovation Some companies are constantly recruiting new talents. One of the important reasons is to expect them to bring new ideas, new ideas and continuous innovation. Its purpose is to carry out innovative training in thinking. 2. Product (service) innovation For industrial enterprises, it is product innovation. For finance, the main service innovation. This can better attract customers. 3. Technological innovation As far as a company is concerned, technological innovation refers not only to the commercial application of independent innovation technology, but also to the innovative application of legally acquired new technologies developed by others or technologies that have entered the public domain to create market advantages. Focusing on tasks and technologies, tasks are re-allocated, updating equipment and technological innovations to achieve organizational innovation. 4. Organization and institutional innovation Typical organizational change and innovation is to enable employees to recognize and realize organizational change and innovation through employee attitudes, values ​​and information exchange. Through corporate organizational change and innovation, people's behavioral style, values, and proficiency can be changed. And at the same time, managers can be changed. == Should policymakers encourage entrepreneurship? == Whether policymakers encourage entrepreneurship is a difficult and complex topic to discuss. There have been numerous life-changing inventions from entrepreneurs, such as Bill Gates and Mark Zuckerberg that have a daily impact on many around the world. In practice, calculating the benefit gained from such entrepreneurs cannot be well-defined and makes it difficult to quantify how much extra effort policymakers should put towards encouraging entrepreneurship. The issue is a contentious area, as many believe enough incentive already exists. Firstly, the pros for incentivising is that entrepreneurship is important for the countries competitiveness, where the benefits could spill over to other industries. In the Australian retail industry, increased competition has lowered net profit margins causing firms to become more efficient by reducing operating cost and ultimately offer consumers lower prices to retain sales<ref>https://www.rba.gov.au/publications/bulletin/2019/jun/competition-and-profit-margins-in-the-retail-trade-sector.html</ref>. An example of the positive spillover effect can be seen by the recent growth in online networking tools such as LinkedIn that create a positive feedback loop where users share knowledge and create a large exchange of ideas from various individuals. This also has the potential to cause a chain reaction with the creation of more industries such as software wanted for the platform or the utilisation of user data. Additionally, if governments were to encourage it, it would allow lower barriers to entry as companies are paid to enter the market, which also increases the competition. However, the main challenge that surrounds entrepreneurship is the riskiness of the practice. The likelihood of a successful invention is less than 1%. This has allowed a dissolution of transparency where we don’t have a sufficient understanding of the true costs as they are all privately born. The public is only exposed to success stories, so any failures are generally unaccounted for. This has created a strong selection bias that highlights only those successful. R&D type industries have far greater transparency in terms of what’s being recorded and tax revenues etc. To answer whether policymakers should encourage entrepreneurship is not straightforward. The trade-off for policymakers is the risk and benefit of generating a successful start-up, the cost of which is not easily identifiable. Other factors that remain influential involve the current competitive environment and whether more competition will enhance or degrade its current economic situation. '''Acquisition as a Form of Investment''' '''What is an acquisition?''' In its most simplistic form, the Oxford dictionary defines an acquisition as “an asset or object bought or obtained” (Oxford Dictionary, 2013). Expanding upon this in an economic context, an acquisition is an agreement in which one company gains control over another, usually through the purchase of shares or assets. There are several types of acquisitions, not all of which are of mutual agreement. The purpose of an acquisition, or better put, the incentive to acquire, differs depending on the type of acquisition. '''Types of Acquisitions''' * ''Horizontal Acquisition:'' A scenario where one company acquires another with similar products or services. The acquiring firm does so to expand their range of products or services without taking on the costs of expanding themselves. The incentive to acquire lies in the synergies between the two companies operations, industry and production stage. The newly combined firm would expect to have gained considerable competitive position due to an increase in market share or scalability, which were only achieved due to the acquisition. * ''Vertical Acquisition'': A scenario where one company acquires another acquires another whom operates in the same industry, but occupies a different position in the supply chain. The incentive to acquire lies in synergies derived from efficiency of production and supply, as well as low cost volatility. The newly combined firm would expect to gain competitive advantage through increased control of the distribution channels (Buzzell, 1993). * ''Conglomerate Acquisition'': A scenario where a company acquires another despite both being involved in unrelated industries. The incentive to acquire lies in the opportunity to reduce capital and overhead costs whilst achieving new efficiencies. Typically, the acquiring firm is seeking to expand into new industries, eliminate redundant activities and/or attain production or distribution synergies (Herger & McCorriston, 2016). * ''Concentric Acquisition'': A scenario where a company acquires another whom they share the same industry but do not directly compete. Typically, concentric acquisitions involve firms who share similar distribution channels or have overlapping systems of production or technology. The incentive to acquire lies in the opportunity to expand product range or boost efficiencies through production or distribution synthesis (Kenton, 2018). '''EVALUATING ACQUISITION SUCCESS''' There are a number methods used to measure the success of an acquisition. Some of the more basic methods include a measure of market share increase versus cost, as well as the gross increase in profit, whilst the more complex methods involve calculating cumulative abnormal returns (CAR). Each method is used for a different purpose, and involves a degree of subjectivity. In an economic context, however, it is most appropriate to measure acquisition success by the total economic impact of the transaction. That is, to measure the impact of the acquisition on both the acquirer and acquiree, rather than the more traditional method of measuring the impact on shareholders of the former. Consider the following formula: [XA – X0] + [YA + Y0] Where, XA = Shareholders value of acquirer after the acquisition YA = Shareholders value of acquiree after the acquisition X0 = Shareholders value of acquirer before the acquisition Y0 = Shareholders value of acquirer before the acquisition In its most basic form, this methodology measures the total economic impact of the acquisition. It is, though, not without assumption. Kaplan (2006) theorises that this formula holds three assumptions. # Firstly, the market must be well informed about the value of the companies before the acquisition; # Secondly, there is suitable information available at the announcement of the acquisition; and # Thirdly, the acquisition is unanticipated. He goes on to consider how this formula could be better arranged to consider the synergies of the acquisition, though acknowledges that, as a foundational economic principle, the formula in its current state is useful. '''Practical Example''' Consider two companies, X and Y, which are worth $10 million each. If X acquires Y, X will gain $1 million in synergies. Based on this, X acquires Y for $15 million. Upon announcement of the acquisition, Y’s value increased by $5 million (from $10 million to $ 15 million) based on X’s decision. Note simplistic models for measuring acquisition success end here. However, X is paying $15 million for $11 million value ($10 million company Y + $1 million synergies), and therefore loses $4 million of its own value. Depending on which perspective you take, you might consider this to be either a good or bad deal. From a combined perspective, though, the formula above dictates: [XA – X0] + [YA + Y0] = [$6mil - $10mil] + [$15mil - $10mil] = -$4mil + $5mil = $1mil Thus, the deals overall economic impact is positive. ==Summary== This section briefly summarises each section above: '''Capital budgeting:''' Used as a process to assess capital expenditure decisions. The process typically involves evaluating the cash flow of several projects before one is chosen based upon a decision rule. Finally, the assumptions of the project are re-evaluated after the project begins to find any potential improvements for future projects. '''Investment Project Evaluation:''' Analysis of the profitability of a project typically using net present value or internal rate of return. '''Net Present Value (NPV):''' Defined as the present value of cash inflows minus the present value of cash outflows over a certain period of time. '''Internal Rate of Return (IRR):''' Used in calculating the profitability of an investment and is a discount rate that sets the net present value of an investment equal to zero. '''Cost-Benefit Analysis (CBA):''' Process used to examine all the associated costs and benefits with a project before undertaking it and takes in account various perspectives of the project. '''Present Value of Perpetuity:''' A never-ending sequence of cash flows that is discounted back to a present value. '''Investing into Research and Development:''' Firms usually invest in to gain a competitive advantage that can be used for additional revenue and market share over competitors. Universities typically use research to create a better reputation and more attractive for students and academics. == Reference/s == <ref> Rangarajan, R. & Lakshmi, R. 2013, "Creativity and Innovation in Entrepreneurship - A Brief Assessment", Sumedha Journal of Management, vol. 2, no. 4, pp. 55-60. </ref> [[Category:Innovation]] [[Category:Investment]] 2fojg4fpp8sfa9rhg3j15ymqdeuo7ec 0 261167 2842430 2842347 2026-10-24T12:06:16Z KyleWii 3113193 2842430 wikitext text/x-wiki {{:{{PAGENAME}}/Sidebar}} {{AP course revamp notice}} Advanced Placement examinations (AP exams) are exams offered in United States by the College Board and are taken each May by students. The tests are the culmination of year-long Advanced Placement (AP) courses. AP exams (with few exceptions) have a multiple-choice section and a free-response section.<ref>[[Wikipedia: Advanced Placement exams]]</ref> The AP Biology exam studies the core scientific principles, theories, and processes that govern living organisms and biological systems, including hands-on laboratory work to investigate natural phenomena.<ref>[https://apstudents.collegeboard.org/courses/ap-biology College Board: AP Biology]</ref> == New Outline == # Chemistry of Life ## [[/Chemistry of Life/Structure of Water and Hydrogen Bonding/|Structure of Water and Hydrogen Bonding]] ## Elements of Life ## Introduction to Biological Macromolecules ## Carbohydrates ## Lipids ## Nucleic Acids ## Proteins # Cell Structure and Function ## Cell Structure: Subcellular Components ## Cell Structure and Function ## Cell Size ## Plasma Membranes ## Membrane Permeability ## Membrane Transport ## Facilitated Diffusion ## Tonicity and Osmoregulation ## Mechanisms of Transport ## Cell Compartmentalization ## Origins of Cell Compartmentalization To be added. == (OLD!) Outline == This is necessary for easy access to the old notes, this will be deleted once its fully revamped. * [[/Chemistry of Life/]] * [[/Cell Structure and Function/]] * [[/Cellular Energetics/]] * [[/Cell Communication and Cell Cycle/]] * [[/Heredity/]] * [[/Gene Expression and Regulation/]] * [[/Natural Selection/]] * [[/Ecology/]] == See Also == * [[Wikipedia: Advanced Placement exams]] == Bibliography == * [https://openstax.org/details/books/biology-ap-courses OpenStax: Biology for AP Courses] == References == {{Reflist}} {{CourseCat}} px7hrb04qzfykrkmwn2f8lu5eo8i0yr 2842431 2842430 2026-10-24T12:08:28Z KyleWii 3113193 2842431 wikitext text/x-wiki {{:{{PAGENAME}}/Sidebar}} {{AP course revamp notice}} Advanced Placement examinations (AP exams) are exams offered in United States by the College Board and are taken each May by students. The tests are the culmination of year-long Advanced Placement (AP) courses. AP exams (with few exceptions) have a multiple-choice section and a free-response section.<ref>[[Wikipedia: Advanced Placement exams]]</ref> The AP Biology exam studies the core scientific principles, theories, and processes that govern living organisms and biological systems, including hands-on laboratory work to investigate natural phenomena.<ref>[https://apstudents.collegeboard.org/courses/ap-biology College Board: AP Biology]</ref> == New Outline == # [[/Chemistry of Life/]] ## [[/Chemistry of Life/Structure of Water and Hydrogen Bonding/|Structure of Water and Hydrogen Bonding]] ## Elements of Life ## Introduction to Biological Macromolecules ## Carbohydrates ## Lipids ## Nucleic Acids ## Proteins # Cell Structure and Function ## Cell Structure: Subcellular Components ## Cell Structure and Function ## Cell Size ## Plasma Membranes ## Membrane Permeability ## Membrane Transport ## Facilitated Diffusion ## Tonicity and Osmoregulation ## Mechanisms of Transport ## Cell Compartmentalization ## Origins of Cell Compartmentalization To be added. == (OLD!) Outline == This is necessary for easy access to the old notes, this will be deleted once its fully revamped. * [[/Chemistry of Life/]] * [[/Cell Structure and Function/]] * [[/Cellular Energetics/]] * [[/Cell Communication and Cell Cycle/]] * [[/Heredity/]] * [[/Gene Expression and Regulation/]] * [[/Natural Selection/]] * [[/Ecology/]] == See Also == * [[Wikipedia: Advanced Placement exams]] == Bibliography == * [https://openstax.org/details/books/biology-ap-courses OpenStax: Biology for AP Courses] == References == {{Reflist}} {{CourseCat}} lqzzmi9fso38p6t9i2ck3t3yml75roj 0 261664 2842432 2811114 2026-10-24T12:14:22Z KyleWii 3113193 Replaced content with "{{:{{BASEPAGENAME}}/Sidebar}} Chemistry of Life covers the foundational chemical and physical principles that make all biological processes possible, it makes up rougly 8% to 11% of the AP exam score. == Topics in this Unit == * Properties of Water ** Polarity and Hydrogen Bonding ** Critical Properties ** Biological Role * Elements of Life ** CHNOPS Elements ** Carbon Versatility ** Funcitonal Groups * Polymerization ** Dehydration Synthesis ** Hydrolysis * The Fo..." 2842432 wikitext text/x-wiki {{:{{BASEPAGENAME}}/Sidebar}} Chemistry of Life covers the foundational chemical and physical principles that make all biological processes possible, it makes up rougly 8% to 11% of the AP exam score. == Topics in this Unit == * Properties of Water ** Polarity and Hydrogen Bonding ** Critical Properties ** Biological Role * Elements of Life ** CHNOPS Elements ** Carbon Versatility ** Funcitonal Groups * Polymerization ** Dehydration Synthesis ** Hydrolysis * The Four Classes of Biological Macromolecueles ** Carbohydrates ** Lipids ** Proteins ** Nucleic Acids == Subunits == {{subpage navbar}} {{CourseCat}} khy2ic3rppqdo840xtqoe6rg6i9mvje 2842433 2842432 2026-10-24T12:15:17Z KyleWii 3113193 2842433 wikitext text/x-wiki {{:{{BASEPAGENAME}}/Sidebar}} Chemistry of Life covers the foundational chemical and physical principles that make all biological processes possible, it makes up rougly 8% to 11% of the AP exam score. == Topics in this Unit == * Properties of Water ** Polarity and Hydrogen Bonding ** Critical Properties ** Biological Role * Elements of Life ** CHNOPS Elements ** Carbon Versatility ** Funcitonal Groups * Polymerization ** Dehydration Synthesis ** Hydrolysis * The Four Classes of Biological Macromolecueles ** Carbohydrates ** Lipids ** Proteins ** Nucleic Acids == Subunits == * [[/Structure of Water and Hydrogen Bonding/]] {{subpage navbar}} {{CourseCat}} 6et3q9ererj4ef9pku9js7onrznzu40 0 274143 2842527 2796473 2026-10-25T06:18:15Z Elominius 2911372 Added expression to escape apostrophe characters in file names. 2842527 wikitext text/x-wiki This lesson teaches the management of digital files and directories. While there exists no ideal method of file management, this resource documents possibly helpful practices and inspiring ideas. This lesson was created under the impression that the average computer and mobile phone user still struggles to keep track of files in the long term despite of all the tools at their disposal. == Descriptive file names == Every file manager<!-- One can safely assume that just about every file manager has this feature. If a file manager without a renaming feature happens to exist, feel free to change it, but it is very unlikely. --> has a "rename" feature, letting you change the names of files and folders. Don't just let this essential feature "rust in the junk yard". Utilize it to make files easily searchable. For example, add a short description after a picture or video file name. If you recorded using a [[smartphone]], you can rename the file on the device itself right afterwards. For example, * <code>VID_{{#time:Ymd_His}} trampoline jumping.mp4</code> It is recommended to add the short description after the file name rather than before, to prevent interference with alphanumerical sorting. Even though file managers can still sort items by date and time, files' time stamp attribute might not always be retained throughout transfers, for example when transferring files through [[:w:Media Transfer Protocol|MTP]], between a smartphone's internal memory and memory card, or uploading files to a cloud storage service. A time stamp at the beginning of the file name allows alphanumerical sorting to act as chronological sorting even if the date and time attribute is lost. Not every single file needs to be given a name manually, since that would take much effort. If your file manager lacks range selection and bulk renaming, meaning the selection and renaming of many items at once, adding a descriptive file name to the first file of similar files will still facilitate finding those files, since the title of the first file in the group covers the remaining files. For example, all four of these imaginary files depict strawberries, but only the first was manually named: * <code>IMG_20230914_090112 strawberries.jpg</code> * <code>IMG_20230914_090116.jpg</code> * <code>IMG_20230914_090119.jpg</code> * <code>IMG_20230914_090124.jpg</code> == Directories == Make use of directories. Categorize files into logically structured directories to facilitate finding them later. Some inexperienced computer users might have a habit of just throwing files into the root directory of their device such as a flash drive. With an increasing count of accumulating files, they would become increasingly difficult to find, especially multimedia content which can not simply be searched for text strings using tools like `grep` but only through metadata. Create a mind map by asking yourself where you would look for these files whenever necessary. Below are two example directory structures for video projects, where the project files of the first project are directly in the project folder, while those of the second are in a subfolder; a matter of preference. <blockquote> * Video projects ** Example project 1 *** Assets ** Example project 2 *** Project files *** Assets </blockquote> Structure files in the order your mind recalls them. For example, the type and source of files is more memorable than the date. For more examples of directory structures, see the [[/directory structures]] sub page. Feel free to add examples to it. For directories where a sufficiently descriptive directory name would be considered too long, consider creating a descriptive text file inside named <code>description.txt</code>, <code>info.txt</code>, or similar. Comments about specific files may be noted in a text file with <code>.meta</code>, <code>.meta.txt</code>, or a similar suffix appended to its name. If files within the same category have wildly varying sizes, such as command line outputs, you might want to move files beyond a size threshold such as 1 MB or 5 MB into a separate folder whose name can be the same with an added "-large" suffix. This allows minimizing the size of compressed snapshots of the directory. Avoid putting sub-directories in directories populated with many files, since they can become a nuisance when wanting to view the lastest file whereas the file manager lists sub-directories at the top, or otherwise be difficult to find between the files when needed. Due to the slow loading times of the [[:w:Media Transfer Protocol|Media Transfer Protocol (MTP)]] which is commonly used to connect mobile devices to desktop/laptop computers, and due to the possibility of file managers trying to determine file types by reading internal file information,<ref>''[https://github.com/linuxmint/nemo/issues/1907 nemo significantly slower opening folder content #1907 - GitHub]'' ([[:w:Nemo (file manager)|Nemo file manager]] reads the [[:w:Header_(computing)|header]] of each file with unknown [[:w:Filename extension|extension]]</ref> it could make navigating more difficult. Therefore, it should be considered to create a separate folder that only contains subfolders, such as <code>Downloads_2</code>. You don't need to spend much effort coming up with good names for files and folders, as you can change them later at any time if you happen to come up with a better idea. == Revision history == When spending much time on a project, write-up, email draft, etc., consider saving a revision history, by selecting "Save as" (also accessible through <kbd>Ctrl</kbd>+<kbd>↑Shift</kbd>+<kbd>S</kbd> keyboard shortcut on some software) and changing the file name through numbering or timestamping, or alternatively creating renamed copies externally using a file manager or command terminal. For possible file naming schemes and variations of numbering and time-stamping, see [[File naming]]. This enables you to revert to an earlier revsion in case of error, prevents total loss in case of failed writes caused by a power outage or software crashing, and later facilitates comprehending the work progression, while only consuming marginial space compared to common data storage, and being efficiently [[Backup#Compressed_archives|compressible]] due to redundancy. A new revision does not necessarily have to be created upon every saving, but whenever the changes since the last revision are major enough at your discretion. Optionally, a short comment summarizing the changes can be added to the file name. For programming code, suffixes like <code>-stable</code> and <code>-unstable</code> may be added after the time stamp, for example: <code>UserScriptName-revision-{{#time:YmdHis}}-stable.js</code>. Separating the number or time stamp allows convenient double-click selection in the file saving dialogue. Revisions can optionally routinely be moved into a separate subfolder. == Dumping ground == For files and folders you are unsure where to put, consider creating a directory on your device named <code>dump</code>, <code>sandbox</code>, or similar. You may wish to categorize those into text, compressed archives, drawings, or by whichever task, or dated [[file naming|folder names]] such as <code>{{#time:Y-m-d}}</code>. When managing files and directories in a command line terminal, an item can temporarily be given a simple and short name such as <code>1</code> to facilitate typing in the commands, that will be changed shortly after. Examples are: ; Output from multiple commands into one log file: <syntaxhighlight lang=sh> $ ls [path] -alR >>1 # Write names and attributes of that folder's content into the file "1". $ find [path] >>1 # Write a list of bare file paths into the file "1". $ mv -n 1 [desired file name] # Rename "1" to desired file name. "-n" prevents accidental overwriting. </syntaxhighlight> The above command could also be done in a single line with <code>(ls ''[path]'' -alR; find ''[path]'') >>''[desired file name]''</code>, but the former might be preferred to save time if the target directory contains a high number of files, as the former command starts immediately after pressing <kbd>⏎ Return</kbd> (also known as "Enter"), or when one wishes to enter commands before thinking of an output file name, while the latter requires typing in the whole command first. ; Moving files from multiple sources into a directory: <syntaxhighlight lang=sh> $ mkdir 1 # Create a folder named "1". $ mv -n *.mp4 *.mkv 1 # Move mp4 and mkv files into it. $ mv -n 1 Videos-$(date +%Y-%m-%d) # Renaming folder to intended name. </syntaxhighlight> Should it be necessary to print out the absolute (full) file path of a file for, for example, quick copying to the clipboard, use the command <code>readlink -f ''[target file]''</code>, and for information about the current mount point, use the <code>findmnt -T .</code> command. == Finished tasks == Files whose task has finished, such as a posted message that was initially drafted offline into a text file, can be moved into a subfolder of the current directory named <code>done</code>, or alternatively in one big shared folder for this purpose. == Storage types == Auxiliary data which is frequently accessed can reside on the operating system drive, on its own partition or a separate one. This especially applies to portable computers and operating systems installed on external media such as a USB flash drive. === Secondary and external storage === On desktop computers, large files can reside on a secondary large hard disk drive or solid state drive, the former of which costs less per space. For laptops, stationary hard drives located at home or portable external hard drives or solid state drives can be used. While working with a laptop as passenger in any moving vehicle such as a bus or tram, solid state memory is preferrable due to sturdiness, as hard disk drives do not like constant physical movement due to mechanical wear. If portable storage necessary, files can be stored on a constantly inserted memory card, which does not compromise ergonomy as they don't physically protrude, or at most a little. In addition, it can be occasionally removed when the data is needed on a different device. SD cards with 1 TB exist since at least 2017, though are expensive.<ref>[https://www.engadget.com/2016-09-20-sandisk-1tb-sd-card.html ''SanDisk outs the 'world's first' 1TB SD card''] (2016-09-20)</ref> A home server adds the convenient benefit of access from all devices at home, and even through the internet if set up by the user ("private cloud"), but typically has longer latency times (access delays) than physically attached storage, and lacks mass storage access that may be necessary for some programs to work properly. In such a case, the files would have to be downloaded first and worked with locally, and uploaded after finishing. === Incubate work on flash storage === If, for example, your computer has a setup with solid state drive for operating system and hard disk as expansion storage, a project may be worked on on the flash storage, and changes can be applied to the hard drive at the end of a day. If the hard drive is set to spin with no or a long timeout such as an hour or more, this may not be as necessary, but for short timeouts, frequent spin-ups would cause mechanical wear and tear in addition to annoying delays. Using flash memory can be particularly useful on battery-powered laptops for power efficiency, though mechanical hard drives are increasingly being usurped by solid state memory in laptops, mainly due to physical robustness, as hard drives on laptops have long served for cost saving, whereas solid state memory is becoming increasingly affordable since the mid-2010s, though external USB hard drives may be used on the go. === Hard drives' purposes === After purchasing a hard drive, choose whether to dedicate it to either auxiliary storage or archival ([[wikipedia: cold data|cold storage]]). Use as auxiliary storage such as for a workstation or server demands the drive be in constant operation, which wears it down over time, making it unsuitable for long-term archival. Use as archival storage demands only sporadic (rare) operation to add or retrieve data, which induces far less mechanical wear. Data retrieved from an archive drive should be copied over to auxiliary storage to avoid needless wear on the former whenever that data is recalled again in near future, as can be expected, and in order to further duplicate files that have proven to be useful, as archival is commonly done with an uncertainty of which files will be useful in future. A functional archive drive may later be repurposed for auxiliary storage after moving the archived data over to a new device. Make a list of data you wish to retrieve the next time you access your archive drive. If you can afford it, mirror the archive to a readily available hard disk or network-attached storage. If your live hard disk or network attached storage runs out of disk space, delete the files that were not accessed for the longest time if they are already backed up somewhere else. To find them, file managers are equipped with the ability to sort files and directories by "last accessed" (or similar) in either direction. === Labels === Consider physically labeling your storage media to facilitate finding data. For example, a date or a short summary of the contents can be written on a label sticker that is applied onto the casing of an external data storage device. For optical discs, a disc marker (also interchangeably known as "CD marker", "DVD marker", etc.) can be used to write notes directly on the disc. If more space is necessary, label stickers on its containing [[:w:Optical_disc_packaging#Jewel_case|jewel case]] or spindle can be used. Many jewel cases include a paper sheet for notes. Do not attach stickers directly onto the disc, as they could disintegrate during high rotation speeds, risking damage to the optical drive's internal components. It is also recommended to change the device's file system label to a year or a short summary of the contents so it can quickly be seen in the file manager's device list (usually a side bar on the left) when plugged in. On data storage devices whose life expectancy appears to be nearing its end, as indicated by [[:w:Self-Monitoring,_Analysis_and_Reporting_Technology|S.M.A.R.T. data]] on hard drives and performance loss on solid state memory. an "expired" or "EOL" (end of life) label can be added. Such devices should at most be used for temporary purposes such as testing. == Data storage size == It is recommended to get a larger storage capacity than one intends to use. This leaves some spare room in the case that more data than expected is created, such as vacation video recordings. On portable devices such as mobile phones and digital cameras, a larger storage capacity lowers the required frequency of file transfers into ones archive. Another benefit of larger storage media is that more data can be readily accessible at a given time, and a needed file can be found faster since fewer devices have to be searched. However, a central file list and [[#Labels|labeling]], as described in other sections, can facilitate finding files as well. If your storage device has, for example, 128GB, it realistically means 100 GB. Not only because of reserved space from file system overhead and possibly operating system files, but because of those 128 GB, the end is less useful, since one might want to put files there that are slightly larger than the remaining space, and one would like to have all those files in one location to avoid confusion and the annoyance of changing media. In other words, if one would like to put 11 GB where only 10 GB are available, those 10 GB are as little useful as 0 GB. == Partitioning == User data may be stored on the same or a separate partition as the operating system. The benefit of a separate partition for user data is that possible file system corruption on the operating system partition would not spill over to user data, though modern file systems such as ''NTFS'' and ''ext4'' protect themselves from damage by [[:w:Journaling file system|journaling]], which allows the file system driver to recover quickly after an unexpected termination of write access caused by an operating system crash or unexpected removal. Infrequently accessed files that may be necessary in near future can be stored there as well, or moved to a secondary drive if their size is significant. Another benefit of a separate user data partition is the smaller backup size of the operating system partition and facilitated recovery in case of a malfunctioning operating system where other means of repair have failed or would be too difficult. Because operating systems are subject to corruption and can at worst become unbootable, it is good practice to [[backup|back up]] their partition regularly into a disk image. A smaller operating system partition can be imaged more quickly and the routine induces less wear on the backup media. Should the operating system malfunction, it can be imaged and then restored more quickly from the functional previous disk image, with less work to merge desirable changes since the last backup. == Packing and archival == Perpetual streams of new files such as web downloads, photos and videos from digital cameras and cell phones, and screen captures can be packed by renaming their parent folder into a uniquely identifying name, such as with date stamp: <code>Camera-{{#time:Y-m-d}}</code>, after which they can be moved to an archive drive at the next backup appointment. The folder's name may also contain a location, device type, and/or short description. If packed more than once on the same day, time stamps or part numbers can be added to the names, or the files can be merged into the same directory. If you no longer intend to change the contents of an archived folder, a file count and byte size can optionally be added to the name to provide a quick overview in file lists and to facilitate verifying whether all data has been transferred without having to navigate back to the source device and wait for loading to view the folder's byte size. An example name is <code>Camera-{{#time:Y_m_d}}-12147367878b-2093items</code>. It is recommended to use the exact byte number to prevent confusion between file size units that are powers of two (KiB, MiB, GiB) and powers of ten (KB, MB, GB). Alternatively to renaming on the source device, the directory with uniquely identifying name can be created on the archive drive first, and files can be moved there out of the source directory. When to pack files is end users' decision, though it is recommended to do so before exhausting free space on the source device. Renaming is not as necessary if new folders are created automatically, like [[:w:Digital_camera#Directory_and_file_structure|some digital camera firmwares do per 999 or 1000 pictures]]. All filled folders can be considered eligible for archival. Larger storage space provides the benefit of more buffer until the next file transfer becomes necessary to clear space, thus it needs to be done less frequently. Individual files needed for a specific purpose such as an impending project can be copied or moved into a separate directory. == Write protection == Write protection may be desirable to defeat the fear of accidental modification of data when not desired by the user. A simple way of achieving write protection in Linux-based operating systems is to mount or re-mount a device or partition as read-only with this command which requires superuser privileges: <code>mount -o remount,ro ''[device or mountpoint]''</code>. If write-protection is not supported by the operating system, an SD card with write protection switch feature can be used. The switch relies on the SD card reader to obey it and deny writing access to the operating system. Some memory card readers, both built-in ones and USB adapters, might not obey the write protection switch. Another way to achieve write protection is finalized write-once optical media or a read-only optical drive with insufficient laser beam power to write data, as described in {{section link||Sensitive environment}}. == File listing == Searches within file lists inside a text file are significantly faster than searches through a file system. See [[Data_recovery#Create_a_file_list|this guide]] on how to create file lists. === File index === As explained in {{section link|File_puzzling#Orphaned_directories}}, some file systems store directories that comprise file paths as "linked lists", meaning distributed over the entire space rather than one index of "nodes" at the beginning, which has both benefits and disadvantages, the latter of which is slower file searching. A searchable file index stored in a text file named "index" created using the <code>find >index</code> command can facilitate finding files, as it contains a list of paths to all files at one place. The index can be updated by running the command again to overwrite the existing one. If the working directory is not the root of the file system, it should be changed to it or the paths need to be specified. The "index" file can be searched with ease using <code>grep -i "searched file name" index</code>, which is typically much faster than directly searching the file system. <code>-i</code> may be left out for case-sensitive search. These commands have additional options, but these are outside the scope of this section. == Time stamp preservation == Some methods of file transfer, such as copying within/onto mobile phone storage, the <code>cp</code> command without activated <code>-p</code> ("preserve") option, and a directory on Unix/Linux not owned by the current user, might discard date and time stamp file attribute(s), resetting it to the current time. To preserve last-modified time stamps over FTP, downloading is preferred, as uploading while preserving it requires both client and server to support the [[:w:FTP#Additional_commands|MDTM (''Modify Fact: Modification Time'') command]], which it is not widely. == High numbers of small files == With an increasing number of files, file searches slow down. High numbers of small files also restrict portability, as they demand more file operations for file transfers, slowing the process down.<ref>Related: [https://www.quora.com/Why-is-copying-1-000-1MB-files-so-much-slower-than-copying-1-1GB-file-given-that-the-same-amount-of-data-is-being-copied/answer/Franklin-Veaux Answer on Quora to "''Why is copying 1,000 1MB files so much slower than copying 1 1GB file, given that the same amount of data is being copied?''" by Franklin Veaux, on January 14th, 2020]</ref> Additionally, higher cluster sizes in combination waste more space to [[File Systems#Cluster_size|cluster overhead]] (unused reserved space). If you happen to have a high number of currently unneeded small files, such as tens of thousands, consider packing them into one big archive file for improved portability. Compression may be considered where efficient, such as in human-readable text files and code, and/or where more necessary, such as online file sharing. Compression ratios of 100 may be achievable by strong compression algorithms on text documents and code. However, it should be taken into consideration that damage magnifies enormously over compressed archives, as demonstrated in {{section link|Backup#Compressed archives}}. Therefore, it is recommended to store compressed archives on at least two devices. Text inside compressed archives can be searched through directly without extracting using tools such as <code>zgrep</code>, <code>xzgrep</code>, <code>bzgrep</code>, and for 7-Zip, <code>7z e -so -bd ''[path]'' |grep ''[query]''</code>. If frequent modification of small files is necessary, an alternative to packed archive files (such as <code>.tar</code>, <code>.zip</code>) are file system containers (''virtual disks'') with a small cluster size. File system containers are manually generated disk image files that can be mounted like usual drives. == Avoid exhausting the operating system's partition == Exhausted space storage should be avoided, especially on an operating system partition, as it could lead to bogus behaviour by software not designed to handle such condition, or other unwanted behaviour. For example, a failed write while saving could blank the target file, causing the loss of work and reset of configuration. A web browser might [[Backup#Browsing_history|automatically delete early browsing history entries]] to make space for new. Even seemingly basic features such as command line parameter completion could malfunction.<ref>{{cite web |url=https://unix.stackexchange.com/questions/277387/tab-completion-errors-bash-cannot-create-temp-file-for-here-document-no-space |date=2016-04-18 |access-date=2022-02-10 |title=Tab completion errors: bash: cannot create temp file for here-document: No space left on device |website=Unix & Linux Stack Exchange }}</ref> On an operating system partition, keeping a safety margin of free storage such as 5% at any time is recommended, and at least 1% on secondary expansion storage. On archival media, a controlled exhaustion of space is less critical, though the readability of the final written files should be verified. Should you still find yourself with 100% exhausted space, first seek few megabytes of files to move out, perhaps temporarily, to improve system stability. This allows you to take time to calmly search for more and/or larger files to move out. == File system repairs == Tools such as <code>CHKDSK</code> on Microsoft Windows and <code>fsck</code> on Linux promise repairing damaged file systems. Logical file system errors may be caused by unexpected power outages or unpluggings. Be careful with file system repairs. It is recommended to back up any device (either to a full-disk image or by copying all files) prior to running one, in order to be able to revert with ease in case of unwanted collateral damage. Detected file system errors may be caused by incompatibile file names across operating systems. For example, Linux allows characters in file and folder names in NTFS that Windows considers invalid, such as a colon (<code>:</code>) and a pipe character (<code>|</code>), as well as case-insensitive file names. Upon detection of invalid characters, the <code>CHKDSK</code> tool moves and renames such items, which leads to the loss of file names and paths. In particular, <code>ChkDsk</code> moves files and folders with invalid names into a directory located at the file system's root named <code>found.000</code>, and renames them to generic names like <code>file00000000.chk</code> and <code>dir_00000000.chk</code>, where the number is hexadecimal and incremented. == Disk usage analysis == [[File:GNOME_Disk_Usage_Analyzer_3.32_screenshot.png|thumb|Disk usage analyzers facilitate finding directories with the largest content size. Some illustrate the directory structure graphically.]] Disk usage analyzers calculate the size of directories on any selected path, allowing the user to easily discover directories which occupy the most space. Large folders not currently needed can be moved over to an archive drive, which clears the most space on the source device. Popular tools for desktop operating systems include ''[[:w:Disk_Usage_Analyzer|Baobab]]'' for Linux (pre-installed on some popular distributions) and ''Xinorbis'' for Windows, both with sophisticated graphical user interfaces. Linux is also equipped with the command-line tool <code>du</code>, which allows outputting results directly into a text file. For mobile (Android OS), [[:w:ES_File_Explorer|ES File Explorer]] is equipped with such functionality, though that application has been subject to controversy and has developed into adware. <div style="float:none; clear:both;"></div> == Deduplication == Several tools to automatically deduplicate files exist: rdfind, fdupes, jdupes, rmlint, dupeGuru, FSlint. Storing one duplicate of files may be desirable in certain situations, such as [[Backup#Compressed_archives|compressed archives]] intended for long-term preservation, where even the slightest damage can render any data after the point of damage unreadable. For accessing the same file from different locations, [[:w:Hard_link|hard link]]s or [[:w:symbolic link|symbolic link]]s can be used. You may want to deduplicate files across two different storage devices without accidentally deleting any files that do not exist on the storage device you want to keep the files on. A manual deduplication may be necessary if you wish to free up space from a device after copying files to a different device or after creating an archive file from those files without accidentally deleting any files that were not copied or archived from the source device, or after a file copying operation was interrupted and you need to clean up without deleting any files that were not copied. The quickest way to accomplish this would be to compare the total count and size of those files in both the source and the target device by selecting the files and opening the "Properties" window. If the byte counts match, the source folder can be safely deleted. However, if the selection contains subdirectories too, differences in file systems mean that the total size might not match due to differences in how directories are stored. Some file managers also don't show the exact byte count. If you want to deduplicate files between two different computers or between a computer and a smartphone, an additional problem is that file managers count files and folders differently. Some file managers only show the number of "items", meaning both files and folders, where as others may only show the number of "files" without counting the folders. If possible, use the same file manager on both devices, and one that shows the exact byte count. A more sophisticated way to manually deduplicate files is creating a temporary script that contains a list of files to be deleted. In Linux, this can be accomplished first changing the working directory to the target folder (where the files are supposed to stay) using <code>cd</code> and then creating a temporary script by running: <syntaxhighlight lang=sh> # list files to be deduplicated find -type f |sed -r "s/'/\\\\'/g" |sed -r "s/(.*)/rm -v '\1'/g" >>temporary.sh # list empty directories to be removed find -type d |sed -r "s/'/\\\\'/g" |sed -r "s/(.*)/rmdir -v '\1'/g" >>temporary.sh # script deletes itself after work done echo "rm temporary.sh" >>temporary.sh </syntaxhighlight> No files have been deleted up to this point. Only the disposable script to do the work has been created. This script contains a list of files to be deleted from the source device, but the list is generated from the target device to avoid containing the name of any files that have not been copied. It is recommended to use a temporary script rather than throwing around commands involving an <code>|xargs</code> pipe, since running <code>|xargs</code> commands in combination with deletion is dangerous since you don't see a list of files before starting the deduplication process. The <code>-v</code> flags are optional and serve to later show the names of the files deleted in the deduplication process in the terminal. Now move the <code>temporary.sh</code> file to the original directory on the source device from which you want to delete the duplicate files. Then change your terminal's working directory to that directory using <code>cd ''[path]''</code>. Before running the script, open it in a text editor and glance over it to verify that it doesn't contain any files you don't intend to delete. Then run it using <code>sh temporary.sh</code>. Do not run the script before changing your working directory to the same directory the script resides in, because the script contains relative paths rather than full (absolute) paths, so the path of your terminal's working directory will be presumed to be infront of the relative paths in the script. If the relative paths to the files that need to be deduplicated have changed but not their names, the <code>find</code> utility can be used to deduplicate files. Make sure to follow the steps exactly as described to avoid deleting files you want to keep. Specifically, double-check your current working directory (type <code>pwd</code> to check if it is not visible in terminal). The working directory should only be a folder dedicated to deduplicating files. <syntaxhighlight lang=sh> cd "/example_target_path/" # Files will not be deleted from this directory. # The following command creates the temporary script that will find and delete the files, but it will not be ran yet. find -printf "%P\n" | sed -r "s/(.*)/find -name \"\\1\" -delete/g " |tail -n +2 > /tmp/deduplicate.sh cd "/example_path/deduplicate" # Don't miss this step or you will delete files from the target path. pwd # Double-check the working directory. Files from this directory will be deduplicated. sh /tmp/deduplicate.sh # Run the script. find -type d -delete # optional: delete the empty directories left behind. </syntaxhighlight> Another way to deduplicate files is with a matching size, though there is a small risk that two files have the identical size. This function generates a temporary script that can be used for deduplicating files with a matching size. <syntaxhighlight lang=sh> # declare function find-dedup() { find -type f -not -empty -printf "%s\n" |sed -r "s/(.*)/find -size \1c -delete/g" |head; } # switching to the where the files stay untouched cd /path/to/target find-dedup > /tmp/deduplicate.sh # creating temporary script # switching to directory where the files should be deduplicated from cd /path/to/source # running temporary script that deletes duplicates based on matching size sh /tmp/deduplicate.sh </syntaxhighlight> == Temporary folder == Programs may use temporary folders such as <code>/tmp/</code> and <code>~/.cache/</code> on Linux and <code>%temp%</code> on Windows to store data such as preview thumbnails and data from the web to reduce loading times. Temporary folders can also be used to protect against accidental deletions of files, by moving those files into a temporary folder whose contents will later be automatically deleted instead of immediately deleting files and folders. For example, on Linux, <code>/tmp/</code> is a virtual garbage truck. The system purges its contents when booting. Files and folders that were moved into it accidentally can be moved out in time. Should the way your operating system or file manager handles its [[:w:Temporary folder|temporary]] or [[:w:Trash (computing)|trash folder]] not suit your needs (e.g. if retention span is unchangeable or limited), you may wish to manually operate such in your user home folder (e.g. <code>~/tmp/</code>, <code>~/trash/</code>). Some software may allow changing the path for the temporary folder. Furthermore, such folders can be used for frequent short-term backups of projects, which can be deleted when free space becomes necessary. In comparison to a traditional "Trash" folder implemented by the operating system or file manager, files can be opened directly as usual, whereas a file manager may disallow directly opening files sitting in a "Trash" folder and show file properties instead, as Windows Explorer does, whereas, for example, the ''Nemo'' file manager for Linux allows directly opening files located in the trash directory. == Trash bin (recycle bin) == When deleting files, you might want to consider using the [[:w:Trash (computing)|trash bin]] (or "recycle bin" and other names) feature, which stores files in a temporary location until they are automatically deleted. This allows reversing unwanted deletions, for example if you selected a file or folder you didn't mean to delete. This lifts the emotional burden and carefulness during deletion, since you know you can undo the deletion for the time being. Without it, deletions are "walks on eggshells", metaphorically speaking. A recycle bin also makes file moving (with copying and deletion from source) safer, because during the deletion step, you have to be less careful to select the exact same files and folders that you moved. If your trash bin does automatic deletion after a certain time, you don't have to bother with cleaning it up later anymore, while you can still bring back the file if necessary in the near future. The files still take up space on your device while residing in the trash bin, but it doesn't matter if you have enough space free. == Verifying copied files == There are several ways to find out whether a selection of files was copied properly from a source to a destination. The fastest way is to count the files by using <code>|wc -l</code> to count the number of lines generated by the <code>find</code> command: <syntaxhighlight lang=sh> find "/path/to/source folder" |wc -l find "/path/to/destination folder" |wc -l </syntaxhighlight> You can also compare the list of files. You need to change the working directory (<code>cd</code>) and sort the output of the <code>find</code> command (<code>|sort</code>) to resolve inconsistencies between the path names that have no effect on the names of the individual files or the file contents. <syntaxhighlight lang=sh> cd "/path/to/source folder" find |sort |md5sum cd "/path/to/destination folder" find |sort |md5sum </syntaxhighlight> If the checksums mismatch, you can write the file lists to temporary text files and then use the <code>diff</code> command to find out which files mismatch: <syntaxhighlight lang=sh> cd "/path/to/source folder" find |sort > /tmp/diff1 cd "/path/to/destination folder" find |sort > /tmp/diff2 diff /tmp/diff[12] </syntaxhighlight> A more thorough way to accomplish this is to compare the file contents. If the beginning of a file was written properly, the rest of the file was very likely written properly too, so here we only compare the beginning of each file. First, we declare a function that will generate a checksum for the beginnings of all files in a directory, including in subdirectories. This can be pasted into the terminal or added to the <code>~/.bashrc</code> file so it is loaded automatically. <syntaxhighlight lang=sh> quickMD5() { find "$@" -type f |sort |xargs -d '\n' -I '{{}}' sh -c "printf \"{{}}: \"; dd if=\"{{}}\" ibs=2K count=1 status=none |md5sum |sed s/\ -//g"; } </syntaxhighlight> Now, we can repeat the same steps from above but with this command. But we don't have to sort the output anymore given that the function already does the work. <syntaxhighlight lang=sh> cd "/path/to/source folder" quickMD5 > /tmp/diff1 cd "/path/to/destination folder" quickMD5 > /tmp/diff2 diff /tmp/diff[12] </syntaxhighlight> These steps can also be combined into another function: <syntaxhighlight lang=sh> md5match() { return_dir="$PWD" # memorize directory before changing to allow specifying relative (non-absolute) paths cd "$1" quickMD5 > /tmp/diff1 cd "$return_dir" cd "$2" quickMD5 > /tmp/diff2 cd "$return_dir" diff /tmp/diff[12] } </syntaxhighlight> This will compare two directories by simply running <code>md5match "/path/to/source folder" "/path/to/destination folder"</code>. To compare the entire contents of all files, we can use this function: <syntaxhighlight lang=sh> superMD5() { find "$@" -type f |sort |xargs -d '\n' md5sum; } </syntaxhighlight> And then we repeat the same steps from above. However, keep in mind that this is slow and usually unnecessary. == Wait for format wars to conclude == After the [[:w:Blu-ray disc|Blu-ray disc]] won the format war against the [[:w:HD DVD|HD DVD]], optical drive manufacturers abruptly discontinued manufacturing drives that can read the HD DVD. Such discontinuations create a risk for losing access to data, see [[:Wikipedia: digital obsolescence]]. Therefore, it is recommended not to store data on formats currently involved in a format war. Store at least one copy of your data on established data storage formats, even if they predate new formats. After support for HD DVD was discontinued by optical drive vendors, formats that predate the HD DVD, the CD and the DVD, were continued to be supported and still are to this day (as of 2025), and will probably be supported for a long time to come given that the data storage format is established, meaning people already own many instances of them and hardware that is able to read from them. == Mobile == === Memory cards === [[File:Huawei_U8950D_no_cover.JPG|thumb|right|Smartphone with inserted memory card (located below camera lens)]] Some smartphones and tablet computers allow the expansion of storage capacity using memory cards, typically MicroSD, which significantly facilitates file management and is user-friendly. Memory cards can be re-used immediately between devices without need for file transferring, and data stored on the memory card is not at the risk of mobile devices' technical defect, as it can be ejected, after which data can be retrieved externally. Mass storage access from an external computer also may allow [[data recovery|recovering some files]] imminently after a deletion accident caused by bogus software<ref>[https://www.reddit.com/r/Android/comments/16e6kc/i_just_deleted_a_random_folder_in_my_internal/ Users report Android devices' entire internal user storage being deleted instantly, caused by poor software design – "''I just deleted a random folder in my internal storage and it wiped my internal storage. What the heck just happened?''" – Reddit.com/r/Android (2013-01-11)]</ref> and/or human error. For huge file transfers, ejecting the memory card and directly transferring to the PC through mass storage may save time compared to MTP (media transfer protocol) through the phone or tablet, as the latter does not handle high counts of files within a directory well. Additionally, memory cards can immediately be reused in a different device without lengthy file transfers. USB-OTG (''On The Go'') may be used as well, connected through an adapter directly to the mobile device, though it might not [[#Time_stamp_preservation|preserve a date and time attribute]]. Tablet computers with desktop operating system are widely equipped with at least one default-sized USB-A port. Additionally, using a memory card takes stress off the device's non-replaceable internal memory, preserving its limited rewrite cycles, which is especially beneficial for repeated heavy tasks such as high-resolution filming and mobile FTP server hosting. <div style="float:none; clear:both;"></div> === Between computer and mobile === ==== Media Transfer Protocol ==== File management on mobile phones and tablet computers with mobile operating systems is more restricted than on desktop/laptop computers and mass storage devices such as USB sticks and memory cards, as the media transfer protocol (MTP), which is used to access files on a mobile device from a computer, lists files slowly, which is problematic for loading directories with high counts of files. As such, it is recommended to manage such directories on the device itself. If transfer between a desktop/laptop computer is desired, handle those files in a little to unpopulated directory. MTP file listing can be sped up by not loading preview thumbnails. Depending on file manager used, this may be done by deactivating preview thumbnails in settings or choosing the "detail" view mode, where files appear in a list instead of a grid which foregoes the loading of preview thumbnails. However, files can be dropped into a directory without opening the target directory, by pasting them through the right-click context menu. A benefit of MTP is it not being prone to file system corruption as a result of unexpected removal, meaning without being "safely unmounted" through the client operating system (i.e. desktop / laptop computer), as it operates through an abstraction layer and the file system is controlled by a driver on a battery-powered host device (i.e. smartphone / tablet). Only a selection of files, no directories, should be moved away from the device, because users have reported files on MTP not being listed properly.<ref>[https://android.stackexchange.com/questions/136839/files-show-up-on-nexus-5-but-not-in-windows-7 ''Files show up on Nexus 5 but not in Windows 7'' – Android Stack Exchange – February 11th, 2016]</ref><ref>[https://android.stackexchange.com/questions/46315/not-all-files-are-visible-over-mtp ''Not all files are visible over MTP'' – Android Stack Exchange – May 29th, 2013] (209,502 views as of September 21st, 2022)</ref><ref>[https://stackoverflow.com/questions/13737261/nexus-4-not-showing-files-via-mtp ''Nexus 4 not showing files via MTP'' – StackOverflow – December 6th, 2012] (67,504 views as of September 21st, 2022)</ref> If a directory is moved away from the device, the computer might delete it from the mobile device without all content having been transferred away. Instead, it should be copied, and the byte size be compared on both the computer and the smartphone itself, where a match indicates a successful transfer, meaning the directory can now be deleted from the mobile device. The only exception where moving folders out is safe is when the number of files within is overseeable, i.e. less than ten, where all files are clearly listed in the computer's file manager. Windows Explorer additionally displays files while listing is in progress, which can be of use when moving files out, since the loading of the file list can be interrupted to allow moving out the displayed files chunk-wise, reducing the number of remaining ones each time. Transferring files onto the device through MTP [[#Time_stamp_preservation|may dismiss their date and time attribute]]. If a file has newly been created on the smartphone while it was connected to the computer through MTP, the computer's file manager could potentially misreport the file size as too small due to having loaded the directory listing in a moment where the file was not complete. Moving the file away from the phone could cause it to be truncated (incomplete) on the target path while being deleted from the source, since the file manager might wrongfully assume that the file has been fully transferred while it hasn't. ==== File Transfer Protocol ==== An alternative to MTP is FTP (file transfer protocol) through ethernet. On the desktop computer, a dedicated and sophisticated FTP client such as FileZilla (open-source) may be used to handle high numbers of files, though FTP is widely supported by file managers and web browsers. FileZilla does not support moving files out of an FTP server, meaning downloading and deleting automatically, whereas moving within a server is supported through the standard rename command. If the intention is moving files out of an FTP server, the highlighted selection of files on the server needs to be deleted after the transfer after verifying that all files have been transferred successfully, meaning no new entries in the "Failed transfers" list. To get a peace of mind that the selection was transferred successfully, try downloading it again while skipping existing local files. If no new files are downloaded, this ensures all files have already been transferred. This might apply to other software as well. FTP server applications for mobile devices may handle file listing differently. Some do not report the year of the file, only day and month, causing the FTP client to insert the current year for files except those last modified at a later time of the year than currently, for which the previous year is inserted instead. Another distinction between FTP server apps is whether they list file and directory names starting with a dot, which is considered ''hidden'' in the Unixverse (i.e. on Unix and Linux-based operating systems, which includes Android OS, the most popular mobile operating system). FTP server applications typically allow the user to select a specific directory to share, rather than the entire storage. This feature has been [[:File:Android extended MTP options concept.png|proposed for MTP]], but never implemented there so far. Two open-source FTP server apps for Android OS are the integrated FTP server of "Amaze File Manager", and the more sophisticated "primitive FTPd", only the latter of which reports files and folders with names starting with a dot. Alternatively, files may be uploaded vice versa from the mobile device to an ethernet FTP site served by a home computer, though as of 2021, no mobile file manager's FTP client supports [[File management#Time stamp preservation|preserving files' date and time stamps]] upon uploading. ==== Handling invalid file names ==== Since the most popular mobile operating system, Android OS, is Linux-based, it supports characters in file names that are unsupported by the most popular desktop operating system, Windows, and by some file systems. These characters include a colon (<code>:</code>), a back slash (<code>\</code>), a vertical pipe (<code>|</code>), a question mark (<code>?</code>), and an asterisk (<code>*</code>). Additionally, file names are case-sensitive, meaning files named "file" and "File" and "FILE" can co-exist within a directory on Linux and Android OS, but not on Windows. While some mobile phone apps disallow the creation of such files and automatically replace characters Windows considers invalid with a substitute character such as an underscore (<code>_</code>), other apps might have created files with names containing aforementioned characters. When copying or moving such files from a mobile device through MTP (Media Transfer Protocol) using Windows, all characters before and including the last invalid character are discarded from the file name on the target location to make the file openable. Therefore directories on a mobile device which contain such files will need to be moved out in two passes to retain the file names. First, an [[:w:archive file|archive file]] such as a ZIP file should be created on the mobile device which can then be moved out before the files themselves can be moved out. The archive file can optionally be created after isolating files with invalid characters in a separate directory to consume less space, though that can be a difficult task on a mobile device due to limited file management software and users' infamiliarity with the Linux terminal which can be accessed on Android OS through third-party applications such as Termux or Jack Palevich's "Android Terminal" app. Another option would be to only create the archive file and not transfer the bare files to the archival media. However, this requires files to be extracted before being opened, which adds a delay for larger files and does not allow for preview thumbails without extracting the entire archive file. === On-device management === Additionally, file access on the most popular mobile operating system has been restricted significantly over time, and to varying degrees per storage type (internal, memory card, and USB-OTG). Such restrictions affect third-party applications installed by the user, including file managers. Pre-installed file managers are usually unaffected, though these tend to be functionally restricted, such as lacking range selection, where only two entries need to be tapped for all inbetween to be marked. [[:File:Android extended storage options concept.png|Options to deactivate these restrictions at user discretion]] were not officially provided, leaving so-called ''[[:w:Android rooting|rooting]]'' as the only possibility of bypassing them. This is a process in which the user unlocks administrative access over the operating system. The operating system vendor claims aforementioned file access restrictions to serve user security, though them being a cloud storage vendor as well suggests a commercial interest that conflicts with end users' desire of freedom, and simultaneously may encourage users to unlock ''root'' access, which is against vendors' recommendation, and where inexperienced tampering can lead to malfunction. == Other ideas == === Archival queue === New files from portable devices which are currently unneeded can be moved into a buffer directory of files ready for archival, which means moving them to a large and stationary hard drive at the next connection to the computer. External flash storage such as USB sticks and solid state drives can also be used to store data for, for example, the duration of a trip or vacation, where they can be moved to an archive hard drive when arriving at home. === Temporary redundant retention after archival === Files that have already been moved to a larger stationary archive drive may be redundantly kept on the smaller portable data storage such as a mobile phone or USB stick, but in a directory in which any file is eligible for deletion, such as at space exhaustion. This would serve as a short-term backup, which could be retrieved from in case anything goes wrong with the archive drive prior to it getting backed up itself. This increases file fragmentation on the portable device, though that does not noticably affect performance on flash storage. === Partition for small files === If your computer setup has no secondary drive and/or partition, you may create a small partition (e.g. 4 GB) with a low cluster size for more efficient storage of small files. Additionally, if space storage happens to be exhausted on the main partition, with software arbitrarily attempting to write to it, files can still be added on the secondary partition without interference. === Sensitive environment === Inside sensitive environment, data may be exchanged through rewritable optical media such as DVD±RW and BD-RE, as these use external storage controllers, making the media itself unable to contain malicious hardware such as so-called ''rubber duckies'' used to simulate keystrokes from a USB keyboard. Additionally, finalized write-once media and/or read-only (''ROM'') optical drives can ensure write protection where necessary, for example in a malware-infested environment. === Copies of description files === Description files such as text files describing the contents of a folder may be stored both inside a folder and a copy in a central location with other description files for easier discovery. The name of that file should contain the name of the folder it is describing. It is recommended to store such a description file inside the folder it is describing rather than along with it in the parent directory, to make it easier to find if the items in the parent directory are not sorted alphabetically, since a file managers might separate files and folders regardless of which sorting method (alphabetical, by size, by name, by last modified) is chosen, which would make the description file not appear next to the folder it is describing. === Handling file conflicts === You might want to merge files from two directories where some file names already exist in the target directory, such as for backup purposes. Some file managers ask the user how to handle file conflicts in the midst of the copying or moving process, which could interrupt the progress while the user is away from the computer, needlessly wasting time. To tell the file manager how to handle file conflicts at the beginning, sort the files in the source directory in a way where a file name that exists in the target directory appears near the top. Then select all files and copy and paste them into the target directory. The file manager will ask you near the beginning of the copying session how you would like it to handle file name conflicts. File managers usually have something like a checkbox to to repeat your chosen action for the rest of the copying session. This way, you can get file conflicts out of the way early. == Observations and tips == === Spare directories === When space on a device or partition is exhausted, no new directories that could be helpful for organizing files can be created, such as for moving files from a highly ''populated'' directory (i.e. with many files, such as a download folder) on a mobile phone in order to skip having to open the populated folder directly through [[#Media Transfer Protocol|MTP (Media Transfer Protocol)]], which notoriously handles long file listings poorly. Prepare for such a situation by creating a reserve of spare empty directories inside one dedicated directory. The spare directories can be moved out of the reserve, and be renamed as necessary, even without space left, which allows organizing files on the go to be able to move them elsewhere immediately when arriving home. === File move behaviour === When the aim is to bring files to an archive, moving files rather than copying and deleting afterwards has the convenience benefit of acting like a check list of files, instead of creating duplicates that would later have to be sorted out without accidentally deleting non-copied files, as well as imminently clearing (freeing) space on the source device. Moving instead of copying and then deleting files also defeats the psychological barrier that may come from the deletion step, as it feels like a destructive action even though it leads to the same result as moving between storage devices. Another barrier is the uncertainty of having inadvertently selected any file not copied. When moving files, some file managers may delete files individually after transfer, while others only delete selected files from source only after the last file has finished transfering. Windows Explorer uses the former method for mass storage devices, but the latter with Media Transfer Protocol. The Linux file manager ''Nemo'' always uses the former method. With the latter method, any interruption would cancel the file transfer without having freed up any space on the source device. === Escape auto-closure === Some file managers such as Windows Explorer close themselves when detecting the removal (unmounting and/or physical unplugging) of a storage device, while others jump to the starting directory. Some file managers might do nothing. The first case may be perceived as annoying, as it forces users to re-open the file manager and navigate all the way back to the previous directory.<ref>[https://www.sevenforums.com/general-discussion/148620-how-stop-windows-explorer-closing-removable-disks.html ''How to stop Windows Explorer closing for removable Disks!'' &ndash; SevenForums]</ref> This can be prevented by opening a different device in the file manager before unplugging. After plugging the device back in, the previously opened directory can be navigated back to immediately through the navigation history, using the on-screen {{button|&larr;}} button or <kbd>Alt</kbd>+<kbd>&larr;</kbd> on the keyboard. === Using the command line === Command-line file operations can be logged for later reference by using the "verbose" switch <code>--verbose</code> or shorthand <code>-v</code> on the <code>cp</code> and <code>mv</code> commands in Linux and redirecting the output into a text file by appending <code>>>path/to/textfile.txt</code>. On Windows, file operations are outputted by default, meaning no "verbose" switch is necessary. On Linux, using <code>| tee -a path/to/textfile.txt</code> allows visibly printing out command line output in real-time (as usual) while logging into a text file simultaneously. Autocompletion using the <kbd>↹Tab</kbd> key is widely supported on both Microsoft Windows and Linux/Unix-based operating systems and facilitates navigation by selecting file and folder names.<ref>[https://www.computerhope.com/tips/tip176.htm Use tab to autocomplete commands in the command line (ComputerHope, 2020-12-31)]</ref> === Shortcuts === Don't hesitate to use bookmarks to frequently accessed directories in your file manager and file picker dialogue (also known as "Open" and "Save as"). If they become unnecessary, they can be removed at any time. For command-line use, familiarize yourself with environment variables, as they allow quicker navigation throughout your directories. They include <code>~</code> on Linux and <code>%user profile</code> (user home directory, equivalent to <code>C:\Users\''Username''\</code>) and <code>%temp%</code> (<code>C:\Users\''Username''\AppData\Local\temp\</code>). Environment variables are also typically recognized by file pickers. Additional shortcuts for command-line use can be specified as variables in the script that runs when starting the terminal, which in Linux is typically located at <code>~/.bashrc</code>. For example, <code>$dl</code> can be set to refer to the download folder by appending <code>dl=~/Downloads</code> to that file. === Spaces in file names === Keep in mind that space characters in file and path names can become a nuisance when entering paths, creating variables, selecting text, and navigating and auto-completing in a command prompt/terminal using the <kbd>↹Tab</kbd> key. === Cleanup === "Cleanup" may refer to moving files scattered around the storage and desktop in folders to improve overview, or deleting files like duplicates to clear free space. In the former case, refer to [[/Directory structures]] and consider moving files you are unsure where to move into a [[#Dumping_ground|dumping ground]]. In the latter case, as with [[Backup#lostBackup|lost backups]], first weigh effort against price, meaning here first consider whether the time and effort spent searching files to clean up really outweighs the storage space price. It is not worth trying to search and eliminate duplicate files for the sake of it if no significant space is cleared. The time and effort necessary to do it might outweigh any benefit from the saved space. However, duplicates may be deleted for organizational purposes, such as duplicate music tracks in the same folder. Note that cleaning does not necessarily speed up the computer noticeably, except if the partition was nearly full, which should be avoided anyway, as described in [[#Avoid_exhausting_the_operating_system's_partition|§ Avoid exhausting the operating system's partition]]. Rather, closing tasks that stress the CPU and take much RAM is effective in speeding up the system. Clearing caches is mostly unnecessary and even disadvantageous, as they provide faster recall of information. For files vulnerable to damage like PNG and TIFF images, and [[Backup#Compressed_archives|compressed archives]], duplicates are beneficial for files not backed up yet. == Tools == * <code>[[:Wikipedia:find (Unix)|find]]</code> - A file searching tool built into Linux distributions, with many filters such as file names and date ranges. * <code>[https://github.com/sharkdp/fd fd]</code> - Lighter alternative with fewer overall features but with colourized output, which the regular <code>find</code> doesn't have. * <code>[https://github.com/tavianator/bfs bfs]</code> - Alternative to <code>find</code> with breadth-first searching instead of depth-first. == Experiences by participants == This section goes deeper into the idea about that "the average computer and mobile phone user still struggles to keep track of files in the long term despite of all the tools at their disposal". Here participants or visitors to this learning resource can add why they need a resource such as this to learn from and/or interact with it. :''Moved to [[Talk:{{PAGENAME}}]].'' == References == <references /> == See also == * [[File naming]] * [[File systems]] * [[Backup]] * [[Data recovery]] == External links == * Comics: ** [https://xkcd.com/949/ File Transfer – xkcd #949, 2011] ([https://www.explainxkcd.com/wiki/index.php/949:_File_Transfer explainer]) ** [https://xkcd.com/1360/ Old Files – xkcd #1360, 2014] ([https://www.explainxkcd.com/wiki/index.php/1360:_Old_Files explainer]) ** [https://xkcd.com/1817/ Incognito Mode – xkcd #1817, 2017] ([https://www.explainxkcd.com/wiki/index.php/1817:_Incognito_Mode explainer]) [[Category:Information technology]] [[Category:Computer_science]] bvlvxgak3x831l83kq22j938286crg7 2 289273 2842445 2842416 2026-10-24T19:03:35Z Dc.samizdat 2856930 /* A theory of the Euclidean cosmos */ 2842445 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. 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.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|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. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. * General relativity describes the rotational motion of objects with mass in four-dimensional Euclidean space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planetary surfaces, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> 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 <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, 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 <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) lies within a smaller 3-sphere shell which bounds it, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of objects large enough to have significant gravity of their own. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at its sphere-like core, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> 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 <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> 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 (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). 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 <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>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.<ref>{{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}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. 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 <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. 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 <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, 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 <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & 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 & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == 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 number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating at the universal rate of quantum events <math>c</math>, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons, considered as single point objects. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...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 in 4-space).{{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 a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of physical processes varies from place to place, and reference frame differences predicted by the theory 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, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == General relativity is rotations in a Euclidean space of four orthogonal dimensions == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective screw translations <math>QT</math>, which are distinct. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation through 4-space. The velocity of his vector of translation <math>T</math> through 4-space remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the surface of the rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacements is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the atomic rotational kinematics of Coxeter groups == ... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, 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 <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's work: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, 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 have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{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}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "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."}} 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.<ref>{{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}}</ref>}} == Boundaries == <blockquote>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.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>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 "Dimensional Analogy."<ref>{{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=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :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. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::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''<ref>{{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|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{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 "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> 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 "left" 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=; "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]."}} 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^o 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^o bend, or as a straight line with one 60^o 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=; "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.)"}} 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 V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ 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 V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ 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 V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ 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}} V₀ and V₂ 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).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & 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 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} 2bwirplhyqti46psud2k0170q0wru31 2842447 2842445 2026-10-24T19:31:35Z Dc.samizdat 2856930 /* A theory of the Euclidean cosmos */ 2842447 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. 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.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|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. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. * General relativity describes the rotational motion of objects with mass in four-dimensional Euclidean space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planetary surfaces, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> 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 <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, 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 <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) lies within a smaller 3-sphere shell which bounds it, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of objects large enough to have significant gravity of their own. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at its sphere-like core, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> 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 <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> 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 (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). 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 <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>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.<ref>{{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}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. 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 <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. 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 <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, 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 <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & 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 & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == 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 number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating at the universal rate of quantum events <math>c</math>, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons, considered as single point objects. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...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 in 4-space).{{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 a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of physical processes varies from place to place, and reference frame differences predicted by the theory 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, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == General relativity is rotations in a Euclidean space of four orthogonal dimensions == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective screw translations <math>QT</math>, which are distinct. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation through 4-space. The velocity of his vector of translation <math>T</math> through 4-space remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the surface of the rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacements is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the atomic rotational kinematics of Coxeter groups == ... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, 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 <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's work: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, 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 have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{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}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "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."}} 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.<ref>{{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}}</ref>}} == Boundaries == <blockquote>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.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>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 "Dimensional Analogy."<ref>{{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=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :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. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::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''<ref>{{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|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{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 "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> 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 "left" 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=; "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]."}} 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^o 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^o bend, or as a straight line with one 60^o 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=; "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.)"}} 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 V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ 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 V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ 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 V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ 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}} V₀ and V₂ 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).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & 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 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} nu0m738sefgu3j2ibcvsy3j10e3coq4 2842448 2842447 2026-10-24T19:48:02Z Dc.samizdat 2856930 /* A theory of the Euclidean cosmos */ 2842448 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. 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.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|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. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. * General relativity describes the rotational motion of objects with mass in four-dimensional Euclidean space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planetary surfaces, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> 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 <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, 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 <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) lies within a smaller 3-sphere which bounds it, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at its sphere-like core, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> 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 <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> 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 (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). 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 <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>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.<ref>{{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}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. 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 <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. 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 <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, 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 <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & 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 & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == 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 number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating at the universal rate of quantum events <math>c</math>, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons, considered as single point objects. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...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 in 4-space).{{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 a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of physical processes varies from place to place, and reference frame differences predicted by the theory 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, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == General relativity is rotations in a Euclidean space of four orthogonal dimensions == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective screw translations <math>QT</math>, which are distinct. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation through 4-space. The velocity of his vector of translation <math>T</math> through 4-space remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the surface of the rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacements is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the atomic rotational kinematics of Coxeter groups == ... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, 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 <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's work: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, 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 have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{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}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "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."}} 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.<ref>{{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}}</ref>}} == Boundaries == <blockquote>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.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>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 "Dimensional Analogy."<ref>{{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=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :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. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::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''<ref>{{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|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{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 "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> 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 "left" 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=; "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]."}} 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^o 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^o bend, or as a straight line with one 60^o 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=; "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.)"}} 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 V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ 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 V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ 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 V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ 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}} V₀ and V₂ 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).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & 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 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} m0aqz7wmf6uib1v06k9dpphl581sc85 2842449 2842448 2026-10-24T19:49:19Z Dc.samizdat 2856930 /* A theory of the Euclidean cosmos */ 2842449 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. 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.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|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. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. * General relativity describes the rotational motion of objects with mass in four-dimensional Euclidean space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planetary surfaces, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> 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 <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, 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 <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) lies within a smaller 3-sphere which bounds it, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to our manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at its sphere-like core, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> 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 <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> 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 (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). 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 <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>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.<ref>{{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}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. 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 <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. 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 <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, 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 <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & 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 & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == 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 number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating at the universal rate of quantum events <math>c</math>, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons, considered as single point objects. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...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 in 4-space).{{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 a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of physical processes varies from place to place, and reference frame differences predicted by the theory 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, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == General relativity is rotations in a Euclidean space of four orthogonal dimensions == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective screw translations <math>QT</math>, which are distinct. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation through 4-space. The velocity of his vector of translation <math>T</math> through 4-space remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the surface of the rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacements is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the atomic rotational kinematics of Coxeter groups == ... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, 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 <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's work: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, 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 have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{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}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "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."}} 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.<ref>{{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}}</ref>}} == Boundaries == <blockquote>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.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>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 "Dimensional Analogy."<ref>{{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=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :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. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::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''<ref>{{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|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{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 "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> 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 "left" 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=; "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]."}} 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^o 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^o bend, or as a straight line with one 60^o 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=; "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.)"}} 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 V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ 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 V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ 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 V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ 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}} V₀ and V₂ 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).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & 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 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} jx7x59zrqrtikuzn5mfycoivz5oglv3 2842451 2842449 2026-10-24T20:04:30Z Dc.samizdat 2856930 /* Symmetries */ 2842451 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. 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.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|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. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. * General relativity describes the rotational motion of objects with mass in four-dimensional Euclidean space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planetary surfaces, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> 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 <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, 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 <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) lies within a smaller 3-sphere which bounds it, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to our manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at its sphere-like core, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> 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 <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> 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 (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Clifford algreba|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 <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>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.<ref>{{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}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. 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 <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. 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 <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, 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 <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & 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 & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == 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 number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating at the universal rate of quantum events <math>c</math>, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons, considered as single point objects. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...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 in 4-space).{{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 a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of physical processes varies from place to place, and reference frame differences predicted by the theory 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, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == General relativity is rotations in a Euclidean space of four orthogonal dimensions == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective screw translations <math>QT</math>, which are distinct. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation through 4-space. The velocity of his vector of translation <math>T</math> through 4-space remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the surface of the rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacements is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the atomic rotational kinematics of Coxeter groups == ... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, 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 <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's work: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, 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 have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{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}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "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."}} 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.<ref>{{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}}</ref>}} == Boundaries == <blockquote>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.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>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 "Dimensional Analogy."<ref>{{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=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :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. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::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''<ref>{{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|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{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 "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> 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 "left" 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=; "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]."}} 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^o 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^o bend, or as a straight line with one 60^o 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=; "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.)"}} 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 V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ 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 V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ 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 V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ 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}} V₀ and V₂ 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).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & 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 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} hzc7i7g1albcxvsnzqwozw32h8e1va7 2842452 2842451 2026-10-24T20:05:40Z Dc.samizdat 2856930 /* Symmetries */ 2842452 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. 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.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|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. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. * General relativity describes the rotational motion of objects with mass in four-dimensional Euclidean space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planetary surfaces, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> 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 <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, 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 <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) lies within a smaller 3-sphere which bounds it, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to our manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at its sphere-like core, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> 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 <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> 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 (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Clifford algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>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.<ref>{{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}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. 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 <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. 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 <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, 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 <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & 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 & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == 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 number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating at the universal rate of quantum events <math>c</math>, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons, considered as single point objects. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...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 in 4-space).{{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 a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of physical processes varies from place to place, and reference frame differences predicted by the theory 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, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == General relativity is rotations in a Euclidean space of four orthogonal dimensions == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective screw translations <math>QT</math>, which are distinct. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation through 4-space. The velocity of his vector of translation <math>T</math> through 4-space remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the surface of the rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacements is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the atomic rotational kinematics of Coxeter groups == ... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, 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 <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's work: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, 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 have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{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}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "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."}} 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.<ref>{{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}}</ref>}} == Boundaries == <blockquote>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.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>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 "Dimensional Analogy."<ref>{{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=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :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. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::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''<ref>{{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|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{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 "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> 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 "left" 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=; "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]."}} 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^o 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^o bend, or as a straight line with one 60^o 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=; "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.)"}} 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 V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ 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 V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ 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 V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ 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}} V₀ and V₂ 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).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & 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 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} ncv6j1khmpimydn7hwrhe8h3z0acivu 2842453 2842452 2026-10-24T20:08:32Z Dc.samizdat 2856930 /* Symmetries */ 2842453 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. 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.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|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. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. * General relativity describes the rotational motion of objects with mass in four-dimensional Euclidean space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planetary surfaces, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> 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 <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, 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 <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) lies within a smaller 3-sphere which bounds it, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to our manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at its sphere-like core, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> 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 <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> 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 (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>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.<ref>{{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}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. 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 <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. 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 <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, 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 <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & 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 & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == 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 number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating at the universal rate of quantum events <math>c</math>, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons, considered as single point objects. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...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 in 4-space).{{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 a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of physical processes varies from place to place, and reference frame differences predicted by the theory 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, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == General relativity is rotations in a Euclidean space of four orthogonal dimensions == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective screw translations <math>QT</math>, which are distinct. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation through 4-space. The velocity of his vector of translation <math>T</math> through 4-space remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the surface of the rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacements is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the atomic rotational kinematics of Coxeter groups == ... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, 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 <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's work: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, 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 have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{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}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "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."}} 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.<ref>{{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}}</ref>}} == Boundaries == <blockquote>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.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>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 "Dimensional Analogy."<ref>{{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=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :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. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::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''<ref>{{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|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{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 "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> 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 "left" 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=; "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]."}} 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^o 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^o bend, or as a straight line with one 60^o 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=; "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.)"}} 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 V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ 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 V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ 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 V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ 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}} V₀ and V₂ 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).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & 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 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} jy63gijcnrhtcxod9lu83eoinwmhz27 2842469 2842453 2026-10-24T23:57:38Z Dc.samizdat 2856930 /* Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions */ 2842469 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. 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.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|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. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. * General relativity describes the rotational motion of objects with mass in four-dimensional Euclidean space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planetary surfaces, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> 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 <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, 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 <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) lies within a smaller 3-sphere which bounds it, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to our manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at its sphere-like core, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> 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 <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> 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 (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>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.<ref>{{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}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. 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 <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. 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 <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, 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 <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & 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 & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == 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 number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating at the universal rate of quantum events <math>c</math>, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons, considered as single point objects. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in Euclidean 4-space == {{Efn|...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 in 4-space).{{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 a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of physical processes varies from place to place, and reference frame differences predicted by the theory 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, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == General relativity is rotations in a Euclidean space of four orthogonal dimensions == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective screw translations <math>QT</math>, which are distinct. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation through 4-space. The velocity of his vector of translation <math>T</math> through 4-space remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the surface of the rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacements is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the atomic rotational kinematics of Coxeter groups == ... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, 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 <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's work: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, 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 have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{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}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "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."}} 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.<ref>{{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}}</ref>}} == Boundaries == <blockquote>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.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>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 "Dimensional Analogy."<ref>{{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=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :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. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::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''<ref>{{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|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{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 "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> 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 "left" 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=; "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]."}} 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^o 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^o bend, or as a straight line with one 60^o 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=; "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.)"}} 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 V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ 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 V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ 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 V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ 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}} V₀ and V₂ 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).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & 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 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} eq902xv122vlmd8yyhmiqsw8cxi04xm 2842470 2842469 2026-10-24T23:59:14Z Dc.samizdat 2856930 /* General relativity is rotations in a Euclidean space of four orthogonal dimensions */ 2842470 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. 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.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|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. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. * General relativity describes the rotational motion of objects with mass in four-dimensional Euclidean space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planetary surfaces, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> 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 <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, 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 <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) lies within a smaller 3-sphere which bounds it, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to our manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at its sphere-like core, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> 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 <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> 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 (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>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.<ref>{{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}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. 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 <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. 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 <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, 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 <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & 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 & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == 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 number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating at the universal rate of quantum events <math>c</math>, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons, considered as single point objects. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in Euclidean 4-space == {{Efn|...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 in 4-space).{{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 a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of physical processes varies from place to place, and reference frame differences predicted by the theory 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, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == General relativity is rotations in a Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective screw translations <math>QT</math>, which are distinct. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation through 4-space. The velocity of his vector of translation <math>T</math> through 4-space remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the surface of the rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacements is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the atomic rotational kinematics of Coxeter groups == ... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, 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 <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's work: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, 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 have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{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}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "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."}} 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.<ref>{{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}}</ref>}} == Boundaries == <blockquote>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.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>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 "Dimensional Analogy."<ref>{{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=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :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. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::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''<ref>{{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|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{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 "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> 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 "left" 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=; "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]."}} 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^o 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^o bend, or as a straight line with one 60^o 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=; "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.)"}} 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 V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ 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 V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ 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 V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ 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}} V₀ and V₂ 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).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & 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 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} mx6alu251brd9k2k41v5nc3kkzvq48o 2842471 2842470 2026-10-25T00:02:12Z Dc.samizdat 2856930 /* Quantum mechanics is the atomic rotational kinematics of Coxeter groups */ 2842471 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. 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.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|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. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. * General relativity describes the rotational motion of objects with mass in four-dimensional Euclidean space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planetary surfaces, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> 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 <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, 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 <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) lies within a smaller 3-sphere which bounds it, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to our manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at its sphere-like core, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> 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 <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> 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 (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>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.<ref>{{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}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. 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 <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. 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 <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, 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 <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & 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 & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == 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 number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating at the universal rate of quantum events <math>c</math>, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons, considered as single point objects. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in Euclidean 4-space == {{Efn|...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 in 4-space).{{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 a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of physical processes varies from place to place, and reference frame differences predicted by the theory 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, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == General relativity is rotations in a Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective screw translations <math>QT</math>, which are distinct. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation through 4-space. The velocity of his vector of translation <math>T</math> through 4-space remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the surface of the rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacements is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in 4-space == ... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, 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 <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's work: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, 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 have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{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}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "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."}} 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.<ref>{{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}}</ref>}} == Boundaries == <blockquote>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.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>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 "Dimensional Analogy."<ref>{{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=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :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. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::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''<ref>{{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|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{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 "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> 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 "left" 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=; "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]."}} 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^o 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^o bend, or as a straight line with one 60^o 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=; "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.)"}} 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 V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ 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 V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ 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 V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ 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}} V₀ and V₂ 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).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & 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 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} gamwy8epj4p7w0lf9m0hs8bqlh802c8 2842472 2842471 2026-10-25T00:03:31Z Dc.samizdat 2856930 /* General relativity is rotations in a Euclidean 4-space */ 2842472 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. 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.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|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. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. * General relativity describes the rotational motion of objects with mass in four-dimensional Euclidean space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planetary surfaces, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> 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 <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, 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 <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) lies within a smaller 3-sphere which bounds it, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to our manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at its sphere-like core, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> 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 <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> 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 (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>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.<ref>{{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}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. 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 <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. 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 <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, 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 <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & 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 & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == 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 number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating at the universal rate of quantum events <math>c</math>, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons, considered as single point objects. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in Euclidean 4-space == {{Efn|...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 in 4-space).{{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 a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of physical processes varies from place to place, and reference frame differences predicted by the theory 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, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == General relativity is the geometry of gravity in 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective screw translations <math>QT</math>, which are distinct. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation through 4-space. The velocity of his vector of translation <math>T</math> through 4-space remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the surface of the rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacements is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in 4-space == ... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, 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 <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's work: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, 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 have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{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}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "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."}} 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.<ref>{{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}}</ref>}} == Boundaries == <blockquote>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.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>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 "Dimensional Analogy."<ref>{{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=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :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. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::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''<ref>{{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|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{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 "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> 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 "left" 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=; "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]."}} 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^o 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^o bend, or as a straight line with one 60^o 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=; "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.)"}} 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 V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ 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 V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ 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 V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ 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}} V₀ and V₂ 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).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & 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 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} l5xwu2610tl3kf8w9i0c3043ouw0io9 2842473 2842472 2026-10-25T00:04:14Z Dc.samizdat 2856930 /* General relativity is the geometry of gravity in 4-space */ 2842473 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. 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.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|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. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. * General relativity describes the rotational motion of objects with mass in four-dimensional Euclidean space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planetary surfaces, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> 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 <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, 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 <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) lies within a smaller 3-sphere which bounds it, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to our manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at its sphere-like core, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> 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 <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> 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 (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>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.<ref>{{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}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. 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 <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. 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 <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, 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 <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & 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 & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == 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 number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating at the universal rate of quantum events <math>c</math>, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons, considered as single point objects. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in Euclidean 4-space == {{Efn|...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 in 4-space).{{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 a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of physical processes varies from place to place, and reference frame differences predicted by the theory 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, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective screw translations <math>QT</math>, which are distinct. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation through 4-space. The velocity of his vector of translation <math>T</math> through 4-space remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the surface of the rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacements is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in 4-space == ... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, 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 <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's work: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, 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 have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{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}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "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."}} 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.<ref>{{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}}</ref>}} == Boundaries == <blockquote>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.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>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 "Dimensional Analogy."<ref>{{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=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :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. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::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''<ref>{{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|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{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 "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> 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 "left" 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=; "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]."}} 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^o 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^o bend, or as a straight line with one 60^o 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=; "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.)"}} 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 V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ 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 V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ 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 V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ 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}} V₀ and V₂ 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).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & 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 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} t4tw8xa0ep75jnj6ptoep4tjoucmj90 2842474 2842473 2026-10-25T00:04:46Z Dc.samizdat 2856930 /* Quantum mechanics is the rotational kinematics of Coxeter groups in 4-space */ 2842474 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. 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.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|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. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. * General relativity describes the rotational motion of objects with mass in four-dimensional Euclidean space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planetary surfaces, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> 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 <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, 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 <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) lies within a smaller 3-sphere which bounds it, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to our manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at its sphere-like core, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> 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 <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> 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 (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>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.<ref>{{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}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. 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 <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. 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 <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, 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 <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & 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 & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == 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 number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating at the universal rate of quantum events <math>c</math>, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons, considered as single point objects. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in Euclidean 4-space == {{Efn|...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 in 4-space).{{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 a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of physical processes varies from place to place, and reference frame differences predicted by the theory 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, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective screw translations <math>QT</math>, which are distinct. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation through 4-space. The velocity of his vector of translation <math>T</math> through 4-space remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the surface of the rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacements is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == ... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, 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 <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's work: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, 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 have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{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}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "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."}} 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.<ref>{{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}}</ref>}} == Boundaries == <blockquote>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.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>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 "Dimensional Analogy."<ref>{{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=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :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. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::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''<ref>{{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|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{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 "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> 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 "left" 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=; "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]."}} 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^o 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^o bend, or as a straight line with one 60^o 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=; "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.)"}} 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 V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ 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 V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ 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 V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ 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}} V₀ and V₂ 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).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & 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 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} 41r5i5a3652gxn3aemvge1e41qjvh3i 2842475 2842474 2026-10-25T00:09:39Z Dc.samizdat 2856930 /* Summary */ 2842475 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. 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.</blockquote> == Summary == We observe that: * Physical space is Euclidean with four perpendicular dimensions. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|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. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planetary surfaces, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> 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 <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, 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 <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) lies within a smaller 3-sphere which bounds it, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to our manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at its sphere-like core, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> 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 <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> 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 (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>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.<ref>{{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}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. 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 <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. 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 <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, 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 <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & 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 & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == 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 number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating at the universal rate of quantum events <math>c</math>, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons, considered as single point objects. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in Euclidean 4-space == {{Efn|...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 in 4-space).{{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 a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of physical processes varies from place to place, and reference frame differences predicted by the theory 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, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective screw translations <math>QT</math>, which are distinct. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation through 4-space. The velocity of his vector of translation <math>T</math> through 4-space remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the surface of the rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacements is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == ... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, 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 <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's work: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, 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 have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{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}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "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."}} 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.<ref>{{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}}</ref>}} == Boundaries == <blockquote>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.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>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 "Dimensional Analogy."<ref>{{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=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :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. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::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''<ref>{{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|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{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 "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> 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 "left" 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=; "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]."}} 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^o 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^o bend, or as a straight line with one 60^o 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=; "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.)"}} 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 V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ 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 V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ 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 V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ 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}} V₀ and V₂ 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).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & 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 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} 9dfqxl7phsylrxca1dta2n4wradjyif 2842476 2842475 2026-10-25T00:11:28Z Dc.samizdat 2856930 2842476 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. 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.</blockquote> == Summary == We observe that: * Physical space is Euclidean with four perpendicular dimensions. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|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. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light <math>c</math>, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planetary surfaces, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> 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 <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, 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 <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) lies within a smaller 3-sphere which bounds it, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to our manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at its sphere-like core, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> 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 <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> 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 (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>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.<ref>{{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}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. 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 <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. 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 <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, 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 <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & 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 & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == 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 number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating at the universal rate of quantum events <math>c</math>, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons, considered as single point objects. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in Euclidean 4-space == {{Efn|...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 in 4-space).{{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 a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of physical processes varies from place to place, and reference frame differences predicted by the theory 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, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective screw translations <math>QT</math>, which are distinct. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation through 4-space. The velocity of his vector of translation <math>T</math> through 4-space remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the surface of the rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacements is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == ... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, 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 <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's work: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, 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 have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{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}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "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."}} 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.<ref>{{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}}</ref>}} == Boundaries == <blockquote>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.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>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 "Dimensional Analogy."<ref>{{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=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :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. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::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''<ref>{{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|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{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 "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> 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 "left" 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=; "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]."}} 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^o 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^o bend, or as a straight line with one 60^o 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=; "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.)"}} 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 V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ 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 V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ 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 V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ 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}} V₀ and V₂ 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).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & 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 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} atrt4ml64ngjb1tpct4ke5zi0dg6ruo 2842477 2842476 2026-10-25T00:12:24Z Dc.samizdat 2856930 /* Summary */ 2842477 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. 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.</blockquote> == Summary == We observe that: * Physical space is Euclidean with four perpendicular dimensions, not three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|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. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light <math>c</math>, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planetary surfaces, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> 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 <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, 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 <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) lies within a smaller 3-sphere which bounds it, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to our manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at its sphere-like core, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> 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 <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> 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 (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>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.<ref>{{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}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. 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 <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. 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 <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, 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 <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & 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 & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == 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 number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating at the universal rate of quantum events <math>c</math>, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons, considered as single point objects. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in Euclidean 4-space == {{Efn|...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 in 4-space).{{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 a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of physical processes varies from place to place, and reference frame differences predicted by the theory 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, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective screw translations <math>QT</math>, which are distinct. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation through 4-space. The velocity of his vector of translation <math>T</math> through 4-space remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the surface of the rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacements is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == ... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, 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 <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's work: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, 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 have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{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}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "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."}} 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.<ref>{{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}}</ref>}} == Boundaries == <blockquote>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.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>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 "Dimensional Analogy."<ref>{{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=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :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. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::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''<ref>{{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|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{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 "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> 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 "left" 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=; "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]."}} 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^o 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^o bend, or as a straight line with one 60^o 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=; "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.)"}} 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 V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ 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 V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ 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 V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ 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}} V₀ and V₂ 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).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & 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 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} cxof3dyrxei63jd346yk2xwc8cmh5nc 2842478 2842477 2026-10-25T00:12:43Z Dc.samizdat 2856930 /* Summary */ 2842478 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. 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.</blockquote> == Summary == We observe that: * Physical space is Euclidean with four perpendicular dimensions. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|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. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light <math>c</math>, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planetary surfaces, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> 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 <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, 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 <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) lies within a smaller 3-sphere which bounds it, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to our manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at its sphere-like core, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> 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 <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> 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 (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>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.<ref>{{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}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. 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 <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. 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 <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, 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 <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & 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 & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == 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 number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating at the universal rate of quantum events <math>c</math>, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons, considered as single point objects. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in Euclidean 4-space == {{Efn|...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 in 4-space).{{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 a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of physical processes varies from place to place, and reference frame differences predicted by the theory 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, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective screw translations <math>QT</math>, which are distinct. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation through 4-space. The velocity of his vector of translation <math>T</math> through 4-space remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the surface of the rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacements is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == ... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, 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 <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's work: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, 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 have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{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}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "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."}} 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.<ref>{{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}}</ref>}} == Boundaries == <blockquote>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.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>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 "Dimensional Analogy."<ref>{{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=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :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. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::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''<ref>{{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|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{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 "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> 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 "left" 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=; "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]."}} 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^o 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^o bend, or as a straight line with one 60^o 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=; "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.)"}} 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 V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ 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 V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ 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 V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ 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}} V₀ and V₂ 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).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & 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 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} atrt4ml64ngjb1tpct4ke5zi0dg6ruo 2842479 2842478 2026-10-25T00:13:28Z Dc.samizdat 2856930 /* Summary */ 2842479 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. 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.</blockquote> == Summary == We observe that: * Physical space is Euclidean with four perpendicular dimensions. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|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. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light <math>c</math>. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planetary surfaces, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> 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 <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, 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 <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) lies within a smaller 3-sphere which bounds it, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to our manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at its sphere-like core, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> 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 <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> 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 (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>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.<ref>{{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}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. 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 <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. 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 <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, 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 <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & 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 & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == 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 number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating at the universal rate of quantum events <math>c</math>, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons, considered as single point objects. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in Euclidean 4-space == {{Efn|...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 in 4-space).{{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 a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of physical processes varies from place to place, and reference frame differences predicted by the theory 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, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective screw translations <math>QT</math>, which are distinct. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation through 4-space. The velocity of his vector of translation <math>T</math> through 4-space remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the surface of the rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacements is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == ... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, 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 <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's work: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, 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 have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{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}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "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."}} 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.<ref>{{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}}</ref>}} == Boundaries == <blockquote>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.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>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 "Dimensional Analogy."<ref>{{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=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :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. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::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''<ref>{{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|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{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 "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> 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 "left" 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=; "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]."}} 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^o 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^o bend, or as a straight line with one 60^o 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=; "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.)"}} 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 V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ 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 V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ 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 V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ 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}} V₀ and V₂ 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).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & 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 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} e6m9h0gbhats8ykq1hnes9r38klgb9o 2842480 2842479 2026-10-25T00:16:27Z Dc.samizdat 2856930 /* Summary */ 2842480 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. 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.</blockquote> == Summary == We observe that: * Physical space is Euclidean with four perpendicular dimensions. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|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 embedded in Euclidean 4-space. * Our entire 3-space manifold is moving linearly at the speed of light <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planetary surfaces, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> 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 <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, 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 <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) lies within a smaller 3-sphere which bounds it, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to our manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at its sphere-like core, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> 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 <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> 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 (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>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.<ref>{{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}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. 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 <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. 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 <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, 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 <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & 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 & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == 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 number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating at the universal rate of quantum events <math>c</math>, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons, considered as single point objects. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in Euclidean 4-space == {{Efn|...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 in 4-space).{{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 a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of physical processes varies from place to place, and reference frame differences predicted by the theory 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, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective screw translations <math>QT</math>, which are distinct. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation through 4-space. The velocity of his vector of translation <math>T</math> through 4-space remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the surface of the rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacements is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == ... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, 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 <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's work: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, 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 have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{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}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "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."}} 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.<ref>{{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}}</ref>}} == Boundaries == <blockquote>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.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>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 "Dimensional Analogy."<ref>{{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=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :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. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::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''<ref>{{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|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{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 "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> 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 "left" 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=; "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]."}} 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^o 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^o bend, or as a straight line with one 60^o 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=; "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.)"}} 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 V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ 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 V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ 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 V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ 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}} V₀ and V₂ 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).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & 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 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} 22wfe73im6av0bi2d0tn2zyx2y3d7o1 2842483 2842480 2026-10-25T00:18:51Z Dc.samizdat 2856930 /* Summary */ 2842483 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. 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.</blockquote> == Summary == We observe that: * Physical space is Euclidean with four perpendicular dimensions. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|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, embedded in Euclidean 4-space. * Our entire 3-space manifold is moving linearly at the speed of light <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planetary surfaces, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> 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 <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, 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 <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) lies within a smaller 3-sphere which bounds it, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to our manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at its sphere-like core, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> 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 <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> 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 (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>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.<ref>{{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}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. 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 <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. 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 <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, 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 <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & 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 & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == 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 number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating at the universal rate of quantum events <math>c</math>, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons, considered as single point objects. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in Euclidean 4-space == {{Efn|...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 in 4-space).{{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 a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of physical processes varies from place to place, and reference frame differences predicted by the theory 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, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective screw translations <math>QT</math>, which are distinct. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation through 4-space. The velocity of his vector of translation <math>T</math> through 4-space remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the surface of the rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacements is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == ... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, 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 <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's work: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, 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 have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{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}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "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."}} 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.<ref>{{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}}</ref>}} == Boundaries == <blockquote>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.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>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 "Dimensional Analogy."<ref>{{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=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :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. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::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''<ref>{{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|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{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 "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> 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 "left" 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=; "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]."}} 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^o 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^o bend, or as a straight line with one 60^o 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=; "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.)"}} 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 V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ 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 V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ 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 V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ 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}} V₀ and V₂ 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).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & 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 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} qw98lf1r8i4wir1isdeiuow8xuksaoh 2842484 2842483 2026-10-25T00:25:24Z Dc.samizdat 2856930 /* A theory of the Euclidean cosmos */ 2842484 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. 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.</blockquote> == Summary == We observe that: * Physical space is Euclidean with four perpendicular dimensions. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|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, embedded in Euclidean 4-space. * Our entire 3-space manifold is moving linearly at the speed of light <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> 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 <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, 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 <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) lies within a smaller 3-sphere which bounds it, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to our manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at its sphere-like core, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> 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 <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> 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 (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>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.<ref>{{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}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. 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 <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. 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 <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, 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 <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & 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 & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == 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 number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating at the universal rate of quantum events <math>c</math>, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons, considered as single point objects. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in Euclidean 4-space == {{Efn|...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 in 4-space).{{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 a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of physical processes varies from place to place, and reference frame differences predicted by the theory 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, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective screw translations <math>QT</math>, which are distinct. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation through 4-space. The velocity of his vector of translation <math>T</math> through 4-space remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the surface of the rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacements is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == ... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, 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 <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's work: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, 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 have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{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}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "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."}} 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.<ref>{{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}}</ref>}} == Boundaries == <blockquote>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.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>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 "Dimensional Analogy."<ref>{{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=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :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. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::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''<ref>{{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|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{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 "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> 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 "left" 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=; "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]."}} 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^o 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^o bend, or as a straight line with one 60^o 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=; "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.)"}} 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 V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ 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 V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ 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 V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ 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}} V₀ and V₂ 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).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & 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 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} t6fkqbk5vr8m65f4h407iy8pp34f4xl 2842486 2842484 2026-10-25T00:33:35Z Dc.samizdat 2856930 /* A theory of the Euclidean cosmos */ 2842486 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. 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.</blockquote> == Summary == We observe that: * Physical space is Euclidean with four perpendicular dimensions. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|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, embedded in Euclidean 4-space. * Our entire 3-space manifold is moving linearly at the speed of light <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> 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 <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, 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 <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) lies within a smaller 3-sphere which bounds it, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to our manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at its sphere-like core, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity <math>c</math>, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> 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 <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> 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 (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>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.<ref>{{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}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. 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 <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. 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 <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, 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 <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & 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 & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == 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 number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating at the universal rate of quantum events <math>c</math>, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons, considered as single point objects. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in Euclidean 4-space == {{Efn|...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 in 4-space).{{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 a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of physical processes varies from place to place, and reference frame differences predicted by the theory 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, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective screw translations <math>QT</math>, which are distinct. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation through 4-space. The velocity of his vector of translation <math>T</math> through 4-space remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the surface of the rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacements is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == ... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, 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 <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's work: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, 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 have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{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}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "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."}} 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.<ref>{{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}}</ref>}} == Boundaries == <blockquote>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.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>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 "Dimensional Analogy."<ref>{{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=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :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. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::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''<ref>{{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|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{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 "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> 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 "left" 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=; "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]."}} 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^o 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^o bend, or as a straight line with one 60^o 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=; "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.)"}} 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 V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ 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 V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ 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 V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ 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}} V₀ and V₂ 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).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & 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 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} 6fwr0tuecwtnp7rhi1zi1ibagbpkp67 2842489 2842486 2026-10-25T00:54:05Z Dc.samizdat 2856930 2842489 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. 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.</blockquote> == Summary == We observe that: * Physical space is Euclidean with four perpendicular dimensions. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|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, embedded in Euclidean 4-space. * Our entire 3-space manifold is moving linearly at the speed of light <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> 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 <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, 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 <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) lies within a smaller 3-sphere which bounds it, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to our manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at its sphere-like core, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity <math>c</math>, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> 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 <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> 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 (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>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.<ref>{{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}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.<ref name=Einstein1923>{{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}}</ref> Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. 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 <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. 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 <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, 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 <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & 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 & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == 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 number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating at the universal rate of quantum events <math>c</math>, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons, considered as single point objects. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in Euclidean 4-space == {{Efn|...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 in 4-space).{{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 a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of physical processes varies from place to place, and reference frame differences predicted by the theory 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, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective screw translations <math>QT</math>, which are distinct. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation through 4-space. The velocity of his vector of translation <math>T</math> through 4-space remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the surface of the rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacements is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == ... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, 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 <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's work: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, 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 have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref name=Einstein1923/> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "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."}} 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.<ref>{{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}}</ref>}} == Boundaries == <blockquote>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.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>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 "Dimensional Analogy."<ref>{{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=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :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. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::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''<ref>{{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|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{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 "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> 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 "left" 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=; "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]."}} 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^o 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^o bend, or as a straight line with one 60^o 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=; "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.)"}} 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 V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ 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 V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ 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 V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ 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}} V₀ and V₂ 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).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & 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 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} oomsr3ty8a38e73xacuxklncnuohays 2842491 2842489 2026-10-25T00:58:42Z Dc.samizdat 2856930 /* Summary */ 2842491 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. 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.</blockquote> == Summary == We observe that: * Physical space is Euclidean with four perpendicular dimensions. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|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 embedded in Euclidean 4-space. * Our entire 3-space manifold is moving linearly at the speed of light <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> 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 <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, 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 <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) lies within a smaller 3-sphere which bounds it, lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to our manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at its sphere-like core, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity <math>c</math>, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> 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 <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> 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 (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>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.<ref>{{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}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.<ref name=Einstein1923>{{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}}</ref> Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. 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 <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. 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 <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, 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 <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & 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 & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == 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 number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating at the universal rate of quantum events <math>c</math>, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons, considered as single point objects. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in Euclidean 4-space == {{Efn|...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 in 4-space).{{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 a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of physical processes varies from place to place, and reference frame differences predicted by the theory 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, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective screw translations <math>QT</math>, which are distinct. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation through 4-space. The velocity of his vector of translation <math>T</math> through 4-space remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the surface of the rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacements is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == ... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, 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 <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's work: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, 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 have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref name=Einstein1923/> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "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."}} 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.<ref>{{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}}</ref>}} == Boundaries == <blockquote>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.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>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 "Dimensional Analogy."<ref>{{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=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :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. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::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''<ref>{{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|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{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 "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> 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 "left" 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=; "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]."}} 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^o 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^o bend, or as a straight line with one 60^o 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=; "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.)"}} 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 V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ 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 V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ 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 V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ 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}} V₀ and V₂ 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).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & 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 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} j4pfy7vjmlfugxk0ikoep5ft1smc2lq 2842493 2842491 2026-10-25T01:12:42Z Dc.samizdat 2856930 /* A theory of the Euclidean cosmos */ 2842493 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. 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.</blockquote> == Summary == We observe that: * Physical space is Euclidean with four perpendicular dimensions. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|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 embedded in Euclidean 4-space. * Our entire 3-space manifold is moving linearly at the speed of light <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> 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 <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, 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 <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to our manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at its sphere-like core, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity <math>c</math>, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> 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 <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> 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 (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>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.<ref>{{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}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.<ref name=Einstein1923>{{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}}</ref> Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. 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 <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. 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 <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, 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 <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & 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 & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == 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 number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating at the universal rate of quantum events <math>c</math>, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons, considered as single point objects. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in Euclidean 4-space == {{Efn|...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 in 4-space).{{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 a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of physical processes varies from place to place, and reference frame differences predicted by the theory 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, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective screw translations <math>QT</math>, which are distinct. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation through 4-space. The velocity of his vector of translation <math>T</math> through 4-space remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the surface of the rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacements is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == ... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, 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 <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's work: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, 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 have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref name=Einstein1923/> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "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."}} 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.<ref>{{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}}</ref>}} == Boundaries == <blockquote>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.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>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 "Dimensional Analogy."<ref>{{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=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :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. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::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''<ref>{{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|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{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 "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> 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 "left" 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=; "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]."}} 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^o 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^o bend, or as a straight line with one 60^o 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=; "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.)"}} 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 V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ 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 V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ 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 V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ 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}} V₀ and V₂ 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).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & 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 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} qxmavxmpbuo42heq7vmhhm82u80x8ab 2842494 2842493 2026-10-25T01:19:51Z Dc.samizdat 2856930 /* A theory of the Euclidean cosmos */ 2842494 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - October 2026}} <blockquote>'''Abstract:''' The physical universe is properly visualized as real Euclidean space <math>\mathbb{R}^4</math> of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are 4-polytopes, small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. We ourselves and the objects we observe near us are only 3-dimensional, but nonetheless we can see into four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. 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.</blockquote> == Summary == We observe that: * Physical space is Euclidean with four perpendicular dimensions. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|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 embedded in Euclidean 4-space. * Our entire 3-space manifold is moving linearly at the speed of light <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets (at their surfaces), are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> 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 <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. Several objects which appear to us to be located at almost the same point in our sky may actually lie in directions many degrees apart, since the entire hyperspherical sky is projected onto an ordinary sphere in our view of it. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, 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 <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around the galaxy's barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system too small to have their own gravity lie entirely within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <math>c</math>. We ourselves live within such an expanding 3-dimensional surface, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience. The earth and all the planets lie on it. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to all three of our 3-manifold's dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. In fewer places still it is occupied by large lumps of matter which exert a significant gravitational attraction, such as planets and their satellites. These round gravitational objects are thicker than one atom in their fourth dimension, but not uniformly so. They are disk-like 4-dimensional objects, 3-balls that are flattish in the fourth dimension. They are typically only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, particularly if their gravity is strong enough to give them a core of superdense matter, like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "heaped up" into larger 4-dimensional objects that form a lump in its moving surface. We have only indirect experience of such 4-dimensional lumps of matter thicker than one atom, since they occur naturally in our vicinity only in regions of superdense matter, such as the core of the earth or the interior of the sun. Deep in the core of the earth under enormous gravitational pressure peculiar states of matter have been observed, defying our understanding of how 3-dimensional crystalline structures behave. That is because they are not 3-dimensional structures, they are 4-dimensional crystals. The earth is a disk-like 4-dimensional object that is thin at its edge but thicker at its sphere-like core, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity <math>c</math>, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us, and the greater their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> 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 <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> 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 (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by [[W:Geometric algebra|Clifford's algebraic methods]]), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>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.<ref>{{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}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.<ref name=Einstein1923>{{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}}</ref> Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. 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 <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. 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 <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, 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 <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & 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 & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == 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 number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object in free fall is translating on a <math>\mathrm{QT}</math> at velocity <math>c</math>. That universal inertial reference frame is not the proper reference frame of any object or observer, wherein the object or observer is stationary and double-rotating at the universal rate of quantum events <math>c</math>, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons, considered as single point objects. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in Euclidean 4-space == {{Efn|...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 in 4-space).{{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 a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of physical processes varies from place to place, and reference frame differences predicted by the theory 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, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. An observer standing on the surface of the earth is accelerated at the rate of 32'/second/second. The observer and the earth are both stationary in their proper inertial reference frames (which are similar except for the rotation of the earth), so the observer's acceleration cannot be perceived as motion in either proper reference frame; it is perceptible there instead as the force of gravity. It is only perceptible as motion in some other reference frame in which the earth and the observer are both in motion, such as the universal common reference frame of Euclidean 4-coordinate space in which they are both in motion at velocity <math>c</math> on their respective screw translations <math>QT</math>, which are distinct. In that universal common reference frame the observer's acceleration of 32'/second/second is the rate of change in the direction of his vector of translation through 4-space. The velocity of his vector of translation <math>T</math> through 4-space remains the constant <math>c</math> throughout; only his direction is changing. A change in the direction of a vector is a rotation <math>Q</math> of a translation <math>T</math>: a screw translation <math>QT</math>. A change in the direction of a vector is a rotation which is an acceleration. We can be precise about the units of measurement of this acceleration at atomic scale. In Coxeter's kinematics rotations of geometric objects occur in discrete increments called rotational displacements, with each distinct kind of rotation performing a characteristic number of its characteristic rotational displacement during a full revolution that returns the object to its starting orientation. There are many distinct kinds of rotation, each with its characteristic rotational displacement and number of displacements per revolution. The general rotation in 4-space is a double rotation <math>Q^2</math> distinguished by two distinct completely orthogonal angles through which the object turns in each rotational displacement. The angles determine not only the distance through which the object turns, but the characteristic trajectory of each point on the surface of the rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference. Many distinct 4-space rotations of interest have a circumference of 720°, and many are larger. In 4-space every object with mass moves on its distinct <math>QT</math> trajectory. Each rotational displacement <math>Q</math> is equivalent to a pair of reflection displacements in two surfaces which intersect each other, at angles related to a distinct star polygon. Each translational displacement <math>T</math> is equivalent to a pair of reflection displacements in two surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <math>c</math>. Every atom translates at velocity <math>c</math>, performing a large number of atomic scale translational displacements per second. Concurrently, the atom rotates at velocity <math>c</math>, performing the same large number of atomic scale rotational displacements per second. Although the number of displacements at atomic scale per unit time is the same universal constant <math>c</math> for all rotations and translations, distinct kinds of rotation revolve at distinct rates, because the number of rotational displacements per complete revolution and the circumference of the complete revolution are distinct characteristics of each kind of rotation. The velocity at which each point on the surface of a rotating object moves, and the frequency of its rotational circling, are functions of the distinct rotation it is performing. However, they are not simple functions of <math>c</math> and the two defining angle parameters of the rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with a unique conservation law. The relationships between those distinct conservation laws are even more mysterious, and have been shown to be chaotic. Steinbach{{Sfn|Steinbach|1997}} discovered the Golden Fields family of ratios which govern distinct rotational displacements related to the regular star polygons at least up to the hendecagon {11}. Kappraff and Adamson{{Sfn|Kappraff|Adamson|2004}} further characterized the families of rotation functions and showed that they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. The inertial tendency of mass is to follow a geodesic. The inertial conservation law of gravity applies to all motions of objects with mass, rotations as well as translations. A translation <math>T</math> is the distinct simple rotation <math>Q</math> in which the characteristic angle of rotational displacement is zero. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacements is the universal constant <math>c</math>, the rate of change of direction (the acceleration of the object) is a function of <math>c</math> and the distinct rotation's conservation law. A observer standing on the surface of the earth is performing a distinct double rotation characterized by two completely orthogonal angles of rotational displacement. The observer's acceleration of 32'/second/second is a function of <math>c</math> and the distinct rotation's conservation law, and a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == ... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <math>n</math>-dimensional congruent transformations in Euclidean space may be called ''dimensional relativity'', since it captures the theories of special and general relativity, and has its roots in dimensional analogy. Dimensional analogy is the exploration of [[w:Hermann_Grassmann#Mathematician|Hermann Grassmann's vector space principle]], in which space cannot be limited to any finite number of dimensions. The geometry of higher-dimensional space is accessible by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated, and by algebraic methods after [[W:William Kingdon Clifford|Clifford]]. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, 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 <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of dimensional relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the Euclidean <math>n</math>-space universe in which they physically locate themself. We see that every observer may also properly view themself as stationary and the universe as an <math>n</math>-sphere with themself at the center observing it, perceptually equidistant from all points on its <math>n</math>-space surface, including their own physical location which is one of those surface points, distinguished to them but moving on the surface, and not the center of anything. This ''polycentric model'' of the universe is a further restatement of the principle of relativity, and of the [[W:Copernican principle|Copernican principle]] that observation is democratic: observers occupy representative viewpoints, not privileged monocentric positions. The polycentric model is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's work: <blockquote>The universe is a sphere whose center is wherever there is intelligence.{{Sfn|Thoreau|1849|p=349|ps=; "The universe is a sphere whose center is wherever there is intelligence." [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polyschemes in any number of dimensions.]}}</blockquote> == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In 1600 [[W:Giordano Bruno|Giordano Bruno]] who first openly advocated these views was burned at the stake in Rome's [[W:Campo de' Fiori|Campo de' Fiori]] by the Catholic inquisition. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <math>c</math>, with their big bang origins occurring at distinct times and places in the ''n''-dimensional universe. The most distant objects we see when we look up at night may not all have the same origin in space and time. As recently as Copernicus we believed all the stars lay on a single 2-sphere embedded in Euclidean 3-space, with our sun at its center. During the enlightenment observers beginning with Bruno dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed that our 3-space could not be Euclidean, that it must be a 3-manifold curved in every place in obedience to Newton's inverse-square law of gravity; and in a geometry related to time, at least, it must be 4-dimensional. In the present work we propose a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, 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 have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <math>n</math>-space as one of many 4-ball galaxies, each translating in a distinct direction through 4-space at velocity <math>c</math>, on more or less divergent paths from each other. But only closer observation will reveal evidence of whether everything we see lies in the same Euclidean 4-space, or if it is distributed in five or more dimensions, and how it is moving there. To remain in agreement with the theory of relativity, the Euclidean four-dimensional viewpoint requires that all mass-carrying objects be in motion in some distinct direction through 4-space at the constant velocity <math>c</math>, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Although their paths away from their origin are not straight lines but various helical isoclines (screw translation spirals), nearby objects must be translating radially at the same average velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw translation has two [[w:Completely_orthogonal|completely orthogonal]] components of motion in 4-space, an orbital rotation (such as the earth's around our sun) and a linear translation of the entire system at velocity <math>c</math> in the direction of the original 3-sphere's radial expansion (along the system's proper time vector). Of course the view from our solar system does not suggest that the 3-sphere enclosing each galaxy is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref name=Einstein1923/> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified non-Euclidean spacetime of four dimensions. Anco and Maghadam found that <math>SO(4)</math> breaks to <math>\mathbb{S}^3</math> (the 3-sphere) if the energy in a Kepler orbit is negative (an elliptical orbit), and to <math>H^3</math> (Minkowski spacetime) if the energy is positive (a hyperbolic orbit). Therefore the fact that the planets orbit on ellipses in our 3-space suggests that they orbit on a 3-sphere embedded in Euclidean 4-space. Thus we can conclude that <math>\mathbb{R}^4</math> is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction of it; the reciprocal of Einstein's disclaimer is the more direct physical model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position as our exclusive conception of our place in space.{{Efn| ...origins of the Euclidean 4-space insight in the observations of Fock, Atkinson, Moser and others.}} The invention of Euclidean geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years, when it was worked out originally by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] before 1853.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; "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."}} 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.<ref>{{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}}</ref>}} == Boundaries == <blockquote>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.<ref>{{Cite book|author=Carlo Rovelli|author-link=W:Carlo Rovelli|title=Seven Brief Lessons on Physics|publisher=Riverhead|year=2016|isbn=978-0399184413}}</ref></blockquote> Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our 3-sphere surface in our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the planet's core and the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest 4-ball object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined objects and motions. So our provincial conception of our place in it now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and no longer a proper explanation. An inertial boundary is a proper explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <blockquote>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 "Dimensional Analogy."<ref>{{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=}}</ref></blockquote> I believe, but I cannot prove, that we live in real space, which is Schläfli's Euclidean space of ''n'' analogous dimensions. As Grassmann showed first, space cannot be limited to any finite number of dimensions. There will always be higher dimensions to discover, first in imagination and then to explore physically, each an astonishing new enlightenment.<ref>{{Cite book|first=T.S.|last=Eliot|title=Little Gidding|volume=Four Quartets|year=1943}}<blockquote> :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. </blockquote></ref> Schläfli discovered every regular convex polytope that exists in any dimension, but that was only the beginning of the story of dimensional analogy, not its end or even the end of its beginning. That project is forever beginning anew. Coxeter discovered that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether discovered all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, so symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a wilderness project, and just at its beginning, but already we observe a Euclidean space of four or more orthogonal spatial dimensions in which all objects with mass move ceaselessly at the constant velocity <math>c</math>, the universal rate at which everything moves, quantum events occur, and each of our proper times evolves. I believe these facts explain the experimentally verified theories of relativity and quantum mechanics, by revealing their unified polycentric geometry, the same way the facts about Copernicus's heliocentric solar system explained the observed motions of the planets, by revealing the geometry of gravity to Kepler. But others will have to do the math, work out the physics, and perform experiments to prove or disprove all of this, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages. <blockquote> ::::::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''<ref>{{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|}}</ref> </blockquote> == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == ... == {{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}} {{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} {{Efn|name=radially equilateral}} {{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as "the convex hull of the neighbouring vertices of a given vertex".{{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 "radius" equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}} {{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example: {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0) {{indent|5}}({{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}({{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|5}}(–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}},–{{sfrac|1|2}}){{spaces|3}}(–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}},–{{sfrac|1|2}},{{spaces|2}}{{sfrac|1|2}}) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)<br> 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 "left" 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=; "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]."}} 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^o 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^o bend, or as a straight line with one 60^o 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=; "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.)"}} 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 V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ 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 V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ 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 V₀ and V₂ are ''two'' {{radic|3}} chords apart{{Efn|V₀ and V₂ 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}} V₀ and V₂ 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).}}, P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V₀ to V₃ is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}} {{Sfn|Mamone, Pileio & 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 𝐹₄.}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|title=A Week on the Concord and Merrimack Rivers|last=Thoreau|first=Henry David|author-link=W:Thoreau|publisher=James Munroe and Company|year=1849|isbn=|location=Boston|ref={{SfnRef|Thoreau|1849}}}} * {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1|ref={{SfnRef|Yamashita|2023}}}} * {{Cite_arXiv | arxiv=2512.02903v2 | date=2 January 2026 | title=Symmetry transformation group arising from the Laplace–Runge–Lenz vector | first1=Stephen C. | last1=Anco | first2=Mahdieh Gol Bashmani | last2=Moghadam | class=math-ph}} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} hfb9lu23pazhaizcd3u5z2z4jxd5a2n 0 305491 2842440 2841633 2026-10-24T15:13:22Z ~2026-56983-83 3113232 /* */ 2842440 wikitext text/x-wiki {{Dynamic List}} {| class="wikitable" !Generation !Year Range |- |Arthurian |1433-1460 (28) |- |Humanist |1461-1482 (22) |- |Reformation |1483-1511 (29) |- |Reprisal |1512-1540 (29) |- |Elizabethan |1541-1565 (25) |- |Parliamentary |1566-1587 (22) |- |Puritan |1588-1617 (30) |- |Cavalier |1618-1647 (30) |- |Glorious |1648-1673 (26) |- |Enlightenment |1674-1700 (27) |- |Awakening |1701-1723 (23) |- |Liberty |1724-1741 (18) |- |Republican |1742-1766 (25) |- |Compromise |1767-1791 (25) |- |Transcendental |1792-1821 (30) |- |Gilded |1822-1842 (21) |- |Progressive |1843-1859 (17) |- |Missionary |1860-1882 (23) |- |Lost |1883-1900 (18) |- |G.I. |1901-1927 (27) |- |Silent |1928-1945 (18) |- |Baby Boom |1946-1964 (19) |- |Generation X |1965-1979 (15) |- |Millennial |1980-1994 (15) |- |Generation Z |1995-2009 (15) |- |Generation Alpha |2010-2024 (15) |- |Generation Beta |2025-2039 (15) |} == Notes == [a] Mccrindle's 'Builders' align best with other models' Silent Generation [b] Mccrindle's 'Generation Y' align best with other models' Millennial Generation [c] Whilst not explicit, any direct successor of Generation Z proposed by Pew Research Center must align best with other Mccrindle's 'Gen Alpha' and S&H's 'Homeland Generation' == References == https://mccrindle.com.au/article/topic/demographics/the-generations-defined/ https://www.pewresearch.org/short-reads/2019/01/17/where-millennials-end-and-generation-z-begins/ https://en.wikipedia.org/wiki/Strauss%E2%80%93Howe_generational_theory https://www.genwhy.com/_files/ugd/5ccc25_695074dbc5724d76ae103635b5d9ab5d.pdf 2hj8dl0iu8vqrhzug36wm04foizw9w1 2842441 2842440 2026-10-24T15:14:47Z ~2026-56983-83 3113232 /* Notes */ 2842441 wikitext text/x-wiki {{Dynamic List}} {| class="wikitable" !Generation !Year Range |- |Arthurian |1433-1460 (28) |- |Humanist |1461-1482 (22) |- |Reformation |1483-1511 (29) |- |Reprisal |1512-1540 (29) |- |Elizabethan |1541-1565 (25) |- |Parliamentary |1566-1587 (22) |- |Puritan |1588-1617 (30) |- |Cavalier |1618-1647 (30) |- |Glorious |1648-1673 (26) |- |Enlightenment |1674-1700 (27) |- |Awakening |1701-1723 (23) |- |Liberty |1724-1741 (18) |- |Republican |1742-1766 (25) |- |Compromise |1767-1791 (25) |- |Transcendental |1792-1821 (30) |- |Gilded |1822-1842 (21) |- |Progressive |1843-1859 (17) |- |Missionary |1860-1882 (23) |- |Lost |1883-1900 (18) |- |G.I. |1901-1927 (27) |- |Silent |1928-1945 (18) |- |Baby Boom |1946-1964 (19) |- |Generation X |1965-1979 (15) |- |Millennial |1980-1994 (15) |- |Generation Z |1995-2009 (15) |- |Generation Alpha |2010-2024 (15) |- |Generation Beta |2025-2039 (15) |} == References == https://mccrindle.com.au/article/topic/demographics/the-generations-defined/ https://www.pewresearch.org/short-reads/2019/01/17/where-millennials-end-and-generation-z-begins/ https://en.wikipedia.org/wiki/Strauss%E2%80%93Howe_generational_theory https://www.genwhy.com/_files/ugd/5ccc25_695074dbc5724d76ae103635b5d9ab5d.pdf 292tc5dcehn5o9i9gyf3ltubllnvoya 2842442 2842441 2026-10-24T15:16:32Z ~2026-56983-83 3113232 /* References */ 2842442 wikitext text/x-wiki {{Dynamic List}} {| class="wikitable" !Generation !Year Range |- |Arthurian |1433-1460 (28) |- |Humanist |1461-1482 (22) |- |Reformation |1483-1511 (29) |- |Reprisal |1512-1540 (29) |- |Elizabethan |1541-1565 (25) |- |Parliamentary |1566-1587 (22) |- |Puritan |1588-1617 (30) |- |Cavalier |1618-1647 (30) |- |Glorious |1648-1673 (26) |- |Enlightenment |1674-1700 (27) |- |Awakening |1701-1723 (23) |- |Liberty |1724-1741 (18) |- |Republican |1742-1766 (25) |- |Compromise |1767-1791 (25) |- |Transcendental |1792-1821 (30) |- |Gilded |1822-1842 (21) |- |Progressive |1843-1859 (17) |- |Missionary |1860-1882 (23) |- |Lost |1883-1900 (18) |- |G.I. |1901-1927 (27) |- |Silent |1928-1945 (18) |- |Baby Boom |1946-1964 (19) |- |Generation X |1965-1979 (15) |- |Millennial |1980-1994 (15) |- |Generation Z |1995-2009 (15) |- |Generation Alpha |2010-2024 (15) |- |Generation Beta |2025-2039 (15) |} giy71tr2p3xbmf5ynqq8tbvjhsd94jp 2842444 2842442 2026-10-24T17:35:39Z ~2026-56867-06 3113236 Gen Alpha starts in 2013 2842444 wikitext text/x-wiki {{Dynamic List}} {| class="wikitable" !Generation !Year Range |- |Arthurian |1433-1460 (28) |- |Humanist |1461-1482 (22) |- |Reformation |1483-1511 (29) |- |Reprisal |1512-1540 (29) |- |Elizabethan |1541-1565 (25) |- |Parliamentary |1566-1587 (22) |- |Puritan |1588-1617 (30) |- |Cavalier |1618-1647 (30) |- |Glorious |1648-1673 (26) |- |Enlightenment |1674-1700 (27) |- |Awakening |1701-1723 (23) |- |Liberty |1724-1741 (18) |- |Republican |1742-1766 (25) |- |Compromise |1767-1791 (25) |- |Transcendental |1792-1821 (30) |- |Gilded |1822-1842 (21) |- |Progressive |1843-1859 (17) |- |Missionary |1860-1882 (23) |- |Lost |1883-1900 (18) |- |G.I. |1901-1927 (27) |- |Silent |1928-1945 (18) |- |Baby Boom |1946-1964 (19) |- |Generation X |1965-1979 (15) |- |Millennial |1980-1994 (15) |- |Generation Z |1995-2012 (17) |- |Generation Alpha |2013-2024 (11) |- |Generation Beta |2025-2039 (15) |} tbphof4sob55od9lqlysg6p6mdqk7hq 2842455 2842444 2026-10-24T20:26:57Z ~2026-56983-83 3113232 /* */ 2842455 wikitext text/x-wiki {{Dynamic List}} {| class="wikitable" !Generation !Year Range |- |Arthurian |1433-1460 (28) |- |Humanist |1461-1482 (22) |- |Reformation |1483-1511 (29) |- |Reprisal |1512-1540 (29) |- |Elizabethan |1541-1565 (25) |- |Parliamentary |1566-1587 (22) |- |Puritan |1588-1617 (30) |- |Cavalier |1618-1647 (30) |- |Glorious |1648-1673 (26) |- |Enlightenment |1674-1700 (27) |- |Awakening |1701-1723 (23) |- |Liberty |1724-1741 (18) |- |Republican |1742-1766 (25) |- |Compromise |1767-1791 (25) |- |Transcendental |1792-1821 (30) |- |Gilded |1822-1842 (21) |- |Progressive |1843-1859 (17) |- |Missionary |1860-1882 (23) |- |Lost |1883-1900 (18) |- |Greatest |1901-1927 (27) |- |Silent |1928-1945 (18) |- |Baby Boom |1946-1964 (19) |- |Generation X |1965-1979 (15) |- |Millennial |1980-1994 (15) |- |Generation Z |1995-2009 (15) |- |Generation Alpha |2010-2024 (15) |- |Generation Beta |2025-2039 (15) |} bovkj4ls1k77ns1fjdrgnhiwimwhvs7 2842555 2842455 2026-10-25T10:45:37Z ~2026-56983-83 3113232 /* */ 2842555 wikitext text/x-wiki {{Dynamic List}} {| class="wikitable" !Generation !Year Range |- |Arthurian |1433-1460 (28) |- |Humanist |1461-1482 (22) |- |Reformation |1483-1511 (29) |- |Reprisal |1512-1540 (29) |- |Elizabethan |1541-1565 (25) |- |Parliamentary |1566-1587 (22) |- |Puritan |1588-1617 (30) |- |Cavalier |1618-1647 (30) |- |Glorious |1648-1673 (26) |- |Enlightenment |1674-1700 (27) |- |Awakening |1701-1723 (23) |- |Liberty |1724-1741 (18) |- |Republican |1742-1766 (25) |- |Compromise |1767-1791 (25) |- |Transcendental |1792-1821 (30) |- |Gilded |1822-1842 (21) |- |Progressive |1843-1859 (17) |- |Missionary |1860-1882 (23) |- |Lost |1883-1900 (18) |- |Interbellum |1901-1927 (27) |- |Silent |1928-1945 (18) |- |Baby Boom |1946-1964 (19) |- |Baby Bust |1965-1979 (15) |- |Millennial |1980-1994 (15) |- |Pluralist |1995-2009 (15) |- |Generation Alpha |2010-2024 (15) |- |Generation Beta |2025-2039 (15) |} ibdhexzhx9ezxuaydz1vh83losnaooy 0 307211 2842519 2841826 2026-10-25T03:17:34Z Jtneill 10242 Copyediting 2842519 wikitext text/x-wiki {{title|Youth environmental activism motivation:<br>What motivates young people to engage in environmental activism?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}} [[File:Chilean people protesting for climate action.jpg|thumb|'''Figure 1.''' Chilian's{{sp}}{{g}} protesting against mine's{{g}} in Chile]] '''Case Study 1: Environmental Activism in Chile''' Recently {{when}} there have been growing concerns about mining and its risks for individuals as well as the environment. The mining of gold,{{g}} has a high environmental impact. This impact has increased over time as new and riskier technologies have been developed (Mudd, 2007). One of the biggest environmental impacts of open pit mining is high water consumption and acid mine drainage (Urkidi, 2010). Socially studies {{g}} have shown that mining areas tend to have comparatively higher levels of economic inequity, AIDS, alcoholism, prostitution and child labour, accentuating poverty and social conflict (Pegg, 2006). In Chile from 2009 to 2019, Scherman and colleagues (2022) wrote about a fluctuation of environmental activism in Chili {{sp}} due to the above conditions. During this period they saw a drastic rise in younger generations dominating among environmental activists (Scherman et al., 2022). This correlated with social media and different ways to protest through instant messaging and sharing ideas. Scherman and colleagues (2022) found social media use was positively associated with participation in environmental issues. Because of the large scale of protesters, they were able to block mining and energy work from being continued. What other motivators do youth have for environmental activism? {{RoundBoxBottom}} Climate change is one of the largest threats to the planet and humankind and can no longer be ignored, therefore urgent climate action is required to transition to more sustainable societies {{f}}. Combating climate change and its impact is now one of the main goals of the UN Agenda 2030 (Romano et al., 2023). One way climate action can be achieved is with environmental activism. Activism is often described as an effort by individuals or a group of people to stand up to injustice or stand for a cause whether it be social, political, or environmental (environmental activism) with an ultimate aim to achieve societal change (Sherrod, 2006). Everyday activism can be understood as changing, adapting or disrupting one’s own and others’ everyday practices in individual and/or collective ways (Walker, 2017). Environmental issues have risen to the top of young people’s priorities (Sloam et al., 2022). This may be due to young people having to in the future experience stronger consequences of climate change than older people are facing today (Wallis & Loy, 2021). This means that youth and young adults play an important role in creating a much needed major cultural shift toward sustainability and environmental justice. More and more young people are joining collective pro-environmental movements, raising their voices against the current inadequate sustainable policies and being the main actors of change {{f}}. The climate strikes that took place across the world in 2018 and 2019 showed that young people are at the leading force of action on climate change (Sloam et al., 2022). Young people have expressed their concern about their futures and their voices are now at the forefront of climate activism, leading them to become official stakeholders at COP27 (Hohenhaus et al., 2023). Existing literature tells us about how young people engage in environmental politics (Wahlström et al. 2019; Pickard, Bowman, and Arya 2022), but we know relatively little about why they do (or do not) become engaged, who these young people are, or the extent to which engagement in environmental issues varies across different countries (Sloam et al., 2022). Thoughout this chapter I will try to explain some of the motivations we can see. Young people are shown to be motivated to engage in environmental activism for multiple reasons. These include rise in the threat climate change, future security, social media, culture, education, school environment, emotions, psychological reasons, self efficacy, Identity, role models, family history, culture, school environment, and exposure to mass media (Ménard 2010; Andolina et al. 2003; Hohenhaus et al., 2023). These motivations will be discussed more throughly in the following chapter. Those participating in climate activism assist in slowing climate change and encouraging others to act. '''Problem statement''': Although Climate change awareness is increasing in all people including youth, the earth is still heating rapidly. How can individuals especially youth help combat this? {{RoundBoxTop|theme=3}} '''Focus questions''' * What is [https://www.liberties.eu/en/stories/activism/44871 activism] {{ic|Move embedded external links to the External inks section}}? * What is [https://www.liberties.eu/en/stories/activism/44871 environmental activism]? * What is [[motivation]]? * Who are considered [https://www.un.org/esa/socdev/documents/youth/fact-sheets/youth-definition.pdf youth?] * Why are more youth participating in activism? * What is motivating youth to engage in activism? * What role does nature, nurture and culture play on youth activism? * What is the relationship between education and environmental activism? * How can we motivate young people to engage in environmental activism? {{RoundBoxBottom}} <quiz display="simple"> Why were there protests in Chili {{sp}}: |type="()"} + Risks associated with mining - Climate change -Bushfires -Political corruption </quiz> <quiz> {The protests in Chili{{sp}} got young people involved due to social media: |type="()"} +True - False </quiz> <quiz> {What motivates young people to participate in environmental activism: |type="()"} - Social media - Education - Rise in the climate change threat + All of the above </quiz> == History of Activism & Youth Environmental Activism == History has a long record of activism and environmental activism that we will brush over to get an idea of why people protest, how long this has being going on and when environmental and youth activism came in. === Early history of activism === The roots of activism can be traced back to the Enlightenment era during the 17th and 18th centuries (Global Society World News., 2024) when individual's{{g}} started thinking for themselves and protesting for better conditions. From there the first form of environmental activism can be traced back to ancient civilizations, where practices like soil conservation were established as early as 2,000 years ago in China, India, and Peru {{f}}. Then the modern environmental movement began to take shape in the late 19th century, primarily in response to pollution and industrialization, leading to significant legislative changes and public awareness (Elliot, Britannica., 2026 {{ic|Use APA style for citations}}). === Environmental activism in the 21st century === After the start of environmental activism in the 19th century {{g}} in 2018 {{ic|Huh? This is the 21st century.}} youth around the world united to demand greater action on climate change (Brugger et al., 2020). A key characteristic of the climate strike movement is that it has been founded and sustained by young people (Brugger et al., 2020).These movements have played a significant role in shaping history and pushing for social and political change (Luttrell, 2023). This is partly due to more in depth knowledge about our environment and climate change being spread rapidly through social media and television that earlier movements did not have. Climate activism has been around for much longer than 2010-2019. However, there was a clear shift in the amount of people listening these years, and the way it was portrayed in the media. Throughout these years it was clear that it {{awkward}} climate change was happening. It wasn't a concern, it wasn't something oncoming, it was here. There was an emergence of not only regular climate protest (See Figure 1), but they {{who}} began to emerge on a global scale (Scherman et al., 2022). Rather than them staying entirely within individual groups, social media forced it to be within the public conscious {{f}}. Now we know how activism and environmental activism began but how did these youth become motivated for change? {{RoundBoxTop|theme=3}} '''Key points''' *Activism has been happening for thousands of years *Environmental activism is relatively new starting in the 19th century *Activism always occurs to help fight some sort of perceived injustice {{f}} *Due to more visible effects of climate change {{g}} *Young people have not always been interested in activism {{f}} {{RoundBoxBottom}} <quiz display="simple"> When was the first form of activism: |type="()"} - Started in 2018 + 2,000 years ago - During the Enlightenment era <quiz> <quiz> {Youth founded the Climate strike movement: |type="()"} +True - False </quiz> == The {{what}} landscape == The current landscape that requires and encourages environmental action is one of rapid climate change, empowered youth, social change and publicized climate action. There has been growing acceptance of the climate emergency and empowered young political representatives such as United States Congresswoman Alexandra Ocasio-Cortez {{ic|Embed link to Wikipedia article}}, whose advocacy for a green deal directly influenced the environmental turn of the Biden administration (Sloam et al., 2022). Youth strikes on climate change have grown rapidly, calling for change and increasing public awareness and concern which have achieved much in a short time such as extensive media attention, support from almost 27,000 scientists, and the foundation of numerous like-minded regional, national, and international groups such as “parents for future” and “artists for future.” (Brugger et al., 2020). The current landscape sees public figures calling for change such as Greta Thunberg and movements such as Fridays for Future (FFF) {{ic|Embed links to relevant Wikipedia article}}. === Fridays for Future === Against this background, large demonstrations and school strikes for climate protection started all over the globe, called the Fridays For Future (FFF) movement (Wallis & Loy, 2021). The movement protested against climate change inaction and was organized by [[wikipedia:Greta_Thunberg|Greta Thunberg]]. The young activists of Fridays For Future (FFF) received attention because of their consistent protests every week in many different locations. At the strikes on 20 and 27 September 2019, more than 3 million people were reported to have participated in the strikes, which took place in more than 150 countries. This shows how serious, united and widespread the young people are doing something themselves not waiting for politicians to catch up. These young activists also received attention because of their age and their speeches, which embarrassed political and economic leaders (Kowasch et al., 2021). === Greta Thunberg === [[File:Thunberg.webp|thumb|'''Figure 2.''' Swedish environmental activist Greta Thunberg|150x150px]] Greta Thunberg (Figure 2) has being instrumental in youth environmental activism due to her outspoken views and coverage. Greta is someone a lot of young activists look up to and who has encouraged climate movements throughout the world. She has told world leaders at Davos Economic Forum in 2019: “I don’t want you to be hopeful, I want you to panic.”. The young Swedish activist has initiated a students’ movement and Fridays for Future, arguing that climate emergency does not allow for further delays: political leaders should take radical social-ecological transformation measures (Kowasch et al., 2021). <quiz display="simple"> Who is Greta Thunberg{{g}}: |type="()"} - The activist behind Fridays For Future - A congresswoman for Bidens{{g}} administration - A Swedish activist who initiated a students movement and who stated a climate emergency does not allow for delays + Both A & C are correct <quiz> {What did FFF {{ic|Explain abbreviation}} protest against: |type="()"} - For climate change + Against climate change inaction </quiz> {{RoundBoxTop|theme=3}} '''Key Points''' *Youth groups formed to encourage activism *Spread worldwide *Role model of Greta Thunberg *Landscape that encourages and needs environmental action *Shift in activism due to social media {{RoundBoxBottom}} == Psychological Motivations behind Youth Environmental Activism == Psychological motivations refer to the motivations a person has that are not always logical but are about how they are feeling and individuals {{g}} behaviours. Two psychological motivations behind youth environmental activism include the Theory of Planned Behaviour (TPB) {{f}} and emotional motivation {{f}}. === Theory of Planned Behaviour === The Theory of Planned Behavior (TPB) is a psychological model developed by Ajzen that links individual belief to human action It states that the best predictor of human behavior is behavioral intention. The theory of planned behavior (TPB) and its extensions are the most widely used models in the study of behavioral intentions (Chai et al., 2020; Varah et al., 2021) and in recent years, TPB has emerged as one of the most popular and reliable theoretical frameworks for forecasting and comprehending behavioral patterns (Varah et al., 2021).For example, TPB can be used to explain why an individual has planned to participate in a climate strike. However, it focuses more on individual rational choices and lacks predictive power for perceptual choices such as habits and emotions that are formed over time (Russell et al., 2017; Fu et al., 2021; Yeow and Loo, 2022). Therefore its cannot always be reliable to explain spur of the moment actions. Previous research has demonstrated that TPB offers unique advantages as a rational choice model for assessing an individual's pro-environmental conduct (Ajzen, 1991; Yeow and Loo, 2022). Hence TPB can be used for explaining an individuals motivations behind youth environmental activism. === Emotional motivation === Emotions are important for motivating an individual as without emotions we would not care enough to do anything let alone activism. Research showed that emotions have the most significant effect on the willingness of individuals to engage in social good behaviours, followed by attitudes toward environmental protection and pro-natural emotion (Fu et al., 2023). Gousse-Lessard (2013) proposed that passion is a strong motivator behind radical environmental activism. This shows that individuals {{g}} emotions can motivate them towards pro-environmental behaviour. For example {{g}} if an individual is told that climate change affects their future adding a personal stake may make them more likely to act. Youth may also participate in climate change protests so they feel they are in control as they are not included in decision making processing regarding sustainability (Romano et al., 2023). Emotions and feelings such as frustration, powerless {{g}}, but also hope and trust, offer some insights {{expand}} into the affective reasons to mobilize, and the role of affective bonds in motivating participation (Kowasch et al., 2021). Therefore both TPB and Emotions can help motivate and explain the motivations behind youth climate activism. <quiz display="simple"> What is the Theory of Planned Behaviour{{g}}: |type="()"} - The theory that states why individuals spontaneously act + The theory that links individual belief to human action </quiz> {{RoundBoxTop|theme=3}}'''Key Points''' *Theory of Planned behaviour helps explain behvaiour and intention *Emotions motivating behaviour {{RoundBoxBottom}} == Influence of Social Identity, Culture and Community on Youth Activism == Social identity, culture and community all have a role in motivating youth activism. This is explained below. === Social Identity as a motivator === Social identity is shaped at young age by family views, norms and beliefs meaning familys{{g}} influence young peoples{{g}} environmental behaviours from a young age. This is supported by Gifford & Nilsson (2014) who state pro-environmental concern and behaviour are influenced by a complex interplay between personal, social, economic and contextual antecedents. Social identity plays a key role in driving and motivating climate movements, both online and on the street, as argued by Doolen {{f}}. Participation is therefore about taking part in a social process and it shapes the identity of the protestors. Identity is “central to participation in meaningful patterns of social and political action” (p. 84) {{f}}. Thus, social identity can be used as a predictor of pro-environmental behaviour. Social identities are shaped and participants became familiar with climate demonstrations, and the need to feel comfortable decreased once the strikes became known as normal and inclusive {{f}}. Meaning some individuals only keep participating as that is a space where they feel safe and that they belong {{g}}. Research in developmental and environmental psychology demonstrates that moral values and identities motivate people’s involvement in environmental commitments, including environmental activism behavior (Jia et al. 2017). Incentives and self-satisfaction are also a motivator for youth activism, motivations related to incentives and self-improvement/self-interest (Fung and Adams, 2017). Therefore {{g}} social identity can be used to motivate pro-environmental behaviour as individuals views are shaped from a young age and participating in climate activism can help individuals feel included {{f}}. === Self-determination and self as a motivator === Self-determination theory behaviour that is experienced as personally important and consistent with one's values (SDT, Deci & Ryan, 1985) can be applied to predicting intentions to engage in pro-environmental behaviour (e.g., De Groot & Steg, 2010). The theory posits that motivation is a multifaceted antecedent of behaviour that can be based on different levels of self-determination. People are more inclined to engage in societally desired behaviours, such as {{missing}} (Gronhoj & Thogersen, 2017). As shown {{where}} SDT can predict pro-environmental behaviour by examining individuals {{g}} determinations. Kramer et al. (2026) tested whether associations between climate-related worry, pro-environmental behaviour and affective wellbeing depend on the quality of youths {{g}} motivation to engage in pro-environmental behaviour. They found that climate-related worry was associated with increased pro-environmental behaviour for youth who acted out of interest and enjoyment (Intrinsic motivation) or internal pressure (introjected motivation) (Kramer et al., 2026). Therefore {{g}} climate worry influences an individuals {{g}} motivation. Youth who had internalized pro-environmental behaviour as personally important (identified motivation) showed high levels of pro-environmental behaviour regardless of worry {{f}}. Climate-related worry was robustly associated with reduced affective wellbeing irrespective of motivation type (Kramer et al., 2026). Young people’s activist motivations are not stand-alone phenomena; they work in tandem with other processes and factors in a dynamic way and are influenced by an individual’s history, previous experiences and passion, a sense of community, existing incentives, and self-satisfaction derived from activist behaviour (Fung & Adams, 2017). Self-determination and self are important motivations and predictors as self-determination can explain how an individual is determined towards climate action and self shows how intrinsic motivation can predict behaviour. === Culture === Cultural factors also play an important role in environmental activism behavior, as does surrounding oneself with like-minded people such as joining activism groups {{f}}. Similarly, exposure to such groups may build upon environmental knowledge and concern (Fung & Adams, 2017). Culture shapes the values young people grow up with, the groups they identify with, and what they see as their responsibilities to other people and to the natural world. For many young people, environmental activism is less an individual lifestyle choice and more an expression of [[w:Cultural identity|cultural identity]] and obligation to family, community and future generations. This is particularly clear for [[w:Indigenous peoples|Indigenous]] young people, whose climate activism is often grounded in relationships with land, water and Country, and in long histories of resistance to dispossession (Ritchie, 2021). Sense of Community is shown to be a strong motivator for pro-environmental behaviour (Fung & Adams, 2017) {{g}} this ties into culture. The following case study shows how culture, identity and collective action can combine to motivate youth climate activism. {{RoundBoxTop|theme=3}} '''Case study 2: Seed Mob – identity, Country and climate justice''' <!-- Figure 3: find a freely licensed Seed Mob / First Nations climate rally image on Wikimedia Commons. Do not upload images from Seed's website or social media (copyright). Then uncomment and update the line below. [[File:FILENAME.jpg|thumb|right|250px|'''Figure 3.''' First Nations young people at a climate justice rally. Collective cultural identity can turn concern into action.]] --> [[w:Seed Mob|Seed Mob]] is Australia's first [[w:Indigenous Australians|Aboriginal and Torres Strait Islander]] youth-led climate network. It was co-founded in 2014 by Amelia Telford, a Bundjalung and South Sea Islander woman, and Larissa Baldwin-Roberts, and began as part of the [[w:Australian Youth Climate Coalition|Australian Youth Climate Coalition]] (AYCC) before becoming an independent, First Nations-governed organisation in 2020 (Hocking, 2020). Seed aims to build the leadership of young Indigenous people through Caring for Country, and has been described as having around 250 young First Nations volunteers and 35,000 supporters (Hohenhaus et al., 2023). Seed Mob's campaigns connect climate change to land, water, health and culture. Members have worked with Traditional Owners to oppose gas fracking in the Beetaloo Basin (Northern Territory, Australia) and the Narrabri gas project in the Pilliga forest on Gomeroi Country (New South Wales). Its 2018 documentary ''Water is Life'' featured community leaders from Borroloola talking about the effects of climate change and fracking on water and bush foods (The Feed, 2020). Hohenhaus et al. (2023) argue that Seed Mob frames climate justice through land rights, self-determination and recognition of the unequal harms that fossil fuel projects place on First Nations communities. '''Why this matters for motivation:''' For Seed Mob members, climate action is not only an environmental choice. It is an expression of who they are and of their responsibilities to Country, community and future generations. {{RoundBoxBottom}} The Seed Mob case study demonstrates how social identity, cultural values and collective strength motivate youth environmental activism. The social identity model of collective action suggests that people are more likely to act together when they identify with another group, perceive injustice and believe they can create change (van Zomeren et al., 2008). Seed combines all three factors through shared First Nations identity, a clear injustice (fossil fuel projects on Country approved without adequate consent), and evidence of collective wins. {{g}} Fritsche et al. (2018). Additionally, self-determined behaviour is more persistent than behaviour driven by external pressure (Ryan & Deci, 2000). When protecting Country is part of who a young person is, climate action becomes an expression of identity rather than a chore. This is consistent with Kramer et al.'s (2026) finding that youth with identified motivation engaged in high levels of pro-environmental behaviour regardless of their level of climate worry Seed Mob also demonstrates [[w:Collective efficacy|collective efficacy]], the shared belief that a group can achieve its goals (Bandura, 2000). Kramer et al. (2026) found {{how}} that young people felt individual action was "useless" alone and preferred collective, systemic action. Through peer leadership, training and visible campaigns, Seed Mob gives young people evidence that acting together can make a difference, which may protect against the hopelessness linked to climate worry. Role models such as Telford {{explain}}, who received the NAIDOC Youth of the Year award in 2014, may further strengthen young people's belief that people like them can lead change. Western motivation theories may also not fully capture Indigenous concepts such as relational responsibility to Country, which Ritchie (2021) suggests is central to Indigenous youth climate activism. Future research {{ic|about what?}} should be led by, or co-designed with, First Nations young people. {{RoundBoxTop|theme=2}} '''Reflection:''' Think of a cultural or community group you strongly identify with. How might that identity shape the causes you care about, and whether you take action? {{RoundBoxBottom}} === Community === Community and Communities help motivate youth climate action. This is demonstrated by the theory of normative social behaviour (TNSB) {{f}}, showing youth had a greater intention to engage in climate change activism if they viewed it to be common and accepted among their peers, if they perceived positive benefits of activism, and if they identified as a climate activist (Gonzalez et al., 2021). Environmental activism also requires a collective environment with other people of shared ideologies about environmental protection (Lee et al. 2014). Individuals are often more empowered to protest when they are with others “together to achieve a common goal” and which may include diverse actions such as signing a petition, street protests, or striking (Brugger et al., 2020). Community motivates environmental actions by the rising trend of young groups being formed with the intent on educating peers and elders on the social and elders on the social and political implications of climate change (Goldman et al., 2014; Feldmann, 2020). Participants of environmental activism can be motivated by helping others in their collective and their relationships (Akiva et al., 2017). In conclusion {{g}} having a community encourages climate activism as youth are motivated by collective because of benefits for motivation ("alone it feels useless"), affective well-being ("doing something together gives me hope") and sustainability ("cheaper public transport would make my choices matter") (Kramer et al., 2026) {{RoundBoxTop|theme=3}} '''Key Points''' *Culture, community and social identity impacting young peoples {{g}} beliefs *How people were raised impacting their activism *Different cultures valuing volunteering and activism more *Cultural identity and connection to Country can motivate climate action (e.g., Seed Mob) *Group identification, perceived injustice and collective efficacy predict collective action {{RoundBoxBottom}} ;<quiz display="simple"> Climate related worry reduces pro-environmental behaviour. |type="()"} - True +False </quiz> <quiz> {According to the theory of normative social behaviour, what can increase young people's intention to participate in climate activism: {{g}} |type="()"} -Avoiding contact with like minded people -Believing all peers oppose climate activism -Believing collective action cannot create change +Believing activism is common and accepted by peers </quiz> == The Role of Education, Social media and Political engagement == Education, social media and political engagement are so important for youth motivation as these factors are where most youth receive their information and knowledge. === Education === Education is shown to motivate youth into pro-environmental behaviour {{f}}. This {{what}} is supported by suggestions that learning rather than maturation is responsible for the higher environmental commitment seen in older generations (Otto & Kaiser, 2014). This confirms the importance of formal sustainability education, but it also suggests a need for increasing young people's exposure to credible information about the state of the natural environment through more informal sources including social media used by the youth (Larsson, Andersson, & Osbeck, 2010). School environments are also supported as important spaces where interest in civic engagement develops (Flanagan and Levine 2010). The undergraduate college experience can be an ideal time and space for individuals to cultivate and foster pro-environmental behavior. Many argue that university students may end up becoming leaders and policy makers of the future, and therefore should be an important focus of activities aimed at sustainable environmental development (Vicente-Molina et al. 2013). Furthermore evidence suggests previous experiences and education are related to awareness (Fung & Adams, 2017). Once a youth are aware of growing environmental threats they have been shown to significantly increase the tendency to engage in environmental activism (Lubell 2002). Risk perceptions may be higher among younger people for at least two reasons. First, young people are more likely to have learnt about climate change through formal education and to be aware of its negative effects. Second, contemporary young people may see climate change as more relevant because they are more at risk to personally experience serious impacts of climate change in the future; indeed, Fisher argues that young people become active on climate change because they recognize their obligations as current and future citizens to address the issue (Brugger et al., 2020). Therefore as youth learn about climate change they become more motivated to protest and are less likely to sit by idly. === Social media === {{ic|Content past this point has been ignored for marking purposes due to being over the maximum word count.}} Social media influences youths environmental action as this is where youth receive their news, information and share movement links. There has been a clear rise in global awareness due to the broad scope of [[wikipedia:Social_media|social media]]. (Scherman et al., 2022). This is due to information being quickly passed around in comparison to previous generation and youth able to see the effects at a rapid pace, rather than articles and TV programs. Young people are also constantly aware of how climate change with effect their futures. Social media plays a crucial role in the sharing of campaigns, movements and platforms as individuals can share information at the click of a button. The current youth have a clear understanding of environmental issues, and have a clear level of self efficacy towards them (Goldman, D. et al). They believe that the engaging with social media, having the conversations and living sustainable lives will make a difference. Therefore social media has a huge role on motivating youth into participating in activism they see videos of protests, injustices and can share this information rapidly. === Politics === Politics are important for motivating youth environmental activism. Environmental movements involving young people of all ages have reshaped the political landscape (della Porta and Diani 2020), bringing the subject to the fore in the media and impacting upon Government policy, such as through commitments to ambitious targets for the reduction of carbon emissions and the creation of citizens’ assemblies on climate and environmental justice. We are seeing a more politically active generation (Scherman et al., 2022). With youth awareness on how policies affect environment (Sloam et al., 2022). Youth engagement with environmental politics in these countries is underpinned by value change the increasing prevalence of post materialist and ‘left-cosmopolitan’ attitudes amongst younger generations, reflected in a prioritisations of environmental, social justice and redistributive policies, with a vision for more connected, humane and equitable societies. In doing so, the authors draw attention to divisions based on gender and educational attainment between those young people who share these values (and who engage with environmental politics) and those who do not (Sloam et al., 2022). {{RoundBoxTop|theme=3}}'''Key Points''' *Education motivating action *Social media sharing information in seconds *Youth more involved in politics now {{RoundBoxBottom}}<quiz display="simple"> What can happen when young people become aware of environmental threats? |type="()"} - They are less likely to participate + They may become more likely to engage in activism -They stop being concerned about the future -They avoid environmental information &lt;/quiz> <quiz> {What is one major way social media supports youth climate activism? |type="()"} - It limits the spread of campaigns - It prevents young people sharing information - It replaces all formal education + It allows campaigns and information to spread rapidly &lt;/quiz> <quiz> {Which factors may influence whether young people engage with environmental politics? |type="()"} + Gender and educational attainment - Favourite social media platform only - Age alone - Access to television only &lt;/quiz> </quiz> == Barriers to Youth Activism and Encouraging Lasting Engagement == There is also a ‘cultural backlash’, driven by older generations, against cosmopolitan values (from opposition to cultural diversity, to the denial of human-made climate change) (Norris and Inglehart 2019). These views make it difficult for youth to get involved and make people second guess themselves. The policing of peaceful youth-led environmental protests is also a growing issue (Sloam et al., 2022). The policing provides a real danger to protesters as youth may feel fear, anxiety and aggression towards police over persecution. Pro-environmental collective actions are also often judged negatively by public opinion, diminishing their effectiveness and potentially impacting youth participation (Romano et al., 2023). This means that youth may not participate due to judgement. ==Conclusion== In conclusion, as climate change has worsened and its effects have become more prominent and well known so have the youth of the world with their climate strikes and activism. These movements are motivated by many reasons including social media, collective action, education, self identity, culture and the effects of climate change being more known. In fighting for environmental action it is important to motivate young people by spreading the message of climate change. There are however limits to youth activism and disadvantages for youth to get involved including cultural backlash from other generations, policing of peaceful protests and being judged negative by the public. ==See also== * [[wikipedia:Climate_movement|Climate Movement]] * [[Climate change|Climate Change]] * [[w:Seed Mob|Seed Mob]] (Wikipedia) * [[w:Social identity theory|Social identity theory]] (Wikipedia) ==References== {{Hanging indent|1= Akiva, T., Carey, R. L., Cross, A. B., Delale-O'Connor, L., & Brown, M. R. (2017). Reasons youth engage in activism programs: Social justice or sanctuary? Journal of Applied Developmental Psychology, 53, 20-30. https://doi.org/https://doi.org/10.1016/j.appdev.2017.08.005 Bandura, A. (2000). Exercise of human agency through collective efficacy. ''Current Directions in Psychological Science, 9''(3), 75–78. https://doi.org/10.1111/1467-8721.00064 Brügger, A., Gubler, M., Steentjes, K., & Capstick, S. B. (2020). Social identity and risk perception explain participation in the Swiss youth climate strikes. Sustainability, 12(24), 10605. Elliott, L. (2026, May 11). environmentalism. Encyclopedia Britannica. https://www.britannica.com/topic/environmentalism Feldman, H. R.. (2020). A rhetorical perspective on youth environmental activism. ''Journal of Science Communication'', ''19''(06), C07. https://doi.org/10.22323/2.19060307 Fritsche, I., Barth, M., Jugert, P., Masson, T., & Reese, G. (2018). A social identity model of pro-environmental action (SIMPEA). ''Psychological Review, 125''(2), 245–269. https://doi.org/10.1037/rev0000090 Fung, C. Y., & Adams, E. A. (2017). What motivates student environmental activists on college campuses? An in-depth qualitative study. Social Sciences, 6(4), 134. Goldman, D., Pe’er, S., & Yavetz, B. (2015). Environmental literacy of youth movement members – is environmentalism a component of their social activism? ''Environmental Education Research'', ''23''(4), 486–514. https://doi.org/10.1080/13504622.2015.1108390 Gousse-Lessard, A.-S., Vallerand, R. J., Carbonneau, N., & Lafrenière, M.-A. K. (2013). The role of passion in mainstream and radical behaviors: A look at environmental activism. ''Journal of Environmental Psychology'', ''35'', 18–29. https://doi.org/10.1016/j.jenvp.2013.03.003 Gonzalez, A. M., Reynolds-Tylus, T., & Skurka, C. (2022). Young adults’ willingness to engage in climate change activism: an application of the theory of normative social behavior. Environmental Communication, 16(3), 388-407. Grønhøj, A., & Thøgersen, J. (2017). Why young people do things for the environment: The role of parenting for adolescents’ motivation to engage in pro-environmental behaviour. Journal of environmental psychology, 54, 11-19. Hocking, R. (2020, December 1). 'First Nations are rising': Seed Mob goes independent. ''NITV''. https://www.sbs.com.au/nitv Hohenhaus, M., Boddy, J., Rutherford, S., Roiko, A., & Hennessey, N. (2023). Engaging young people in climate change action: A scoping review of sustainability programs. Sustainability, 15(5), 4259. Hohenhaus, M., Rutherford, S., Boddy, J., & Borkoles, E. (2023). Climate warriors down under: Contextualising Australia's youth climate justice movement. ''npj Climate Action, 2'', Article 45. https://doi.org/10.1038/s44168-023-00085-y Kowasch, M., Cruz, J. P., Reis, P., Gericke, N., & Kicker, K. (2021). Climate youth activism initiatives: Motivations and aims, and the potential to integrate climate activism into ESD and transformative learning. Sustainability, 13(21), 11581. Kramer, A. W., Wetering, J. V. D., Groep, S. V. D., Brinke, L. W. T., Toenders, Y. J., Green, K. H., & Crone, E. A. (2026). Youth Perspectives on the Climate Crisis: Motivation and Action Pathways. Developmental Science, 29(4), e70217. Luttrell, R. (2023). Historical Roots and Modern Movements: A Framework for Activism. In Strategic Social Media as Activism (pp. 5-39). Routledge. News, G. S. W. (2024). The origins of the activism movement: A historical and global perspective. https://www.globalsociety.earth/post/the-origins-of-the-activism-movement-a-historical-and-global-perspective Pickard, S. (2022). Young environmental activists and Do-It-Ourselves (DIO) politics: Collective engagement, generational agency, efficacy, belonging and hope. Journal of Youth Studies, 25(6), 730-750. Ritchie, J. (2021). Movement from the margins to global recognition: Climate change activism by young people and in particular Indigenous youth. ''International Studies in Sociology of Education, 30''(1–2), 53–72. https://doi.org/10.1080/09620214.2020.1854830 Romano, L., Russo, C., Gladwin, T. E., & Panno, A. (2024). Adolescents and young adults’ participation in pro-environmental movements: A systematic review. The Journal of Genetic Psychology, 185(5), 373-398. Ryan, R. M., & Deci, E. L. (2000). Self-determination theory and the facilitation of intrinsic motivation, social development, and well-being. ''American Psychologist, 55''(1), 68–78. https://doi.org/10.1037/0003-066X.55.1.68 Scherman, A., Valenzuela, S., & Rivera, S.. (2022). Youth environmental activism in the age of social media: the case of Chile (2009-2019). ''Journal of Youth Studies'', ''25''(6), 751–770. https://doi.org/10.1080/13676261.2021.2010691 Sloam, J., Pickard, S., & Henn, M.. (2022). ‘Young People and Environmental Activism: The Transformation of Democratic Politics’. ''Journal of Youth Studies'', ''25''(6), 683–691. https://doi.org/10.1080/13676261.2022.2056678 The Feed. (2020, November 11). The young Indigenous woman fighting fracking in remote NT. ''SBS''. https://www.sbs.com.au/news/the-feed/the-young-indigenous-woman-fighting-fracking-in-remote-nt Urkidi, L. (2010). A glocal environmental movement against gold mining: Pascua–Lama in Chile. Ecological Economics, 70(2), 219-227. https://doi.org/https://doi.org/10.1016/j.ecolecon.2010.05.004 van Zomeren, M., Postmes, T., & Spears, R. (2008). Toward an integrative social identity model of collective action: A quantitative research synthesis of three socio-psychological perspectives. ''Psychological Bulletin, 134''(4), 504–535. https://doi.org/10.1037/0033-2909.134.4.504 Wallis, H., & Loy, L. S. (2021). What drives pro-environmental activism of young people? A survey study on the Fridays For Future movement. Journal of Environmental Psychology, 74, 101581. https://doi.org/https://doi.org/10.1016/j.jenvp.2021.101581 Wallis, H., & Loy, L. S. (2021). What drives pro-environmental activism of young people? A survey study on the Fridays For Future movement. Journal of environmental psychology, 74, 101581. }} ==External links== * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) * [https://www.seedmob.org.au/ Seed Indigenous Youth Climate Network] (Seed Mob) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Activism]] [[Category:Motivation and emotion/Book/Environment]] [[Category:Motivation and emotion/Book/Youth]] dxskwja1jyra44lgwvxjcbqv65ixyhq 1 307307 2842520 2841177 2026-10-25T03:18:07Z Jtneill 10242 Book chapter feedback 2842520 wikitext text/x-wiki ==Suggestions== Hi, This is looking great and it has been really interesting reading about this topic and the sections on social media and politics. I almost picked this one actually, since it overlaps with some of my Global Studies research. It has given me useful context for why youth activism has become increasingly visible. If I had to be picky, one area that could be further developed is the motivation section. While a lot has been identified, it is not completely clear how these factors actually explain why some people move from concern about climate change to participating in action. Maybe organising this section around some psychological mechanisms rather than treating each factor separately. For instance, self-efficacy could explain whether young people believe their actions. I think this would help make the chapter feel a bit more psychologically focused, and to answer the motivational part of the research question. An article that could be of use: Why do youth participate in climate activism? A mixed-methods investigation of the #FridaysForFuture climate protests: 10.1016/j.jenvp.2021.101647 - Jack4234 == Some research to look at.. == Hey there. Your topic of youth environmental activism motivation is a really interesting topic for your book chapter assignment. I've done a bit of research for you on some research literature on this topic. I hope you might find these resources useful for your book chapter. - Cloughton, I. (2021). Global Youth Activism on Climate Change. ''Social Work & Policy Studies: Social Justice, Practice and Theory'', ''4''(1). https://openjournals.library.sydney.edu.au/SWPS/article/view/14960 - Feldman, H. R. (2020). A rhetorical perspective on youth environmental activism. ''Journal of Science Communication'', ''19''(06), C07–C07. https://doi.org/10.22323/2.19060307 - O'brien, K., Selboe, E., & Hayward, B. M. (2018). Exploring youth activism on climate change. ''Ecology and Society'', ''23''(3).https://www.jstor.org/stable/26799169 I Hope you find these resources helpful for your book chapter, good luck! Kind regards. [[User:HassanAlsamara|HassanAlsamara]] ([[User talk:HassanAlsamara|discuss]] • [[Special:Contributions/HassanAlsamara|contribs]]) 22:48, 15 August 2024 (UTC) == Heading casing == {| style="float: center; background:transparent;" |- | [[File:Crystal Clear app ktip.svg|48px|left]] | {{#if:Josephpat03|Hi [[User:Josephpat03|Josephpat03]].|}} FYI, the recommended [[Wikiversity]] heading style uses [[w:Letter case#Sentence_case|sentence casing]]. For example:<br> <big><big>Self-determination theory</big></big> rather than <big><big>Self-Determination Theory</big></big> Here's an example chapter with correct heading casing: [[Motivation and emotion/Book/2019/Growth mindset development|Growth mindset development]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 02:17, 2 September 2024 (UTC) |} <!-- Official topic development feedback --> {{METF/2024 |1= <!-- Title --> # The title and/or sub-title were not correctly worded and/or formatted (fixed) |2= <!-- Headings --> # See earlier comment about [[#heading casing|Heading casing]] # Insufficient # There is no indication of psychological science/research in the headings <!-- Alignment with focus questions --> # Adopt closer alignment between sub-title, focus questions, and top-level headings <!-- Other ---> # Aim for 3 to 6 top-level headings between the Overview and Conclusion, with up to a similar number of sub-headings for large sections |3= <!-- Overview--> # Insufficient <!-- Scenario --> # A scenario or case study is planned # Add an image to the scenario to help attract reader interest <!-- Description --> # Add a brief, evocative description of the problem/topic <!-- Focus questions --> # Present focus questions in a feature box at the end of this section |4= <!-- Key points--> # Partial development of key points <!-- Citations --> # Insufficient use of citations <!-- Theory and research --> # Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples # It is unclear whether the best available psychological theory and research has been consulted in the preparation of this plan <!-- Style --> # Use APA style 7th edition for citations (e.g., do not include author initials) <!-- Other --> # ''Avoid providing too much background information''. Aim to briefly summarise general concepts and provide internal links to relevant book chapters and/or Wikipedia pages for further information. Then focus most of the content on ''directly answering the core question(s)'' posed by the chapter sub-title. # Consider using the [https://unicanberra.instructure.com/courses/15707/external_tools/262?display=borderless Studiosity] service and/or a service like [https://www.grammarly.com/ Grammarly] to help improve the quality of written expression such as checking grammatical and spelling errors <!-- Conclusion --> # Conclusion (the most important section) hasn't been developed # What might the take-home, practical messages be? (What are the answer(s) to the question(s) in the sub-title and/or focus questions?) |5= <!-- Figure --> # One or more relevant figure(s) presented and captioned <!-- Caption --> # The figure caption(s) could better explain how the image connects to key points being made in the main text # Use APA style for figure captions <!-- Cite --> # Figure(s) are cited at least once in the main text # Cite each figure at least once in the main text using APA style <!-- Size --> # Consider increasing image size from to make it easier to view |6= <!-- Learning feature --> <!-- Interwiki links ---> # Include in-text [[m:Help:Interwiki linking|interwiki links]] for the first mention of key terms to relevant Wikipedia articles and/or to other relevant book chapters (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Examples/case studies --> # Placeholder use of one or more scenarios/examples/case studies <!-- Quiz --> # Consider including one or more quiz question(s) about the take-home messages <!-- Tables --> # Also consider using one or more tables to summarise key information |7= <!-- References --> <!-- Overall --> # Good <!-- Systematic reviews --> # Are there any systematic reviews about this topic? <!-- Suggestions --> # Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]: ## capitalisation ## make doi hyperlinks active (i.e., clickable) ## remove extra apostrophes |8= <!-- Resources --> <!-- See also --> # See also ## OK ## One of two link types provided ### Also link to related book chapters ## Use [[w:Letter case#Sentence casing|sentence casing]] ## Include source in brackets after link (e.g., (Wikipedia) or (Book chapter, year) for Wikiversity book chapters) ## Use alphabetical order <!-- External links --> # External links ## Not developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) |9= <!-- User page --> # Basic <!-- Description about self --> # Very brief description about self – consider expanding <!-- Links to profile(s) --> # Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks. <!-- Link to book chapter --> # A link to the book chapter is provided |10= <!-- Social contribution --> # One out of three types of contributions made with with direct link(s) to evidence. The other types of contribution are making: #* direct improvements to other [[Motivation and emotion/Book|chapters (past or current)]] #* posts about the unit or project on other platforms such as the {{Motivation and emotion/Canvas}} discussion forum or on [https://x.com X] using the {{Motivation and emotion/Hashtag}} }} <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]] |2= <!-- Headings --> <!-- Heading structure --> <!-- 2-level --> # Basic [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] – could benefit from further development (expand to provide more descriptive, incisive headings) # I've removed the "Key Points" headings to tidy # Remove citations from headings <!-- Alignment with focus questions --> # Insufficient alignment between sub-title, focus questions, and top-level headings <!-- GenAI ---> # Are the headings based on [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. |3= <!-- Overview--> # Excellent – Scenario, image, evocative description of the problem/topic, and focus questions # Very good # Good # Basic # Insufficient # Hasn't been developed – Needs scenario, image, evocative description of the problem/topic, and focus questions <!-- GenAI ---> # Does this section include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, it needs to be acknowledged as such in the edit summaries, otherwise it violates academic integrity. <!-- Scenario --> # Consider providing a more user-friendly, real-life scenario to attract reader interest; this research study could be moved into the most relevant subsequent section # Also include an image <!-- Description --> # A basic description of the problem/topic is planned or presented # The description should more clearly focus on the motivational aspects of YEA <!-- Focus questions --> # Focus questions are aligned with sub-title # Develop closer alignment between the sub-title, focus questions, and top-level headings |4= <!-- Key points--> <!-- Overall --> # Basic development # Keep the history section brief; concentrate on motivational factors involved in contemporary YEA <!-- Theory and research --> # Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples. <!-- Citations --> # Basic use of citations <!-- Conclusion --> # Conclusion hasn't been developed # What are the practical, take-home messages? (address the focus questions) |5= <!-- Figure --> # Relevant figure(s) are presented <!-- Caption --> # Figure caption(s) could better explain how the image connects to key points being made in the main text # Figure caption(s) should include '''Figure X'''. ... <!-- Cite --> # Cite each figure at least once in the main text using APA style (e.g., see Figure 1) |6= <!-- Learning feature --> <!-- Interwiki links ---> # Promising in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] <!-- Scenarios/examples/case studies --> # Consider use of more scenarios/examples/case studies <!-- Quiz --> # Promising use of quiz question(s) # Place quiz each question in the most relevant section <!-- Tables --> # Also consider using tables to summarise key information |7= <!-- References --> <!-- Overall --> # Good <!-- Systematic reviews --> # At least one relevant systematic review and/or meta-analysis has been identified <!-- APA style --> # Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]: ## capitalisation ## [[Help:Wikitext quick reference|italicisation]] ## [https://apastyle.apa.org/instructional-aids/reference-guide.pdf doi formatting] ## make doi hyperlinks active (i.e., clickable) (fixed) |8= <!-- Resources --> <!-- See also --> # See also ## Basic ## One of two link types provided ### Also link to related [[Motivation and emotion/Book|motivation and emotion book chapters]] ## Use alphabetical order <!-- External links --> # External links ## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) |9= <!-- User page --> # Good <!-- Description about self --> # Brief description about self – consider expanding <!-- Links to profile(s) --> # Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks. <!-- Link to book chapter --> # A link to the book chapter is provided |10= <!-- Social contribution --> # Excellent – at least three different types of contributions with direct link(s) to evidence -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 21:50, 31 August 2026 (UTC) -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 02:17, 2 September 2024 (UTC) == Some Research on Environmental Activism and Suggestions for Chapter == Hi! I think you've chosen a really interesting topic for this assessment and I've found some sources that might assist in some potential areas (political engagement for climate activism) for your textbook chapter. 1. Fisher, D.R., Nasrin, S. (2020) Climate activism and its effects. ''WIREs Climate Change'' 12, 1-11. DOI: 10.1002/wcc.683 Article looks at interaction of climate activism and political engagement. Touches on the impacts of the coronavirus and subsequent impacts on online mediums regarding climate activism. Also addresses the current state of climate activism and addresses some areas of future research. 2. Neas, S., Ward, A., Bowman, B. (2022) Young people’s climate activism: A review of the literature. ''Secular Political Participation'' 4. https://doi.org/10.3389/fpos.2022.940876 Considers a large number of sources that might be useful for this section on political engagement. Observes trends in youth climate activism and addresses youth voices in social media and news outlets. Also looks at some areas for future research which might be a good heading/section in your textbook chapter. '''Greta Thunberg''' is looked at in detail which might be a good case study or learning theory idea. Hopefully these sources and suggestions are helpful! [[User:CharlotteLane55|CharlotteLane55]] ([[User talk:CharlotteLane55|discuss]] • [[Special:Contributions/CharlotteLane55|contribs]]) 02:17, 2 October 2024 (UTC) == Seed Mob case study added to "Does culture motivate youth activism?" == Hi all, I've added a second case study on Seed Mob, Australia's first Indigenous youth-led climate network, to the ''Does culture motivate youth activism?'' section of the chapter. The aim is to give the chapter an Australian, First Nations example that connects culture and identity to youth climate activism, and to respond to the topic development feedback about using more case studies, quiz questions and external links. Please note, if you don't like the inclusions, I invite you to delete them. Also, my email is u3284454@uni.canberra.edu.au if you're having some issues. I can provide my phone number via email if you reach out and need to talk through anything. Apologies in advance if I've caused any confusion, particularly given the time constrains to get this assessment in. Writing this was equally beneficial for me, because it provided me an opportunity to get my social contributions in for my assessment grades too, while ''hopefully'' adding value to your assessment while trying to stay aligned with your voice and plan as well. Summary of what I did below. Gen AI acknowledgement for the below. It was easier for me to ask Claude to provide a summary of the work than me individually write it out. However, I reviewed (and edited a bit of this info below) - just leaning on the side of more than less regarding generative AI acknowledgement and use :) Warmly Joel ____ '''What I added''' * An introductory paragraph on how culture and connection to Country shape youth activism, replacing the template tips in that section. * A case study feature box covering Seed's history, campaigns (Beetaloo Basin, Pilliga/Narrabri, ''Water is Life'') and how it frames climate justice. * Linking paragraphs that apply theory to the case study: ** Social identity and collective action – SIMCA (van Zomeren et al., 2008) and SIMPEA (Fritsche et al., 2018): group identity, perceived injustice and collective efficacy. ** Self-determination theory – cultural identity as a source of identified motivation, linked to Kramer et al. (2026) (Ryan & Deci, 2000). ** Collective efficacy – peer leadership and collective wins as a buffer against climate worry (Bandura, 2000). ** Critical perspective – most evidence on Seed comes from media rather than psychological research, and Western motivation theories may not fully capture Indigenous relational concepts (Ritchie, 2021). * A reflection box and two new key points. * A cross-reference from "Identity as a motivator" to the case study. * A new quiz question on SIMCA. * 8 new APA references, plus See also (Seed Mob, Social identity theory) and External links (Seed) entries. '''Generative AI acknowledgement''' Generative AI (Claude) was used to help research, draft some sections based off key points and prompts I provided (particularly enhancing readability and cross-referencing with what was already in your chapter so it flowed with your existing key points), and to format these additions. Particularly for formatting, Claude was used to translate my word document language to the Wiki source code language for better formatting of the case study and reflection box. I reviewed and edited the content and checked the sources before publishing. '''Any issues?''' If there are any issues, please email me on u3284454@uni.canberra.edu.au and I can delete those additions (if you don't want them and are having some issues deleting them). [[User:Jshottt|Jshottt]] ([[User talk:Jshottt|discuss]] • [[Special:Contributions/Jshottt|contribs]]) 01:32, 10 October 2026 (UTC) :Hey Joel, :Thank you for your help. :Kind regards, :Steph [[User:SJPiper|SJPiper]] ([[User talk:SJPiper|discuss]] • [[Special:Contributions/SJPiper|contribs]]) 03:32, 10 October 2026 (UTC) <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is a basic chapter # The main area(s) for potential improvement: #* use the best psychological theory about the topic #* more detailed review of the best psychological research about the topic #* quality of written expression #* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]] #* integrate more strongly with other chapters by using embedded links to related topics <!-- Overall – Citations --> # Good use of academic, peer-reviewed citations to support claims <!-- Overall – Word count --> # Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes. <!-- Overall – Copyedits --> # For additional feedback, see the following comments and [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FYouth_environmental_activism_motivation&diff=2842519&oldid=2841826 these copyedits] |2= <!-- Overview – Comments... --> # Insufficient (too long); needs redrafting <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Reasonably engaging scenario or case study <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box <!-- Overview – Explains problem --> # Description of problem is too long/overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections. <!-- Overview – Focus questions --> # There are too many overlapping focus questions; integrate and select the main three to five questions |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement # A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic # Reduce general theoretical background (e.g., history). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question). <!-- Theory – Builds on --> # Builds somewhat on other [[w:|Wikipedia]] articles but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Reasonably good but somewhat wandering, ponderous depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Insufficient use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Examples --> # Good use of examples to illustrate theoretical concepts # The Seed Mob example is useful, but overly long |4= <!-- Research comments... --> <!-- Research – Key findings --> # Basic review of relevant research # Insufficient use of relevant systematic reviews and/or meta-analyses # More detail about key studies would be ideal <!-- Research – Critical thinking --> # Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out specific critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Reasonably good integration between the most relevant theory(ies) and the best research # The chapter places more emphasis on theory than on research; strive for an integrated balance # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> <!-- Conclusion – Word count --> # Not counted for marking purposes due to the chapter being over the maximum word count |7= <!-- Written expression – Style comments... --> <!-- Written expression – Written expression --> # Written expression ## The quality of written expression is OK but there are several aspects which are below professional standard <!-- Written expression – Sentences --> ## Some sentences could be explained more clearly (e.g., see the {{explain}} and {{rewrite}} tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression ## Some sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+. ## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence. <!-- Written expression – Language --> ## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., "it") instead of 1st (e.g., "we") or 2nd person (e.g., "you") in the main text. 1st or 2nd person can work well for case studies or feature boxes. <!-- Written expression – Layout --> # Layout ## The headings could be more clearly aligned with the focus questions ## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings ## See earlier comment about [[#Heading casing|heading casing]] <!-- Written expression – Grammar --> # Grammar ## The grammar for many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback ## Improve the clarity and flow of the writing by checking and correct use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas]. ## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats') ## Use past tense when describing research studies, although implications of findings could be in the present tense <!-- Written expression – Abbreviations --> ## Abbreviations ### Only use abbreviations such as e.g., i.e., et al., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out ### Check and correct formatting of abbreviations (such as e.g., i.e., etc.) ### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms. ### Only introduce abbreviations which are subsequently used <!-- Written expression – Spelling --> # Spelling ## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour) ## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages. <!-- Written expression – Proofreading --> # Proofreading ## More proofreading is needed (e.g., fix punctuation and typographical errors) to bring the quality of written expression closer to a professional standard ## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info] <!-- Written expression – APA style --> # APA style ## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] ## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.] ## Direct quotes need page numbers—even better, communicate about concepts in your own words <!-- Written expression – Figures --> ## Figures ### Reasonably well captioned <!-- Written expression – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style <!-- Written expression – Citations --> # Citations use reasonably good [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags) ## If there are three or more authors, cite the first author followed by et al., then year. 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See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence. }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 03:18, 25 October 2026 (UTC) 6ume57kz0bt1i0ggqok4b74gq08lihr 2 325867 2842459 2834875 2026-10-24T21:19:28Z Dc.samizdat 2856930 /* General relativity is just Galilean relativity in a general space of four orthogonal dimensions */ 2842459 wikitext text/x-wiki == Symmetry transformation group arising from the Laplace–Runge–Lenz vector == *{{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}} <blockquote>Through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which is well known in the literature. However, the physically relevant part of the LRL vector is its direction angle in the plane of motion (since its magnitude is just a function of energy and angular momentum). The ... infinitesimal dynamical symmetry corresponding to the direction part of the LRL vector ... when combined with the rotation symmetries ... is shown to be the semi-direct product of <math>SO(3)</math> and <math>\mathbb{R^3}</math>. This stands in contrast to the <math>SO(4)</math> symmetry group generated by the LRL symmetries and the rotations.</blockquote> In other words, bodies in Kepler orbits are translating through 4-space via <math>QT</math>, breaking their stationary <math>Q^2</math> symmetry. The LRL direction vector is their <math>T</math>, their proper time vector. The Kepler problem in Newtonian mechanics has been widely studied and encompasses both a particle moving under inverse-square force and two bodies moving under mutual gravitational attraction. One of its most interesting features is the existence of the conserved Laplace–Runge–Lenz (LRL) vector (see e.g.  and references therein). It lies in the plane of motion, which is orthogonal to the angular momentum vector, and points in the direction given by the periapsis in the case of inverse-square motion and the position of closest separation in the case of two-body motion. This conserved vector has a long history, going back over three centuries, and plays a key role in understanding the shape and orientation of astronomical orbits in the solar system. Its quantum mechanical analog was important historically in the initial derivation of energy levels in the hydrogen atom (see e.g.  and references therein). In contrast to the conservation of energy and angular momentum in the Kepler problem, which come directly from Noether’s theorem using the manifest time-translation and rotation symmetries of Newtonian space, the LRL vector does not arise from any kinematic symmetry principle. Nevertheless, Noether’s theorem in reverse  states that every conserved quantity is generated by some infinitesimal symmetry transformation that leaves invariant the action principle for the Kepler problem. Since the LRL vector comprises three independent conserved quantities, its symmetry generator consists of three independent infinitesimal transformations, which were first worked out in Ref.. They turn out to involve the velocity variable, unlike the infinitesimal time-translation and rotation transformations which involve only the time and position variables. In particular, the LRL infinitesimal symmetry transformations are primary examples of what are called ''dynamical symmetries'', in contradistinction to ''point symmetries'' which are typified by infinitesimal time-translations and rotations. The only additional point symmetry admitted by the Kepler problem consists of a scaling of the time and position variables, but this scaling transformation does not leave the action principle invariant. The five point symmetries together with the three LRL symmetries in the Kepler problem form a 6-dimensional Lie subalgebra when the LRL symmetries are normalized to have the same physical dimension as the rotations. This subalgebra’s structure turns out to be <math>\mathfrak{so}(4)</math> when the energy is negative, and <math>\mathfrak{so}(3,1)</math> when the energy is positive . These Lie algebras are isomorphic to the symmetry algebra of isometries of a 3-sphere <math>S^3</math> in <math>\mathbb{R^4}</math> and a 3-hyperboloid <math>H^3</math> in Minkowski space <math>\mathbb{R^{3+1}}</math>. One explanation for the appearance of four-dimensional spaces is provided by Moser’s observation  that, in the negative energy case, the Kepler problem is equivalent to non-affine geodesic motion on <math>S^3</math> in a four-dimensional auxiliary space under stereographic projection. A kinematical description of the projection is given by identifying the stereographic coordinates as the components of the Kepler velocity variable. The rotations in the auxiliary space correspond in the Kepler problem to changing the angular momentum while keeping the energy unchanged. These rotations constitute a <math>SO(4)</math> transformation group that combines the ordinary physical <math>SO(3)</math> rotations and the LRL symmetry transformations. A similar equivalence holds in the positive energy case by considering non-affine geodesic motion on <math>H(3)</math>. One remaining question is how does the group generated by the LRL infinitesimal symmetry transformations act just on the time, position, and velocity variables in the Kepler problem. It is hard to find an explicit description of this group in the literature (although a description in terms of the coordinates in the auxiliary 4-dimensional space is known ). The first main result of the present work will be to derive the explicit form of the LRL symmetry transformation group in terms of the kinematic physical variables. As a by-product, the action of the new infinitesimal symmetries on all of the conserved quantities is obtained. The results are given in terms of the physical kinematical variables in the Kepler problem, rather than in an enlarged auxiliary space in which the LRL symmetries are usually stated. Another remaining question, which is not commonly discussed in the literature, arises from the magnitude of LRL vector. This quantity is given by a scalar expression in terms of the energy and magnitude of the angular momentum. The corresponding infinitesimal symmetry transformation, coming from Noether’s theorem in reverse, is just a linear combination of the generators of time-translation and rotation in the plane of motion. Namely it is a point symmetry, involving only the time and position variables. On the one hand, this choice of normalization of the LRL vector is essential for the subalgebra generated by the LRL symmetries and rotations to be when energy is negative and when energy is positive. Yet on the other hand, it is a free choice, which raises the question of whether a different or perhaps simpler structure is possible for the resulting symmetry subalgebra if some other choice of normalization is made. A natural choice is to normalize it to be a unit vector, retaining only the physically relevant direction information contained in the LRL vector. This will be called the LRL direction vector. As a second main result, the components of the LRL direction vector will be shown to generate three infinitesimal symmetry transformations that commute with themselves. == 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 their Hubble redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of these three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity <math>c</math>. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance. That small separation might not make much difference in our view of the night sky, but their actual separation in the fourth dimension may be much greater, large enough to significantly transform our map of the heavens. That is because it is unlikely that the stars in the galaxy all lie exactly on the surface of the expanding universal 3-sphere, after millions of years of expansion....{{Efn| 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? That would be an expected consequence of the special rotational symmetry group of 4-space <math>SO(4)</math>, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. Also of interest would be whether the central region of the galaxy is a 4-ball or a 4-ellipsoid.}} == Why the Equivalence Principle symmetry == How is the Weak Equivalence Principle (of Einstein) derived from Coxeter's group-theoretic formulation of the principle of relativity (his relation of congruent transformations in n-dimensional Euclidean space)? That is, what is the equivalence between rotations and translations, via their common basis in reflections? What exactly is the relative (i.e. symmetrical, interchangeable) relation between a translation and a rotation? == Light == === Polarization === How is polarized light oriented with respect to the proper time dimension? Is the magnetic field simply a fourth orthogonal orientation of the electric field? What is its orientation with respect to the proper time dimension? === Sails === On 3-sphere vs. 2-sphere - can you perhaps tack through 4-space on light pressure, if you can distinguish left and right hand radiation - relation to polarizations of light. Light pressure vs. particle pressure - <math>c\prime = 2c</math>. === Propagation kinematics through 4-space === 11-cell's role. Geodesics of truncated 5-cells. == 137 === * <math>137 = 5! + 4! - 3! - 2! + 1!</math> * a [[W:Pythagorean prime|Pythagorean prime]]: a prime number of the form <math>4n+1</math>, where <math>n=34</math>&nbsp;(<math>137=4\times 34+1</math>) or the sum of two squares <math>11^{2}+4^{2} = (121+16)</math>. * a combination of three terms <math>4^{3}+3^{4}-2^{3} = (64+81-8)</math>, cube of 4 + [[W:Triangular number|triangular number]] T4+T2 on each cube face (along 3 axes) - peaks (single 6th peak as free link) * The "73 calculator trick" involves picking any four-digit number, entering it twice to create an eight-digit number (e.g., 1234 becomes 12341234), then dividing by 137, and finally dividing that result by your original four-digit number to always get 73 as the answer. This works because repeating a four-digit decimal number N is the same as multiplying N by 10001, and 10001 is the product of 73 and 137 (73×137). == Galilean relativity in a space of four orthogonal dimensions == The paragraph ending in "That is all of physics" came to me on my mother's 101st birthday, just over a year after she died, like an example of rebirth in the spring. I say rebirth because it is probably not original, and even if it is, life must necessarily forget it someday, and just as necessarily rediscover it again someday afterwards. It seems to me to be as close an example to ''immortal life'' as I have encountered. That state, oxymoronic in nearly all senses I fear, may be said to exist where life discovers an immortal principle, and so (like the 11-cells) many lives distinct but not disjoint combine to give existence to an immortal object, where none could exist at all in singularity. [[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 19:44, 14 March 2024 (UTC) Writing this on the morning after my mother's 103rd birthday, after several related ideas finally assembled themselves in my mind in sleep last night, for the first time. This then is my first written observation of these several ideas that I have been pondering for years. The Higgs boson, the largest mediator of the fundamental force, has the mass/energy of the 120-cell. The photon as particle (as distinct from the free radiating photon as wave), the mediating boson of the electron, I think will turn out to have the mass/energy of the 11-cell. The 1/137 mass/energy relationship behind the ''e'' constant is due to the subsumption relationship between these two 4-polytopes and a third 4-polytope, the 4-simplex or 5-cell, which is the proton. Mass/energy is proportional to 4-content, and also a function of the isoclinic angle of rotation. [[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 03:57, 16 March 2026 (UTC) == Galaxies == The spacetime of general relativity is often illustrated as a projection to a curved 2D surface in which large gravitational objects make gravity wells or dimples in the surface. In the Euclidean 4D view of the universe the 3D surface of a large cosmic object such as a galaxy surrounds an empty 4D space, and large gravitational objects within the galaxy must make dimples in its surface. But should we see them as dimples exactly? Would they dimple inwards or outwards? In the spacetime illustrations they are naturally always shown as dimpling downwards, which is somewhat disingenuous, strongly suggesting to the viewer that the reason for gravity is that it flows downhill - the original tautology we are trying to surmount! In the Euclidean 4D galaxy the dimple, if it is one, must be either inward or outward, and which it is matters since the dimple is flying outward at velocity {{mvar|c}}. The universe is not collapsing inward. Is a large gravitational mass (such as a star) ''ahead'' of the smaller masses orbiting around it (such as its planets), or is it ''behind'' them, as they fly through 4-space on their Clifford parallel trajectories? The answer is ''both'' of course, because a star is not a dimple, it is a 4-ball, and it dimples the 3D surface both inwards and outwards. It is a thick place in the 3D surface. We should view it as having its gravitational center precisely at the surface of the expanding 3-sphere. What is a black hole? It is the hollow four-dimensional space that a galaxy is the three-dimensional surface of. When we view another galaxy, such as Andromeda, we are seeing that whole galaxy from a distance, the way the moon astronauts looked back at the whole earth. We see our own Milky Way galaxy from where we are on its surface, the way we see the earth from its surface, except that the earth is solid, but the galaxy is hollow and transparent. We can look across its empty center and see all the other stars also on its surface, including those opposite ours on the far side of its 3-sphere. The thicker band of stars we see in our night sky and identify as the Milky Way is not our whole galaxy; the majority of the other stars visible to the naked eye also lie in our galaxy. That dense band is not thicker and brighter than other parts of our galaxy because it lies toward a dense galactic center (our galaxy has an empty center), but for the opposite reason: those apparently more thickly clustered stars lie all around us on the galaxy's surface, in the nearest region of 4-space surrounding us. They appear to be densely packed only because we are looking at them "edge on". Actually, we are looking into this nearby apparently dense region ''face on'', not edge on, because it is a round sphere of nearby space surrounding us. In contrast, stars in our galaxy outside that bright band lie farther off from us, across the empty center of the galaxy, and we see them spread out as they actually are, instead of "edge on" where they appear to be more densely clustered. The "dense band" covers only an equatorial band of the night sky instead of all the sky, because when we look out into the four-dimensional space around us, we can see stars above and below our three-dimensional hyperplane in four-dimensional space. Everything in our solar system lies in our hyperplane, and the nearby stars around us in our galaxy are near our hyperplane (just slightly below it). All the other, more distant stars in our galaxy are also below our hyperplane. We can see objects outside our galaxy, such as other galaxies, both above and below our hyperplane, because we receive light signals from all directions in 4-space. We can see distant objects all around us above our hyperplane (looking up from the galactic surface into the fourth dimension), and all around us below our hyperplane (looking down through our transparent 4-dimensional galaxy and out the other side). But we do not observe (or measure) the angles of objects above or below our hyperplane, because we conceptualize the volume of space around us as a 3-ball when it is actually a 4-ball. We unknowingly conflate all the objects we receive light from, projecting them into a single 3-ball, whether they are above, below or in our hyperplane. == The dimpled fabric of spacetime == The spacetime of general relativity is often illustrated as a projection to a curved 2D surface in which large gravitational objects make gravity wells or dimples in the surface. In the Euclidean 4D view of the universe the 3D surface of a large cosmic object such as a galaxy surrounds an empty 4D space, and large gravitational objects within the galaxy must make dimples in its surface. But should we see them as dimples exactly? Would they dimple inwards or outwards? In the spacetime illustrations they are naturally always shown as dimpling downwards, which is somewhat disingenuous, strongly suggesting to the viewer that the reason for gravity is that it flows downhill - the original tautology we are trying to surmount! In the Euclidean 4D galaxy the dimple, if it is one, must be either inward or outward, and which it is matters. Is a large gravitational mass (such as a star) ''ahead'' of the smaller masses orbiting around it (such as its planets), or is it ''behind'' them, as they fly through 4-space on their Clifford parallel trajectories? The answer is ''both'' of course, because a star is not a dimple, it is a 4-ball, and it dimples the 3D surface both inwards and outwards. It is a thick place in the 3D surface. We should view it as having its gravitational center precisely at the surface of a 3-sphere expanding radially at velocity <math>c</math>. [[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 18:15, 10 November 2024 (UTC) == John Stewart Bell by Adam Becker == A roulette wheel has three orthogonal dimensions of binary parity: high/low (a finite integer set partitioned in two), even/odd (a different partitioning of the integer set), and a third dimension red/black (which belong to a distinct sequence, and I dare say a distinct group, than the other two parities). "Polarization is a property of light -- light is an electromagnetic wave, and the polarization is the direction that the wave is doing its waving in. Polarization is sort of like a little arrow that each photon carries with it that can point in different dirdctions. But ... we can't actually tell what direction a photon's polarization arrow is pointing. All we can do is measure a photon's polarization along one particular axis at a time in a somewhat indirect [binary] manner ...." [[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 21:36, 31 December 2024 (UTC) == Minkowski spacetime and Euclidean 4-space in relativity == ... Is there a configuration in Euclidean space where every observer, and every observed object, is moving at velocity <math>c</math>? Yes, there is one such configuration, in 4-dimensional Euclidean space. This configuration must be like the one described above, in our ''Abstract''. == The rate of atomic symmetry operations == ... == General relativity describes a curved 3-dimensional manifold embedded in Euclidean 4-dimensional space == ... == The geometry of the atomic nucleus == In [[W:Euclidean 4-space|Euclidean four dimensional space]], an [[W:atomic nucleus|atomic nucleus]] is a one or more concentric 4-polytopes of increasing radius. Each concentric shell is a single or compound [[24-cell]], the regular 4-polytope with [[W:Coxeter group#Symmetry groups of regular polytopes|𝔽₄ symmetry]]. Nuclear shells are concentric [[W:3-sphere|3-sphere]]s occupied (fully or partially) by the orbits of this 24-point [[#The 6 regular convex 4-polytopes|regular convex 4-polytope]]. An actual atomic nucleus is a rotating four dimensional object. It is not a ''rigid'' rotating object, it is a kinematic one, because the nucleus of an actual atom of a distinct [[W:nucleon number|nucleon number]] contains a distinct number of orbiting 24-cell vertices which may be in different isoclinic rotational orbits. These moving vertices never describe a compound of static 24-cells at any single instant in time, though their orbits do all the time. The physical configuration of a nucleus as concentric 24-cells can be reduced to the [[W:kinematics|kinematics]] of the orbits of its constituent protons and neutrons. The geometry of the atomic nucleus is therefore strictly [[W:Euclidean geometry#19th century|Euclidean]] in four dimensional space. === 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#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 (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 precise [[24-cell|detailed description]] enabling the reader to properly visualize it runs to many pages and illustrations, with many accompanying pages of explanatory notes on basic phenomena that arise only in 4-dimensional space: [[24-cell#Squares|completely orthogonal planes]], [[24-cell#Hexagons|Clifford parallelism]] and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Helical hexagrams and their isoclines|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 surprise. [[#The 6 regular convex 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 (generally), 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. [[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 is much harder for us to visualize, because the only way we can experience it is in our imaginations; we have no body of sensory experience in 4-dimensional space to draw upon. For that reason, 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. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than 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 any other at any time. That is still an example of a rigid object in a single distinct isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing the characteristic rotation of the 24-cell. But 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 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 shell without collisions? In adjacent 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 questions of this kind of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[W:kinetics|kinetics]]. In four dimensions, we find that space is more commodious than we thought it was, and 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 more symmetric packings. Sadoc studied twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack perfectly in 4-space without exhibiting any torsion, although their packing in 3-space was imperfect, "frustrated" by their torsion. <blockquote>The frustration, which arises when the molecular orientation is transported along the two [circular] AB paths of figure 1 [helix], is imposed by the very topological nature of the Euclidean space R³. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S³, 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]]....{{Efn|name=Petrie polygon of a honeycomb}} The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S³, 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_∞ 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 S³.{{Efn|name=dense fabric of pole-circles}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc & Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings ("frustrations") in 3-dimensional space.}}</blockquote> Of course we do not find molecules condensing to close-pack the 3-sphere in our experience. We find 3-spheres in the atomic realm (atoms are 4-polytopes), and in cosmologic realm (at the surface boundaries of stars, and perhaps at the surface masses of hollow galaxies). But in the molecular realm we find only objects which 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 wholly imaginary one, though no less illuminating for the symmetry it reveals not having a realization as a 3-sphere of actual molecules. And might it not have a realization in the atomic realm? We know that atoms have their own complex internal structure, which we are unable to model geometrically in flat 3-dimensional space. Suppose that the constituent particles of atoms actually close-pack the 3-sphere as helices, in the manner of Sadoc's imaginary twisted molecules, that is, as discrete Hopf fibrations of the atomic 4-polytope, where each discrete Hopf fibration corresponds to the distinct isoclinic rotation characteristic of a constituent particle. [[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 19:28, 8 November 2025 (UTC) === Isospin === A [[W:Nucleon|nucleon]] is a [[W:proton|proton]] or a [[W:neutron|neutron]]. The proton carries a positive net [[W:Electric charge|charge]], and the neutron carries a zero net charge. The proton's [[W:Mass|mass]] is only about 0.13% less than the neutron's, and since they are observed to be identical in other respects, they can be viewed as two states of the same nucleon, together forming an isospin doublet ({{nowrap|''I'' {{=}} {{sfrac|1|2}}}}). In isospin space, neutrons can be transformed into protons and conversely by actions of the [[W:SU(2)|SU(2)]] symmetry group. In nature, protons are very stable (the most stable particle known); a proton and a neutron are a stable nuclide; but free neutrons decay into protons in about 10 or 15 seconds. According to the [[W:Noether theorem|Noether theorem]], [[W:Isospin|isospin]] is conserved with respect to the [[W:strong interaction|strong interaction]].<ref name=Griffiths2008>{{cite book |author=Griffiths, David J. |title=Introduction to Elementary Particles |edition=2nd revised |publisher=WILEY-VCH |year=2008 |isbn=978-3-527-40601-2}}</ref>{{rp|129–130}} Nucleons are acted upon equally by the strong interaction, which is invariant under rotation in isospin space. Isospin was introduced as a concept in 1932 by [[W:Werner Heisenberg|Werner Heisenberg]],<ref> {{cite journal |last=Heisenberg |first=W. |author-link=W:Werner Heisenberg |year=1932 |title=Über den Bau der Atomkerne |journal=[[W:Zeitschrift für Physik|Zeitschrift für Physik]] |volume=77 |issue=1–2 |pages=1–11 |doi=10.1007/BF01342433 |bibcode = 1932ZPhy...77....1H |s2cid=186218053 |language=de}}</ref> well before the 1960s development of the [[W:quark model|quark model]], to explain the symmetry of the proton and the then newly discovered neutron. Heisenberg introduced the concept of another conserved quantity that would cause the proton to turn into a neutron and vice versa. In 1937, [[W:Eugene Wigner|Eugene Wigner]] introduced the term "isospin" to indicate how the new quantity is similar to spin in behavior, but otherwise unrelated.<ref> {{cite journal |last=Wigner |first=E. |author-link=W:Eugene Wigner |year=1937 |title=On the Consequences of the Symmetry of the Nuclear Hamiltonian on the Spectroscopy of Nuclei |journal=[[W:Physical Review|Physical Review]] |volume=51 |pages=106–119 |doi=10.1103/PhysRev.51.106 |bibcode = 1937PhRv...51..106W |issue=2 }}</ref> Similar to a spin-1/2 particle, which has two states, protons and neutrons were said to be of isospin 1/2. The proton and neutron were then associated with different isospin projections ''I''₃&nbsp;=&nbsp;+1/2 and −1/2 respectively. Isospin is a different kind of rotation entirely than the ordinary spin which objects undergo when they rotate in three-dimensional space. Isospin does not correspond to a [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] in any space (of any number of dimensions). However, it does seem to correspond exactly to an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]] in a Euclidean space of four dimensions. Isospin space resembles the [[W:3-sphere|3-sphere]], the [[W:Elliptical space#Elliptic space (the 3D case)|curved 3-dimensional space]] that is the surface of a [[W:4-ball (mathematics)#In Euclidean space|4-dimensional ball]]. === Spinors === [[File:Spinor on the circle.png|thumb|upright=1.5|A spinor visualized as a vector pointing along the [[W:Möbius band|Möbius band]], exhibiting a sign inversion when the circle (the "physical system") is continuously rotated through a full turn of 360°.]][[W:Spinors|Spinors]] are [[W:representation of a Lie group|representations]] of a [[W:spin group|spin group]], which are [[W:Double covering group|double cover]]s of the [[W:special orthogonal group|special orthogonal groups]]. The spin group Spin(4) is the double cover of [[W:SO(4)|SO(4)]], the group of rotations in 4-dimensional Euclidean space. [[600-cell#Fibrations of isocline polygrams|Isoclines]], the helical geodesic paths followed by points under isoclinic rotation, correspond to spinors representing Spin(4). Spinors can be viewed as the "square roots" of [[W:Section (fiber bundle)|cross sections]] of [[W:vector bundle|vector bundle]]s; in this correspondence, a fiber bundle of isoclines (of a distinct isoclinic rotation) is a cross section (inverse bundle) of a fibration of great circles (in the invariant planes of that rotation). A spinor can be visualized as a moving vector on a Möbius strip which transforms to its negative when continuously rotated through 360°, just as [[24-cell#Helical hexagrams and their isoclines|an isocline can be visualized as a Möbius strip]] winding twice around the 3-sphere, during which [[24-cell#Isoclinic rotations|720° isoclinic rotation]] the rigid 4-polytope turns itself inside-out twice.{{Sfn|Goucher|2019|loc=Spin Groups}} Under isoclinic rotation, a rigid 4-polytope is an isospin-1/2 object with two states. === Isoclinic rotations in the nucleus === Isospin is regarded as a symmetry of the strong interaction under the [[W:Group action (mathematics)|action]] of the [[W:Lie group|Lie group]] [[W:SU(2)|SU(2)]], the two [[W:eigenstate|states]] being the [[W:Up quark|up flavour]] and [[W:Down quark|down flavour]]. A 360° isoclinic rotation of a rigid [[W:nuclide|nuclide]] would transform its protons into neutrons and vice versa, exchanging the up and down flavours of their constituent [[W:quarks|quarks]], by turning the nuclide and all its parts inside-out (or perhaps we should say upside-down). Because we never observe this, we know that the nucleus is not a ''rigid'' polytope undergoing isoclinic rotation. If the nucleus ''were'' a rigid object, nuclides that were isospin-rotated 360° would be isoclinic mirror images of each other, isospin +1/2 and isospin −1/2 states of the whole nucleus. We don't see whole nuclides rotating as a rigid object, but considering what would happen if they ''were'' rigid tells us something about the geometry we must expect inside the nucleons. One way that an isospin-rotated neutron could become a proton would be if the up quark and down quark were a left and right mirror-image pair of the same object; exchanging them in place would turn each down-down-up neutron into an up-up-down proton. But the case cannot be quite that simple, because the up quark and the down quark are not mirror-images of the same object: they have very different mass and other incongruities. Another way an isospin-rotated neutron could be a proton would be if the up and down quarks were asymmetrical kinematic polytopes (not indirectly congruent mirror-images, and not rigid polytopes), rotating within the nucleus in different ''hybrid'' orbits. By that we mean that they may have vertices orbiting in rotations characteristic of more than one 4-polytope, so they may change shape as they rotate. In that case their composites (protons and neutrons) could have a symmetry not manifest in their components, but emerging from their combination. .... === Hybrid isoclinic rotations === The 24-cell has [[24-cell#Isoclinic rotations|its own characteristic isoclinic rotations]] in 4 Clifford parallel hexagonal planes (each intersecting 6 vertices), and also inherits the [[16-cell#Rotations|characteristic isoclinic rotations of its 3 Clifford parallel constituent 16-cells]] in 6 Clifford parallel square planes (each intersecting 4 vertices). The twisted circular paths followed by vertices in these two different kinds of rotation have entirely different geometries. Vertices rotating in hexagonal invariant planes follow [[24-cell#Helical hexagrams and their isoclines|helical geodesic curves whose chords form hexagrams]], and vertices rotating in square invariant planes follow [[24-cell#Helical octagrams and their isoclines|helical geodesic curves whose chords form octagrams]]. In a rigid isoclinic rotation, ''all'' the [[24-cell#Geodesics|great circle polygons]] move, in any kind of rotation. What distinguishes the hexagonal and square isoclinic rotations is the invariant planes of rotation the vertices stay in. The rotation described [[#Rotations|above]] (of 8 vertices rotating in 4 Clifford parallel hexagonal planes) is a single hexagonal isoclinic rotation, not a kinematic or hybrid rotation. A ''kinematic'' isoclinic rotation in the 24-cell is any subset of the 24 vertices rotating through the same angle in the same time, but independently with respect to the choice of a Clifford parallel set of invariant planes of rotation and the chirality (left or right) of the rotation. A ''hybrid'' isoclinic rotation combines moving vertices from different kinds of isoclinic rotations, characteristic of different regular 4-polytopes. For example, if at least one vertex rotates in a square plane and at least one vertex rotates in a hexagonal plane, the kinematic rotation is a hybrid rotation, combining rotations characteristic of the 16-cell and characteristic of the 24-cell. As an example of the simplest hybrid isoclinic rotation, consider a 24-cell vertex rotating in a square plane, and a second vertex, initially one 24-cell edge-length distant, rotating in a hexagonal plane. Rotating isoclinically at the same rate, the two moving vertices will never collide where their paths intersect, so this is a ''valid'' hybrid rotation. To understand hybrid rotations in the 24-cell more generally, visualize the relationship between great squares and great hexagons. The [[24-cell#Squares|18 great squares]] occur as three sets of 6 orthogonal great squares,{{Efn|name=six orthogonal planes of the Cartesian basis}} each [[16-cell#Coordinates|forming a 16-cell]]. The three 16-cells are completely disjoint{{Efn|name=completely disjoint}} and [[24-cell#Clifford parallel polytopes|Clifford parallel]]: each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}).{{Efn|name=three isoclinic 16-cells}} The 18 square great circles are crossed by 16 hexagonal great circles; each [[24-cell#Hexagons|hexagon]] has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two [[24-cell#Triangles|great triangles]] inscribed in each great hexagon (occupying its alternate vertices, with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking three completely disjoint great squares, one from each of the three completely disjoint 16-cells''.{{Efn|There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}} Isoclinic rotations take the elements of the 4-polytope to congruent [[24-cell#Clifford parallel polytopes|Clifford parallel elements]] elsewhere in the 4-polytope. The square rotations do this ''locally'', confined within each 16-cell: for example, they take great squares to other great squares within the same 16-cell. The hexagonal rotations act ''globally'' within the entire 24-cell: for example, they take great squares to other great squares in ''different'' 16-cells. The [[16-cell#Helical construction|chords of the square rotations]] bind the 16-cells together internally, and the [[24-cell#Helical hexagrams and their isoclines|chords of the hexagonal rotations]] bind the three 16-cells together. .... === Charge === Positive and negative charge are [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#The perfection of Fuller's cyclic design|compression and tension]], respectively. ... === Color === When the existence of quarks was suspected in 1964, [[W:Oscar W. Greenberg|Greenberg]] introduced the notion of color charge to explain how quarks could coexist inside some [[W:hadron|hadron]]s in [[W:quark model#The discovery of color|otherwise identical quantum states]] without violating the [[W:Pauli exclusion principle|Pauli exclusion principle]]. The modern concept of [[W:color charge|color charge]] completely commuting with all other charges and providing the strong force charge was articulated in 1973, by [[W:William A. Bardeen|William Bardeen]], [[W:de:Harald Fritzsch|Harald Fritzsch]], and [[W:Murray Gell-Mann|Murray Gell-Mann]].<ref>{{cite conference |author1=Bardeen, W. |author2=Fritzsch, H. |author3=Gell-Mann, M. |year=1973 |title=Light cone current algebra, ''π''⁰ decay, and ''e''⁺ ''e''^&minus; annihilation |arxiv=hep-ph/0211388 |editor=Gatto, R. |book-title=Scale and conformal symmetry in hadron physics |page=[https://archive.org/details/scaleconformalsy0000unse/page/139 139] |publisher=[[W:John Wiley & Sons|John Wiley & Sons]] |isbn=0-471-29292-3 |bibcode=2002hep.ph...11388B |url-access=registration |url=https://archive.org/details/scaleconformalsy0000unse/page/139 }}</ref><ref>{{cite journal |title=Advantages of the color octet gluon picture |journal=[[W:Physics Letters B|Physics Letters B]] |volume=47 |issue=4 |page=365 |year=1973 |last1=Fritzsch |first1=H. |last2=Gell-Mann |first2=M. |last3=Leutwyler |first3=H. |doi=10.1016/0370-2693(73)90625-4 |bibcode=1973PhLB...47..365F |citeseerx=10.1.1.453.4712}}</ref> Color charge is not [[W:electric charge|electric charge]]; the whole point of it is that it is a quantum of something different. But it is related to electric charge, through the way in which the three different-colored quarks combine to contribute fractional quantities of electric charge to a nucleon. As we shall see, color is not really a separate kind of charge at all, but a partitioning of the electric charge into [[24-cell#Clifford parallel polytopes|Clifford parallel subspaces]]. The [[W:Color charge#Red, green, and blue|three different colors]] of quark charge might correspond to three different 16-cells, such as the three disjoint 16-cells inscribed in the 24-cell. Each color might be a disjoint domain in isospin space (the space of points on the 3-sphere).{{Efn|The 8 vertices of each disjoint 16-cell constitute an independent [[16-cell#Coordinates|orthonormal basis for a coordinate reference frame]].}} Alternatively, the three colors might correspond to three different fibrations of the same isospin space: three different ''sequences'' of the same total set of discrete points on the 3-sphere. These alternative possibilities constrain possible representations of the nuclides themselves, for example if we try to represent nuclides as particular rotating 4-polytopes. If the neutron is a (8-point) 16-cell, either of the two color possibilities might somehow make sense as far as the neutron is concerned. But if the proton is a (5-point) 5-cell, only the latter color possibility makes sense, because fibrations (which correspond to distinct isoclinic left-and-right rigid rotations) are the ''only'' thing the 5-cell has three of. Both the 5-cell and the 16-cell have three discrete rotational fibrations. Moreover, in the case of a rigid, isoclinically rotating 4-polytope, those three fibrations always come one-of-a-kind and two-of-a-kind, in at least two different ways. First, one fibration is the set of invariant planes currently being rotated through, and the other two are not. Second, when one considers the three fibrations of each of these 4-polytopes, in each fibration two isoclines carry the left and right rotations respectively, and the third isocline acts simply as a Petrie polygon, the difference between the fibrations being the role assigned to each isocline. If we associate each quark with one or more isoclinic rotations in which the moving vertices belong to different 16-cells of the 24-cell, and the sign (plus or minus) of the electric charge with the chirality (right or left) of isoclinic rotations generally, we can configure nucleons of three quarks, two performing rotations of one chirality and one performing rotations of the other chirality. The configuration will be a valid kinematic rotation because the completely disjoint 16-cells can rotate independently; their vertices would never collide even if the 16-cells were performing different rigid square isoclinic rotations (all 8 vertices rotating in unison). But we need not associate a quark with a [[16-cell#Rotations|rigidly rotating 16-cell]], or with a single distinct square rotation. Minimally, we must associate each quark with at least one moving vertex in each of three different 16-cells, following the twisted geodesic isocline of an isoclinic rotation. In the up quark, that could be the isocline of a right rotation; and in the down quark, the isocline of a left rotation. The chirality accounts for the sign of the electric charge (we have said conventionally as +right, −left), but we must also account for the quantity of charge: +{{sfrac|2|3}} in an up quark, and −{{sfrac|1|3}} in a down quark. One way to do that would be to give the three distinct quarks moving vertices of {{sfrac|1|3}} charge in different 16-cells, but provide up quarks with twice as many vertices moving on +right isoclines as down quarks have vertices moving on −left isoclines (assuming the correct chiral pairing is up+right, down−left). Minimally, an up quark requires two moving vertices (of the up+right chirality).{{Efn|Two moving vertices in one quark could belong to the same 16-cell. A 16-cell may have two vertices moving in the same isoclinic square (octagram) orbit, such as an antipodal pair (a rotating dipole), or two vertices moving in different square orbits of the same up+right chirality.{{Efn|There is only one [[16-cell#Helical construction|octagram orbit]] of each chirality in each fibration of the 16-cell, so two octagram orbits of the same chirality cannot be Clifford parallel (part of the same distinct rotation). Two vertices right-moving on different octagram isoclines in the same 16-cell is a combination of two distinct rotations, whose isoclines will intersect: a kinematic rotation. It can be a valid kinematic rotation if the moving vertices will never pass through a point of intersection at the same time. Octagram isoclines pass through all 8 vertices of the 16-cell, and all eight isoclines (the left and right isoclines of four different fibrations) intersect at ''every'' vertex.}} However, the theory of [[W:Color confinement|color confinement]] may not require that two moving vertices in one quark belong to the same 16-cell; like the moving vertices of different quarks, they could be drawn from the disjoint vertex sets of two different 16-cells.}} Minimally, a down quark requires one moving vertex (of the down−left chirality). In these minimal quark configurations, a proton would have 5 moving vertices and a neutron would have 4. .... === Nucleons === [[File:Symmetrical_5-set_Venn_diagram.svg|thumb|[[W:Branko Grünbaum|Grünbaum's]] rotationally symmetrical 5-set Venn diagram, 1975. It is an illustration of the [[5-cell]] labeling each of its <math>2^5</math> elements. Think of it as an [[W:Nuclear magnetic resonance|NMR image]] of the 4-dimensional proton in projection to the plane.]] The proton is a very stable mass particle. Is there a stable orbit of 5 moving vertices in 4-dimensional Euclidean space? There are few known solutions to the 5-body problem, and fewer still to the [[W:n-body problem|{{mvar|n}}-body problem]], but one is known: the ''central configuration'' of {{mvar|n}} bodies in a space of dimension {{mvar|n}}-1. A [[W:Central configuration|central configuration]] is a system of [[W:Point particle|point masses]] with the property that each mass is pulled by the combined attractive force of the system directly towards the [[W:Center of mass|center of mass]], with acceleration proportional to its distance from the center. Placing three masses in an equilateral triangle, four at the vertices of a regular [[W:Tetrahedron|tetrahedron]], five at the vertices of a regular [[5-cell]], or more generally {{mvar|n}} masses at the vertices of a regular [[W:Simplex|simplex]] produces a central configuration [[W:Central configuration#Examples|even when the masses are not equal]]. In an isoclinic rotation, all the moving vertices orbit at the same radius and the same speed. Therefore if any 5 bodies are orbiting as an isoclinically rotating regular 5-cell (a rigid 4-simplex figure undergoing isoclinic rotation), they maintain a central configuration, describing 5 mutually stable orbits. Unlike the proton, the neutron is not always a stable particle; a free neutron will decay into a proton. A deficiency of the minimal configurations is that there is no way for this [[W:beta minus decay|beta minus decay]] to occur. The minimal neutron of 4 moving vertices described [[#Color|above]] cannot possibly decay into a proton by losing moving vertices, because it does not possess the four up+right moving vertices required in a proton. This deficiency could be remedied by giving the neutron configuration 8 moving vertices instead of 4: four down−left and four up+right moving vertices. Then by losing 3 down−left moving vertices the neutron could decay into the 5 vertex up-down-up proton configuration.{{Efn|Although protons are very stable, during [[W:stellar nucleosynthesis|stellar nucleosynthesis]] two H₁ protons are fused into an H₂ nucleus consisting of a proton and a neutron. This [[W:beta plus decay|beta plus "decay"]] of a proton into a neutron is actually the result of a rare high-energy collision between the two protons, in which a neutron is constructed. With respect to our nucleon configurations of moving vertices, it has to be explained as the conversion of two 5-point 5-cells into a 5-point 5-cell and an 8-point 16-cell, emitting two decay products of at least 1-point each. Thus it must involve the creation of moving vertices, by the conversion of kinetic energy to point-masses.}} A neutron configuration of 8 moving vertices could occur as the 8-point 16-cell, the second-smallest regular 4-polytope after the 5-point 5-cell (the hypothesized proton configuration). It is possible to double the neutron configuration in this way, without destroying the charge balance that defines the nucleons, by giving down quarks three moving vertices instead of just one: two −left vertices and one +right vertex. The net charge on the down quark remains −{{sfrac|1|3}}, but the down quark becomes heavier (at least in vertex count) than the up quark, as in fact its mass is measured to be. A nucleon's quark configuration is only a partial specification of its properties. There is much more to a nucleon than what is contained within its three quarks, which contribute only about 1% of the nucleon's energy. The additional 99% of the nucleon mass is said to be associated with the force that binds the three quarks together, rather than being intrinsic to the individual quarks separately. In the case of the proton, 5 moving vertices in the stable orbits of a central configuration (in one of the [[5-cell#Geodesics and rotations|isoclinic rotations characteristic of the regular 5-cell]]) might be sufficient to account for the stability of the proton, but not to account for most of the proton's energy. It is not the point-masses of the moving vertices themselves which constitute most of the mass of the nucleon; if mass is a consequence of geometry, we must look to the larger geometric elements of these polytopes as their major mass contributors. The quark configurations are thus incomplete specifications of the geometry of the nucleons, predictive of only some of the nucleon's properties, such as charge.{{Efn|Notice that by giving the down quark three moving vertices, we seem to have changed the quark model's prediction of the proton's number of moving vertices from 5 to 7, which would be incompatible with our theory that the proton configuration is a rotating regular 5-cell in a central configuration of 5 stable orbits. Fortunately, the actual quark model has nothing at all to say about moving vertices, so we may choose to regard that number as one of the geometric properties the quark model does not specify.}} In particular, they do not account for the forces binding the nucleon together. Moreover, if the rotating regular 5-cell is the proton configuration and the rotating regular 16-cell is the neutron configuration, then a nucleus is a complex of rotating 5-cells and 16-cells, and we must look to the geometric relationship between those two very different regular 4-polytopes for an understanding of the nuclear force binding them together. The most direct [[120-cell#Relationships among interior polytopes|geometric relationship among stationary regular 4-polytopes]] is the way they occupy a common 3-sphere together. Multiple 16-cells of equal radius can be compounded to form each of the larger regular 4-polytopes, the 8-cell, 24-cell, 600-cell, and 120-cell, but it is noteworthy that multiple regular 5-cells of equal radius cannot be compounded to form any of the other 4-polytopes except the largest, the 120-cell. The 120-cell is the unique intersection of the regular 5-cell and 16-cell: it is a compound of 120 regular 5-cells, and also a compound of 75 16-cells. All regular 4-polytopes except the 5-cell are compounds of 16-cells, but none of them except the largest, the 120-cell, contains any regular 5-cells. So in any compound of equal-radius 16-cells which also contains a regular 5-cell, whether that compound forms some single larger regular 4-polytope or does not, no two of the regular 5-cell's five vertices ever lie in the same 16-cell. So the geometric relationship between the regular 5-cell (our proton candidate) and the regular 16-cell (our neutron candidate) is quite a distant one: they are much more exclusive of each other's elements than they are distantly related, despite their complementary three-quark configurations and other similarities as nucleons. The relationship between a regular 5-cell and a regular 16-cell of equal radius is manifest only in the 120-cell, the most complex regular 4-polytope, which [[120-cell#Geometry|uniquely embodies all the containment relationships]] among all the regular 4-polytopes and their elements. If the nucleus is a complex of 5-cells (protons) and 16-cells (neutrons) rotating isoclinically around a common center, then its overall motion is a hybrid isoclinic rotation, because the 5-cell and the 16-cell have different characteristic isoclinic rotations, and they have no isoclinic rotation in common.{{Efn|The regular 5-cell does not occur inscribed in any other regular 4-polytope except one, the 600-vertex 120-cell. No two of the 5 vertices of a regular 5-cell can be vertices of the same 16-cell, 8-cell, 24-cell, or 600-cell. The isoclinic rotations characteristic of the regular 5-cell maintain the separation of its 5 moving vertices in 5 disjoint Clifford-parallel subspaces at all times. The [[16-cell#Rotations|isoclinic rotation characteristic of the 16-cell]] maintains the separation of its 8 moving vertices in 2 disjoint Clifford-parallel subspaces (completely orthogonal great square planes) at all times. Therefore, in any hybrid rotation of a concentric 5-cell and 16-cell, at most one 5-cell subspace (containing 1 vertex) might be synchronized with one 16-cell subspace (containing 4 vertices), such that the 1 + 4 vertices they jointly contain occupy the same moving subspace continually, forming a rigid 5-vertex polytope undergoing some kind of rotation. If in fact it existed, this 5-vertex rotating rigid polytope would not be [[5-cell#Geometry|not a 5-cell, since 4 of its vertices are coplanar]]; it is not a 4-polytope but merely a polyhedron, a [[W:square pyramid|square pyramid]].}} .... === Nuclides === ... === Quantum phenomena === The Bell-Kochen-Specker (BKS) theorem rules out the existence of deterministic noncontextual hidden variables theories. A proof of the theorem in a space of three or more dimensions can be given by exhibiting a finite set of lines through the origin that cannot each be colored black or white in such a way that (i) no two orthogonal lines are both black, and (ii) not all members of a set of ''d'' mutually orthogonal lines are white.{{Efn|"The Bell-Kochen-Specker theorem rules out the existence of deterministic noncontextual hidden variables theories. A proof of the theorem in a Hilbert space of dimension d ≥ 3 can be given by exhibiting a finite set of rays [9] that cannot each be assigned the value 0 or 1 in such a way that (i) no two orthogonal rays are both assigned the value 1, and (ii) not all members of a set of d mutually orthogonal rays are assigned the value 0."{{Sfn|Waegell & Aravind|2009|loc=2. The Bell-Kochen-Specker (BKS) theorem}}|name=BKS theorem}} .... === Motion === What does it mean to say that an object moves through space? Coxeter group theory provides precise answers to questions of this kind. A rigid object (polytope) moves by distinct transformations, changing itself in each discrete step into a congruent object in a different orientation and position.{{Efn|name=transformations}} .... === General relativity is just Galilean relativity in a general space of four orthogonal dimensions === .... Coxeter orthogonal dimensional relativity furnishes the geometric spatial definition of the inertial-gravitational force. It does so by limiting free motion (in the sense of free-falling) to one rotation and one translation, making double rotations accelerations which involve an energy transition. ...I now don't think that is quite right, rather: It does so by limiting free motion (in the sense of free-falling) to a double rotation (or one rotation and one translation) at conserved angles, making changes to the angles of rotation accelerations which involve an energy transition. === Physics === .... === Spherical relativity === We began our inquiry by wondering why physical space should be limited to just three dimensions (why ''three''). By visualizing the universe as a Euclidian space of four dimensions, we recognize that relativistic and quantum phenomena are natural consequences of symmetry group operations (including reflections and rotations) in four orthogonal dimensions. We are then led naturally to question whether the universe has only four dimensions. Physical space must bear as many dimensions as we need to ascribe to it, though the distinct phenomena for which we find a need to do so, in order to explain them, seem to be fewer and fewer as we consider higher and higher dimensions. To laws of physics generally, such as the principle of relativity in particular, we should always append the phrase "in Euclidean spaces of any number of dimensions". Laws of physics should operate in any flat Euclidean space <math>R^n</math> and in its corresponding spherical space <math>S^n</math>. The first and simplest sense in which we are forced to contemplate a fifth dimension is to accommodate our normal idea of time. Just as Einstein was forced to admit time as a dimension, in his four-dimensional spacetime of three spatial dimensions plus time, for some purposes we require a fifth time dimension to accompany our four spatial dimensions, when our purpose is orthogonal to (in the sense of independent of) the four spatial dimensions. For example, if we theorize that we observe a finite homogeneous universe, and that it is a Euclidean 4-space overall, we may prefer not to have to identify any distinct place within that 4-space as the center where the universe began in a big bang. To avoid having to pick a distinct place as the center of the universe, our model of it must be expanded, at least to be a ''spherical'' 4-dimensional space with the fifth radial dimension as time. Essentially, we require the fifth dimension in order to make our homogeneous 4-space finite, by wrapping it around into a 4-sphere. But perhaps we can still resist admitting the fifth radial dimension as a full-fledged Euclidean spatial dimension, at least so long as we have not observed how any naturally occurring object configurations are best described as 5-polytopes. One phenomenon which resists explanation in a space of just four dimensions is the propagation of light in a vacuum. The propagation of mass-carrying particles is explained as the consequence of their rotations in closed, curved spaces (3-spheres) of finite size, moving through four-dimensional Euclidean space at a universal constant speed, the speed of light. But an apparent paradox remains that light must seemingly propagate through four-dimensional Euclidean space at more than the speed of light. From an ''n''-dimensional viewpoint, this apparent paradox can be resolved, and in retrospect it is clear how massless particles can translate through four-dimensional space at twice the speed constant, since they are not simultaneously rotating. Another phenomenon justifying a five-dimensional view of space is the relation between the the 5-cell proton and the 16-cell neutron (the 4-simplex and 4-orthoplex polytopes). Their indirect relationship can be observed in the 4-600-point polytope (the 120-cell), and in its 11-cells,{{Sfn|Christie|2025|loc="A symmetrical arrangement of eleven 11-cells"}} but it is only directly observed (absent a 120-cell) in a five-dimensional reference frame. === Nuclear geometry === We have seen how isoclinic rotations (Clifford displacements) relate the orbits in the atomic nucleus to each other, just as they relate the regular convex 4-polytopes to each other, in a sequence of nested objects of increasing complexity. We have identified the proton as a 5-point, 5-cell 4-simplex 𝜶₄, the neutron as an 8-point, 16-cell 4-orthoplex 𝛽₄, and the shell of the atomic nucleus as a 24-point 24-cell. As Coxeter noted, that unique 24-point object stands quite alone in four dimensions, having no analogue above or below. === Atomic geometry === I'm on a plane flying to Eugene to visit Catalin, we'll talk after I arrive. I've been working on both my unpublished papers, the one going put for pre-publication review soon about 4D geometry, and the big one not going out soon about the 4D sun, 4D atoms, and 4D galaxies and n-D universe. I'vd just added the following paragraph to that big paper: Atomic geometry The force binding the protons and neutrons of the nucleus together into a distinct element is specifically an expression of the 11-cell 4-polytope, itself an expression of the pyritohedral symmetry, which binds the distinct 4-polytopes to each other, and relates the n-polytopes to their neighbors of different n by dimensional analogy. flying over mt shasta out my right-side window at the moment, that last text showing "not delivered" yet because there's no wifi on this plane, gazing at that great peak of the world and feeling as if i've just made the first ascent of it === Atomic elements === There are 120 5-cell protons in the 120-cell, and so there are 120 distinct atomic elements on the 3-sphere. === Molecular geometry === Molecules are 3-dimensional structures that live in the thin film of 3-membrane only one atom thick in most places that is our ordinary space, but since that is a significantly curved 3-dimensional space at the scale of a molecule, the way the molecule's covalent bonds form is influenced by the local curvature in 4-dimensions at that point. In the water molecule, there is a reason why the hydrogen atoms are attached to the oxygen atom at an angle of 104.45° in 3-dimensional space, and at root it must be the same symmetry that locates any two of the hydrogen proton's five vertices 104.45° apart on a great circle arc of its tiny 3-sphere. === Cosmology === ==== Solar systems ==== ===== Stars ===== ... ===== The Kepler problem ===== In the proper reference frame of some hypothetical observer in 4-dimensional Euclidean space, the Kepler problem has a solution in which all the planetary orbits are circles. This occurs because all bodies are always orbiting, each in some distinct isoclinic (equi-angled double) rotation. All atoms are rotating isoclinically at (double) the velocity <math>c</math>. In the observer's proper reference frame, the isoclinic double rotation of each of his own atoms is most conveniently represented as an equivalent screw displacement: a linear translation at velocity <math>c</math> (along the observer's proper time axis through 4-space), combined with a simple rotation at velocity <math>c</math> (of the atom internally). This representation, though proper, is subject to the over-simplification of special relativity, because it ignores the fact that the observer is himself moving on some circular orbit; he is not linearly translating through 4-space. When the observer's own orbit is also represented, the screw displacement becomes an elliptical double rotation, with a very slow, very large-radius simple circular orbit (almost a linear translation), combined with a very fast, very small-radius simple circular orbit of each atom internally. This is the observer's proper reference frame according to general relativity. An observer moving differently will perceive the motion of the observer and his atoms to be a different elliptical double rotation. We can find a proper reference frame, of some hypothetical observer moving hypothetically, in which the ratio of the radii of the two orthogonal rotations of the atom becomes any value we choose between 1 and <math>c</math>. In particular there exists, conceptually accessible to all observers independent of their motion, a distinct reference frame in which an atom is stationary, rotating isoclinically at (double) the velocity <math>c</math>. That is simply the stationary reference frame of the atom itself, such as the stationary reference frame of any observer and his own atoms. In 3-dimensional physics, it is known as the observer's proper inertial reference frame. But in 4-dimensional physics, that is ''not'' the most convenient or sensible reference frame in which to consider objects moving differently, in different proper reference frames. In 4-dimensional physics, a reference frame in which an observer and his atoms are translating linearly at velocity <math>c</math> is known as the observer's proper inertial reference frame. Any such proper reference frame makes a great deal more sense for the purpose of considering the motion of objects moving differently in 4-space, just as a heliocentric solar system makes a great deal more sense than a geocentric solar system for the purpose of considering the motion of the planets. ... ==Appendix== {{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=2}} {{Regular convex 4-polytopes|columns=7|wiki=W:|radius=1}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|last=Mac Gregor|first=Malcolm H.|title=The energy packet generator 137 and the key energy equation E_w + E_z = E_top|date=2017|publisher=El Mac Books|place=Santa Cruz, CA|author-link=W:Fine-structure constant|ref={{Mac Gregor 137|2017}}}} {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} hivx7ekie27i3qck71mw053bcfh2u7x 2842460 2842459 2026-10-24T21:30:48Z Dc.samizdat 2856930 /* The Kepler problem */ 2842460 wikitext text/x-wiki == Symmetry transformation group arising from the Laplace–Runge–Lenz vector == *{{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}} <blockquote>Through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which is well known in the literature. However, the physically relevant part of the LRL vector is its direction angle in the plane of motion (since its magnitude is just a function of energy and angular momentum). The ... infinitesimal dynamical symmetry corresponding to the direction part of the LRL vector ... when combined with the rotation symmetries ... is shown to be the semi-direct product of <math>SO(3)</math> and <math>\mathbb{R^3}</math>. This stands in contrast to the <math>SO(4)</math> symmetry group generated by the LRL symmetries and the rotations.</blockquote> In other words, bodies in Kepler orbits are translating through 4-space via <math>QT</math>, breaking their stationary <math>Q^2</math> symmetry. The LRL direction vector is their <math>T</math>, their proper time vector. The Kepler problem in Newtonian mechanics has been widely studied and encompasses both a particle moving under inverse-square force and two bodies moving under mutual gravitational attraction. One of its most interesting features is the existence of the conserved Laplace–Runge–Lenz (LRL) vector (see e.g.  and references therein). It lies in the plane of motion, which is orthogonal to the angular momentum vector, and points in the direction given by the periapsis in the case of inverse-square motion and the position of closest separation in the case of two-body motion. This conserved vector has a long history, going back over three centuries, and plays a key role in understanding the shape and orientation of astronomical orbits in the solar system. Its quantum mechanical analog was important historically in the initial derivation of energy levels in the hydrogen atom (see e.g.  and references therein). In contrast to the conservation of energy and angular momentum in the Kepler problem, which come directly from Noether’s theorem using the manifest time-translation and rotation symmetries of Newtonian space, the LRL vector does not arise from any kinematic symmetry principle. Nevertheless, Noether’s theorem in reverse  states that every conserved quantity is generated by some infinitesimal symmetry transformation that leaves invariant the action principle for the Kepler problem. Since the LRL vector comprises three independent conserved quantities, its symmetry generator consists of three independent infinitesimal transformations, which were first worked out in Ref.. They turn out to involve the velocity variable, unlike the infinitesimal time-translation and rotation transformations which involve only the time and position variables. In particular, the LRL infinitesimal symmetry transformations are primary examples of what are called ''dynamical symmetries'', in contradistinction to ''point symmetries'' which are typified by infinitesimal time-translations and rotations. The only additional point symmetry admitted by the Kepler problem consists of a scaling of the time and position variables, but this scaling transformation does not leave the action principle invariant. The five point symmetries together with the three LRL symmetries in the Kepler problem form a 6-dimensional Lie subalgebra when the LRL symmetries are normalized to have the same physical dimension as the rotations. This subalgebra’s structure turns out to be <math>\mathfrak{so}(4)</math> when the energy is negative, and <math>\mathfrak{so}(3,1)</math> when the energy is positive . These Lie algebras are isomorphic to the symmetry algebra of isometries of a 3-sphere <math>S^3</math> in <math>\mathbb{R^4}</math> and a 3-hyperboloid <math>H^3</math> in Minkowski space <math>\mathbb{R^{3+1}}</math>. One explanation for the appearance of four-dimensional spaces is provided by Moser’s observation  that, in the negative energy case, the Kepler problem is equivalent to non-affine geodesic motion on <math>S^3</math> in a four-dimensional auxiliary space under stereographic projection. A kinematical description of the projection is given by identifying the stereographic coordinates as the components of the Kepler velocity variable. The rotations in the auxiliary space correspond in the Kepler problem to changing the angular momentum while keeping the energy unchanged. These rotations constitute a <math>SO(4)</math> transformation group that combines the ordinary physical <math>SO(3)</math> rotations and the LRL symmetry transformations. A similar equivalence holds in the positive energy case by considering non-affine geodesic motion on <math>H(3)</math>. One remaining question is how does the group generated by the LRL infinitesimal symmetry transformations act just on the time, position, and velocity variables in the Kepler problem. It is hard to find an explicit description of this group in the literature (although a description in terms of the coordinates in the auxiliary 4-dimensional space is known ). The first main result of the present work will be to derive the explicit form of the LRL symmetry transformation group in terms of the kinematic physical variables. As a by-product, the action of the new infinitesimal symmetries on all of the conserved quantities is obtained. The results are given in terms of the physical kinematical variables in the Kepler problem, rather than in an enlarged auxiliary space in which the LRL symmetries are usually stated. Another remaining question, which is not commonly discussed in the literature, arises from the magnitude of LRL vector. This quantity is given by a scalar expression in terms of the energy and magnitude of the angular momentum. The corresponding infinitesimal symmetry transformation, coming from Noether’s theorem in reverse, is just a linear combination of the generators of time-translation and rotation in the plane of motion. Namely it is a point symmetry, involving only the time and position variables. On the one hand, this choice of normalization of the LRL vector is essential for the subalgebra generated by the LRL symmetries and rotations to be when energy is negative and when energy is positive. Yet on the other hand, it is a free choice, which raises the question of whether a different or perhaps simpler structure is possible for the resulting symmetry subalgebra if some other choice of normalization is made. A natural choice is to normalize it to be a unit vector, retaining only the physically relevant direction information contained in the LRL vector. This will be called the LRL direction vector. As a second main result, the components of the LRL direction vector will be shown to generate three infinitesimal symmetry transformations that commute with themselves. == 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 their Hubble redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of these three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity <math>c</math>. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance. That small separation might not make much difference in our view of the night sky, but their actual separation in the fourth dimension may be much greater, large enough to significantly transform our map of the heavens. That is because it is unlikely that the stars in the galaxy all lie exactly on the surface of the expanding universal 3-sphere, after millions of years of expansion....{{Efn| 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? That would be an expected consequence of the special rotational symmetry group of 4-space <math>SO(4)</math>, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. Also of interest would be whether the central region of the galaxy is a 4-ball or a 4-ellipsoid.}} == Why the Equivalence Principle symmetry == How is the Weak Equivalence Principle (of Einstein) derived from Coxeter's group-theoretic formulation of the principle of relativity (his relation of congruent transformations in n-dimensional Euclidean space)? That is, what is the equivalence between rotations and translations, via their common basis in reflections? What exactly is the relative (i.e. symmetrical, interchangeable) relation between a translation and a rotation? == Light == === Polarization === How is polarized light oriented with respect to the proper time dimension? Is the magnetic field simply a fourth orthogonal orientation of the electric field? What is its orientation with respect to the proper time dimension? === Sails === On 3-sphere vs. 2-sphere - can you perhaps tack through 4-space on light pressure, if you can distinguish left and right hand radiation - relation to polarizations of light. Light pressure vs. particle pressure - <math>c\prime = 2c</math>. === Propagation kinematics through 4-space === 11-cell's role. Geodesics of truncated 5-cells. == 137 === * <math>137 = 5! + 4! - 3! - 2! + 1!</math> * a [[W:Pythagorean prime|Pythagorean prime]]: a prime number of the form <math>4n+1</math>, where <math>n=34</math>&nbsp;(<math>137=4\times 34+1</math>) or the sum of two squares <math>11^{2}+4^{2} = (121+16)</math>. * a combination of three terms <math>4^{3}+3^{4}-2^{3} = (64+81-8)</math>, cube of 4 + [[W:Triangular number|triangular number]] T4+T2 on each cube face (along 3 axes) - peaks (single 6th peak as free link) * The "73 calculator trick" involves picking any four-digit number, entering it twice to create an eight-digit number (e.g., 1234 becomes 12341234), then dividing by 137, and finally dividing that result by your original four-digit number to always get 73 as the answer. This works because repeating a four-digit decimal number N is the same as multiplying N by 10001, and 10001 is the product of 73 and 137 (73×137). == Galilean relativity in a space of four orthogonal dimensions == The paragraph ending in "That is all of physics" came to me on my mother's 101st birthday, just over a year after she died, like an example of rebirth in the spring. I say rebirth because it is probably not original, and even if it is, life must necessarily forget it someday, and just as necessarily rediscover it again someday afterwards. It seems to me to be as close an example to ''immortal life'' as I have encountered. That state, oxymoronic in nearly all senses I fear, may be said to exist where life discovers an immortal principle, and so (like the 11-cells) many lives distinct but not disjoint combine to give existence to an immortal object, where none could exist at all in singularity. [[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 19:44, 14 March 2024 (UTC) Writing this on the morning after my mother's 103rd birthday, after several related ideas finally assembled themselves in my mind in sleep last night, for the first time. This then is my first written observation of these several ideas that I have been pondering for years. The Higgs boson, the largest mediator of the fundamental force, has the mass/energy of the 120-cell. The photon as particle (as distinct from the free radiating photon as wave), the mediating boson of the electron, I think will turn out to have the mass/energy of the 11-cell. The 1/137 mass/energy relationship behind the ''e'' constant is due to the subsumption relationship between these two 4-polytopes and a third 4-polytope, the 4-simplex or 5-cell, which is the proton. Mass/energy is proportional to 4-content, and also a function of the isoclinic angle of rotation. [[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 03:57, 16 March 2026 (UTC) == Galaxies == The spacetime of general relativity is often illustrated as a projection to a curved 2D surface in which large gravitational objects make gravity wells or dimples in the surface. In the Euclidean 4D view of the universe the 3D surface of a large cosmic object such as a galaxy surrounds an empty 4D space, and large gravitational objects within the galaxy must make dimples in its surface. But should we see them as dimples exactly? Would they dimple inwards or outwards? In the spacetime illustrations they are naturally always shown as dimpling downwards, which is somewhat disingenuous, strongly suggesting to the viewer that the reason for gravity is that it flows downhill - the original tautology we are trying to surmount! In the Euclidean 4D galaxy the dimple, if it is one, must be either inward or outward, and which it is matters since the dimple is flying outward at velocity {{mvar|c}}. The universe is not collapsing inward. Is a large gravitational mass (such as a star) ''ahead'' of the smaller masses orbiting around it (such as its planets), or is it ''behind'' them, as they fly through 4-space on their Clifford parallel trajectories? The answer is ''both'' of course, because a star is not a dimple, it is a 4-ball, and it dimples the 3D surface both inwards and outwards. It is a thick place in the 3D surface. We should view it as having its gravitational center precisely at the surface of the expanding 3-sphere. What is a black hole? It is the hollow four-dimensional space that a galaxy is the three-dimensional surface of. When we view another galaxy, such as Andromeda, we are seeing that whole galaxy from a distance, the way the moon astronauts looked back at the whole earth. We see our own Milky Way galaxy from where we are on its surface, the way we see the earth from its surface, except that the earth is solid, but the galaxy is hollow and transparent. We can look across its empty center and see all the other stars also on its surface, including those opposite ours on the far side of its 3-sphere. The thicker band of stars we see in our night sky and identify as the Milky Way is not our whole galaxy; the majority of the other stars visible to the naked eye also lie in our galaxy. That dense band is not thicker and brighter than other parts of our galaxy because it lies toward a dense galactic center (our galaxy has an empty center), but for the opposite reason: those apparently more thickly clustered stars lie all around us on the galaxy's surface, in the nearest region of 4-space surrounding us. They appear to be densely packed only because we are looking at them "edge on". Actually, we are looking into this nearby apparently dense region ''face on'', not edge on, because it is a round sphere of nearby space surrounding us. In contrast, stars in our galaxy outside that bright band lie farther off from us, across the empty center of the galaxy, and we see them spread out as they actually are, instead of "edge on" where they appear to be more densely clustered. The "dense band" covers only an equatorial band of the night sky instead of all the sky, because when we look out into the four-dimensional space around us, we can see stars above and below our three-dimensional hyperplane in four-dimensional space. Everything in our solar system lies in our hyperplane, and the nearby stars around us in our galaxy are near our hyperplane (just slightly below it). All the other, more distant stars in our galaxy are also below our hyperplane. We can see objects outside our galaxy, such as other galaxies, both above and below our hyperplane, because we receive light signals from all directions in 4-space. We can see distant objects all around us above our hyperplane (looking up from the galactic surface into the fourth dimension), and all around us below our hyperplane (looking down through our transparent 4-dimensional galaxy and out the other side). But we do not observe (or measure) the angles of objects above or below our hyperplane, because we conceptualize the volume of space around us as a 3-ball when it is actually a 4-ball. We unknowingly conflate all the objects we receive light from, projecting them into a single 3-ball, whether they are above, below or in our hyperplane. == The dimpled fabric of spacetime == The spacetime of general relativity is often illustrated as a projection to a curved 2D surface in which large gravitational objects make gravity wells or dimples in the surface. In the Euclidean 4D view of the universe the 3D surface of a large cosmic object such as a galaxy surrounds an empty 4D space, and large gravitational objects within the galaxy must make dimples in its surface. But should we see them as dimples exactly? Would they dimple inwards or outwards? In the spacetime illustrations they are naturally always shown as dimpling downwards, which is somewhat disingenuous, strongly suggesting to the viewer that the reason for gravity is that it flows downhill - the original tautology we are trying to surmount! In the Euclidean 4D galaxy the dimple, if it is one, must be either inward or outward, and which it is matters. Is a large gravitational mass (such as a star) ''ahead'' of the smaller masses orbiting around it (such as its planets), or is it ''behind'' them, as they fly through 4-space on their Clifford parallel trajectories? The answer is ''both'' of course, because a star is not a dimple, it is a 4-ball, and it dimples the 3D surface both inwards and outwards. It is a thick place in the 3D surface. We should view it as having its gravitational center precisely at the surface of a 3-sphere expanding radially at velocity <math>c</math>. [[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 18:15, 10 November 2024 (UTC) == John Stewart Bell by Adam Becker == A roulette wheel has three orthogonal dimensions of binary parity: high/low (a finite integer set partitioned in two), even/odd (a different partitioning of the integer set), and a third dimension red/black (which belong to a distinct sequence, and I dare say a distinct group, than the other two parities). "Polarization is a property of light -- light is an electromagnetic wave, and the polarization is the direction that the wave is doing its waving in. Polarization is sort of like a little arrow that each photon carries with it that can point in different dirdctions. But ... we can't actually tell what direction a photon's polarization arrow is pointing. All we can do is measure a photon's polarization along one particular axis at a time in a somewhat indirect [binary] manner ...." [[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 21:36, 31 December 2024 (UTC) == Minkowski spacetime and Euclidean 4-space in relativity == ... Is there a configuration in Euclidean space where every observer, and every observed object, is moving at velocity <math>c</math>? Yes, there is one such configuration, in 4-dimensional Euclidean space. This configuration must be like the one described above, in our ''Abstract''. == The rate of atomic symmetry operations == ... == General relativity describes a curved 3-dimensional manifold embedded in Euclidean 4-dimensional space == ... == The geometry of the atomic nucleus == In [[W:Euclidean 4-space|Euclidean four dimensional space]], an [[W:atomic nucleus|atomic nucleus]] is a one or more concentric 4-polytopes of increasing radius. Each concentric shell is a single or compound [[24-cell]], the regular 4-polytope with [[W:Coxeter group#Symmetry groups of regular polytopes|𝔽₄ symmetry]]. Nuclear shells are concentric [[W:3-sphere|3-sphere]]s occupied (fully or partially) by the orbits of this 24-point [[#The 6 regular convex 4-polytopes|regular convex 4-polytope]]. An actual atomic nucleus is a rotating four dimensional object. It is not a ''rigid'' rotating object, it is a kinematic one, because the nucleus of an actual atom of a distinct [[W:nucleon number|nucleon number]] contains a distinct number of orbiting 24-cell vertices which may be in different isoclinic rotational orbits. These moving vertices never describe a compound of static 24-cells at any single instant in time, though their orbits do all the time. The physical configuration of a nucleus as concentric 24-cells can be reduced to the [[W:kinematics|kinematics]] of the orbits of its constituent protons and neutrons. The geometry of the atomic nucleus is therefore strictly [[W:Euclidean geometry#19th century|Euclidean]] in four dimensional space. === 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#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 (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 precise [[24-cell|detailed description]] enabling the reader to properly visualize it runs to many pages and illustrations, with many accompanying pages of explanatory notes on basic phenomena that arise only in 4-dimensional space: [[24-cell#Squares|completely orthogonal planes]], [[24-cell#Hexagons|Clifford parallelism]] and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Helical hexagrams and their isoclines|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 surprise. [[#The 6 regular convex 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 (generally), 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. [[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 is much harder for us to visualize, because the only way we can experience it is in our imaginations; we have no body of sensory experience in 4-dimensional space to draw upon. For that reason, 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. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than 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 any other at any time. That is still an example of a rigid object in a single distinct isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing the characteristic rotation of the 24-cell. But 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 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 shell without collisions? In adjacent 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 questions of this kind of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[W:kinetics|kinetics]]. In four dimensions, we find that space is more commodious than we thought it was, and 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 more symmetric packings. Sadoc studied twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack perfectly in 4-space without exhibiting any torsion, although their packing in 3-space was imperfect, "frustrated" by their torsion. <blockquote>The frustration, which arises when the molecular orientation is transported along the two [circular] AB paths of figure 1 [helix], is imposed by the very topological nature of the Euclidean space R³. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S³, 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]]....{{Efn|name=Petrie polygon of a honeycomb}} The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S³, 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_∞ 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 S³.{{Efn|name=dense fabric of pole-circles}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc & Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings ("frustrations") in 3-dimensional space.}}</blockquote> Of course we do not find molecules condensing to close-pack the 3-sphere in our experience. We find 3-spheres in the atomic realm (atoms are 4-polytopes), and in cosmologic realm (at the surface boundaries of stars, and perhaps at the surface masses of hollow galaxies). But in the molecular realm we find only objects which 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 wholly imaginary one, though no less illuminating for the symmetry it reveals not having a realization as a 3-sphere of actual molecules. And might it not have a realization in the atomic realm? We know that atoms have their own complex internal structure, which we are unable to model geometrically in flat 3-dimensional space. Suppose that the constituent particles of atoms actually close-pack the 3-sphere as helices, in the manner of Sadoc's imaginary twisted molecules, that is, as discrete Hopf fibrations of the atomic 4-polytope, where each discrete Hopf fibration corresponds to the distinct isoclinic rotation characteristic of a constituent particle. [[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 19:28, 8 November 2025 (UTC) === Isospin === A [[W:Nucleon|nucleon]] is a [[W:proton|proton]] or a [[W:neutron|neutron]]. The proton carries a positive net [[W:Electric charge|charge]], and the neutron carries a zero net charge. The proton's [[W:Mass|mass]] is only about 0.13% less than the neutron's, and since they are observed to be identical in other respects, they can be viewed as two states of the same nucleon, together forming an isospin doublet ({{nowrap|''I'' {{=}} {{sfrac|1|2}}}}). In isospin space, neutrons can be transformed into protons and conversely by actions of the [[W:SU(2)|SU(2)]] symmetry group. In nature, protons are very stable (the most stable particle known); a proton and a neutron are a stable nuclide; but free neutrons decay into protons in about 10 or 15 seconds. According to the [[W:Noether theorem|Noether theorem]], [[W:Isospin|isospin]] is conserved with respect to the [[W:strong interaction|strong interaction]].<ref name=Griffiths2008>{{cite book |author=Griffiths, David J. |title=Introduction to Elementary Particles |edition=2nd revised |publisher=WILEY-VCH |year=2008 |isbn=978-3-527-40601-2}}</ref>{{rp|129–130}} Nucleons are acted upon equally by the strong interaction, which is invariant under rotation in isospin space. Isospin was introduced as a concept in 1932 by [[W:Werner Heisenberg|Werner Heisenberg]],<ref> {{cite journal |last=Heisenberg |first=W. |author-link=W:Werner Heisenberg |year=1932 |title=Über den Bau der Atomkerne |journal=[[W:Zeitschrift für Physik|Zeitschrift für Physik]] |volume=77 |issue=1–2 |pages=1–11 |doi=10.1007/BF01342433 |bibcode = 1932ZPhy...77....1H |s2cid=186218053 |language=de}}</ref> well before the 1960s development of the [[W:quark model|quark model]], to explain the symmetry of the proton and the then newly discovered neutron. Heisenberg introduced the concept of another conserved quantity that would cause the proton to turn into a neutron and vice versa. In 1937, [[W:Eugene Wigner|Eugene Wigner]] introduced the term "isospin" to indicate how the new quantity is similar to spin in behavior, but otherwise unrelated.<ref> {{cite journal |last=Wigner |first=E. |author-link=W:Eugene Wigner |year=1937 |title=On the Consequences of the Symmetry of the Nuclear Hamiltonian on the Spectroscopy of Nuclei |journal=[[W:Physical Review|Physical Review]] |volume=51 |pages=106–119 |doi=10.1103/PhysRev.51.106 |bibcode = 1937PhRv...51..106W |issue=2 }}</ref> Similar to a spin-1/2 particle, which has two states, protons and neutrons were said to be of isospin 1/2. The proton and neutron were then associated with different isospin projections ''I''₃&nbsp;=&nbsp;+1/2 and −1/2 respectively. Isospin is a different kind of rotation entirely than the ordinary spin which objects undergo when they rotate in three-dimensional space. Isospin does not correspond to a [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] in any space (of any number of dimensions). However, it does seem to correspond exactly to an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]] in a Euclidean space of four dimensions. Isospin space resembles the [[W:3-sphere|3-sphere]], the [[W:Elliptical space#Elliptic space (the 3D case)|curved 3-dimensional space]] that is the surface of a [[W:4-ball (mathematics)#In Euclidean space|4-dimensional ball]]. === Spinors === [[File:Spinor on the circle.png|thumb|upright=1.5|A spinor visualized as a vector pointing along the [[W:Möbius band|Möbius band]], exhibiting a sign inversion when the circle (the "physical system") is continuously rotated through a full turn of 360°.]][[W:Spinors|Spinors]] are [[W:representation of a Lie group|representations]] of a [[W:spin group|spin group]], which are [[W:Double covering group|double cover]]s of the [[W:special orthogonal group|special orthogonal groups]]. The spin group Spin(4) is the double cover of [[W:SO(4)|SO(4)]], the group of rotations in 4-dimensional Euclidean space. [[600-cell#Fibrations of isocline polygrams|Isoclines]], the helical geodesic paths followed by points under isoclinic rotation, correspond to spinors representing Spin(4). Spinors can be viewed as the "square roots" of [[W:Section (fiber bundle)|cross sections]] of [[W:vector bundle|vector bundle]]s; in this correspondence, a fiber bundle of isoclines (of a distinct isoclinic rotation) is a cross section (inverse bundle) of a fibration of great circles (in the invariant planes of that rotation). A spinor can be visualized as a moving vector on a Möbius strip which transforms to its negative when continuously rotated through 360°, just as [[24-cell#Helical hexagrams and their isoclines|an isocline can be visualized as a Möbius strip]] winding twice around the 3-sphere, during which [[24-cell#Isoclinic rotations|720° isoclinic rotation]] the rigid 4-polytope turns itself inside-out twice.{{Sfn|Goucher|2019|loc=Spin Groups}} Under isoclinic rotation, a rigid 4-polytope is an isospin-1/2 object with two states. === Isoclinic rotations in the nucleus === Isospin is regarded as a symmetry of the strong interaction under the [[W:Group action (mathematics)|action]] of the [[W:Lie group|Lie group]] [[W:SU(2)|SU(2)]], the two [[W:eigenstate|states]] being the [[W:Up quark|up flavour]] and [[W:Down quark|down flavour]]. A 360° isoclinic rotation of a rigid [[W:nuclide|nuclide]] would transform its protons into neutrons and vice versa, exchanging the up and down flavours of their constituent [[W:quarks|quarks]], by turning the nuclide and all its parts inside-out (or perhaps we should say upside-down). Because we never observe this, we know that the nucleus is not a ''rigid'' polytope undergoing isoclinic rotation. If the nucleus ''were'' a rigid object, nuclides that were isospin-rotated 360° would be isoclinic mirror images of each other, isospin +1/2 and isospin −1/2 states of the whole nucleus. We don't see whole nuclides rotating as a rigid object, but considering what would happen if they ''were'' rigid tells us something about the geometry we must expect inside the nucleons. One way that an isospin-rotated neutron could become a proton would be if the up quark and down quark were a left and right mirror-image pair of the same object; exchanging them in place would turn each down-down-up neutron into an up-up-down proton. But the case cannot be quite that simple, because the up quark and the down quark are not mirror-images of the same object: they have very different mass and other incongruities. Another way an isospin-rotated neutron could be a proton would be if the up and down quarks were asymmetrical kinematic polytopes (not indirectly congruent mirror-images, and not rigid polytopes), rotating within the nucleus in different ''hybrid'' orbits. By that we mean that they may have vertices orbiting in rotations characteristic of more than one 4-polytope, so they may change shape as they rotate. In that case their composites (protons and neutrons) could have a symmetry not manifest in their components, but emerging from their combination. .... === Hybrid isoclinic rotations === The 24-cell has [[24-cell#Isoclinic rotations|its own characteristic isoclinic rotations]] in 4 Clifford parallel hexagonal planes (each intersecting 6 vertices), and also inherits the [[16-cell#Rotations|characteristic isoclinic rotations of its 3 Clifford parallel constituent 16-cells]] in 6 Clifford parallel square planes (each intersecting 4 vertices). The twisted circular paths followed by vertices in these two different kinds of rotation have entirely different geometries. Vertices rotating in hexagonal invariant planes follow [[24-cell#Helical hexagrams and their isoclines|helical geodesic curves whose chords form hexagrams]], and vertices rotating in square invariant planes follow [[24-cell#Helical octagrams and their isoclines|helical geodesic curves whose chords form octagrams]]. In a rigid isoclinic rotation, ''all'' the [[24-cell#Geodesics|great circle polygons]] move, in any kind of rotation. What distinguishes the hexagonal and square isoclinic rotations is the invariant planes of rotation the vertices stay in. The rotation described [[#Rotations|above]] (of 8 vertices rotating in 4 Clifford parallel hexagonal planes) is a single hexagonal isoclinic rotation, not a kinematic or hybrid rotation. A ''kinematic'' isoclinic rotation in the 24-cell is any subset of the 24 vertices rotating through the same angle in the same time, but independently with respect to the choice of a Clifford parallel set of invariant planes of rotation and the chirality (left or right) of the rotation. A ''hybrid'' isoclinic rotation combines moving vertices from different kinds of isoclinic rotations, characteristic of different regular 4-polytopes. For example, if at least one vertex rotates in a square plane and at least one vertex rotates in a hexagonal plane, the kinematic rotation is a hybrid rotation, combining rotations characteristic of the 16-cell and characteristic of the 24-cell. As an example of the simplest hybrid isoclinic rotation, consider a 24-cell vertex rotating in a square plane, and a second vertex, initially one 24-cell edge-length distant, rotating in a hexagonal plane. Rotating isoclinically at the same rate, the two moving vertices will never collide where their paths intersect, so this is a ''valid'' hybrid rotation. To understand hybrid rotations in the 24-cell more generally, visualize the relationship between great squares and great hexagons. The [[24-cell#Squares|18 great squares]] occur as three sets of 6 orthogonal great squares,{{Efn|name=six orthogonal planes of the Cartesian basis}} each [[16-cell#Coordinates|forming a 16-cell]]. The three 16-cells are completely disjoint{{Efn|name=completely disjoint}} and [[24-cell#Clifford parallel polytopes|Clifford parallel]]: each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}).{{Efn|name=three isoclinic 16-cells}} The 18 square great circles are crossed by 16 hexagonal great circles; each [[24-cell#Hexagons|hexagon]] has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two [[24-cell#Triangles|great triangles]] inscribed in each great hexagon (occupying its alternate vertices, with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking three completely disjoint great squares, one from each of the three completely disjoint 16-cells''.{{Efn|There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}} Isoclinic rotations take the elements of the 4-polytope to congruent [[24-cell#Clifford parallel polytopes|Clifford parallel elements]] elsewhere in the 4-polytope. The square rotations do this ''locally'', confined within each 16-cell: for example, they take great squares to other great squares within the same 16-cell. The hexagonal rotations act ''globally'' within the entire 24-cell: for example, they take great squares to other great squares in ''different'' 16-cells. The [[16-cell#Helical construction|chords of the square rotations]] bind the 16-cells together internally, and the [[24-cell#Helical hexagrams and their isoclines|chords of the hexagonal rotations]] bind the three 16-cells together. .... === Charge === Positive and negative charge are [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#The perfection of Fuller's cyclic design|compression and tension]], respectively. ... === Color === When the existence of quarks was suspected in 1964, [[W:Oscar W. Greenberg|Greenberg]] introduced the notion of color charge to explain how quarks could coexist inside some [[W:hadron|hadron]]s in [[W:quark model#The discovery of color|otherwise identical quantum states]] without violating the [[W:Pauli exclusion principle|Pauli exclusion principle]]. The modern concept of [[W:color charge|color charge]] completely commuting with all other charges and providing the strong force charge was articulated in 1973, by [[W:William A. Bardeen|William Bardeen]], [[W:de:Harald Fritzsch|Harald Fritzsch]], and [[W:Murray Gell-Mann|Murray Gell-Mann]].<ref>{{cite conference |author1=Bardeen, W. |author2=Fritzsch, H. |author3=Gell-Mann, M. |year=1973 |title=Light cone current algebra, ''π''⁰ decay, and ''e''⁺ ''e''^&minus; annihilation |arxiv=hep-ph/0211388 |editor=Gatto, R. |book-title=Scale and conformal symmetry in hadron physics |page=[https://archive.org/details/scaleconformalsy0000unse/page/139 139] |publisher=[[W:John Wiley & Sons|John Wiley & Sons]] |isbn=0-471-29292-3 |bibcode=2002hep.ph...11388B |url-access=registration |url=https://archive.org/details/scaleconformalsy0000unse/page/139 }}</ref><ref>{{cite journal |title=Advantages of the color octet gluon picture |journal=[[W:Physics Letters B|Physics Letters B]] |volume=47 |issue=4 |page=365 |year=1973 |last1=Fritzsch |first1=H. |last2=Gell-Mann |first2=M. |last3=Leutwyler |first3=H. |doi=10.1016/0370-2693(73)90625-4 |bibcode=1973PhLB...47..365F |citeseerx=10.1.1.453.4712}}</ref> Color charge is not [[W:electric charge|electric charge]]; the whole point of it is that it is a quantum of something different. But it is related to electric charge, through the way in which the three different-colored quarks combine to contribute fractional quantities of electric charge to a nucleon. As we shall see, color is not really a separate kind of charge at all, but a partitioning of the electric charge into [[24-cell#Clifford parallel polytopes|Clifford parallel subspaces]]. The [[W:Color charge#Red, green, and blue|three different colors]] of quark charge might correspond to three different 16-cells, such as the three disjoint 16-cells inscribed in the 24-cell. Each color might be a disjoint domain in isospin space (the space of points on the 3-sphere).{{Efn|The 8 vertices of each disjoint 16-cell constitute an independent [[16-cell#Coordinates|orthonormal basis for a coordinate reference frame]].}} Alternatively, the three colors might correspond to three different fibrations of the same isospin space: three different ''sequences'' of the same total set of discrete points on the 3-sphere. These alternative possibilities constrain possible representations of the nuclides themselves, for example if we try to represent nuclides as particular rotating 4-polytopes. If the neutron is a (8-point) 16-cell, either of the two color possibilities might somehow make sense as far as the neutron is concerned. But if the proton is a (5-point) 5-cell, only the latter color possibility makes sense, because fibrations (which correspond to distinct isoclinic left-and-right rigid rotations) are the ''only'' thing the 5-cell has three of. Both the 5-cell and the 16-cell have three discrete rotational fibrations. Moreover, in the case of a rigid, isoclinically rotating 4-polytope, those three fibrations always come one-of-a-kind and two-of-a-kind, in at least two different ways. First, one fibration is the set of invariant planes currently being rotated through, and the other two are not. Second, when one considers the three fibrations of each of these 4-polytopes, in each fibration two isoclines carry the left and right rotations respectively, and the third isocline acts simply as a Petrie polygon, the difference between the fibrations being the role assigned to each isocline. If we associate each quark with one or more isoclinic rotations in which the moving vertices belong to different 16-cells of the 24-cell, and the sign (plus or minus) of the electric charge with the chirality (right or left) of isoclinic rotations generally, we can configure nucleons of three quarks, two performing rotations of one chirality and one performing rotations of the other chirality. The configuration will be a valid kinematic rotation because the completely disjoint 16-cells can rotate independently; their vertices would never collide even if the 16-cells were performing different rigid square isoclinic rotations (all 8 vertices rotating in unison). But we need not associate a quark with a [[16-cell#Rotations|rigidly rotating 16-cell]], or with a single distinct square rotation. Minimally, we must associate each quark with at least one moving vertex in each of three different 16-cells, following the twisted geodesic isocline of an isoclinic rotation. In the up quark, that could be the isocline of a right rotation; and in the down quark, the isocline of a left rotation. The chirality accounts for the sign of the electric charge (we have said conventionally as +right, −left), but we must also account for the quantity of charge: +{{sfrac|2|3}} in an up quark, and −{{sfrac|1|3}} in a down quark. One way to do that would be to give the three distinct quarks moving vertices of {{sfrac|1|3}} charge in different 16-cells, but provide up quarks with twice as many vertices moving on +right isoclines as down quarks have vertices moving on −left isoclines (assuming the correct chiral pairing is up+right, down−left). Minimally, an up quark requires two moving vertices (of the up+right chirality).{{Efn|Two moving vertices in one quark could belong to the same 16-cell. A 16-cell may have two vertices moving in the same isoclinic square (octagram) orbit, such as an antipodal pair (a rotating dipole), or two vertices moving in different square orbits of the same up+right chirality.{{Efn|There is only one [[16-cell#Helical construction|octagram orbit]] of each chirality in each fibration of the 16-cell, so two octagram orbits of the same chirality cannot be Clifford parallel (part of the same distinct rotation). Two vertices right-moving on different octagram isoclines in the same 16-cell is a combination of two distinct rotations, whose isoclines will intersect: a kinematic rotation. It can be a valid kinematic rotation if the moving vertices will never pass through a point of intersection at the same time. Octagram isoclines pass through all 8 vertices of the 16-cell, and all eight isoclines (the left and right isoclines of four different fibrations) intersect at ''every'' vertex.}} However, the theory of [[W:Color confinement|color confinement]] may not require that two moving vertices in one quark belong to the same 16-cell; like the moving vertices of different quarks, they could be drawn from the disjoint vertex sets of two different 16-cells.}} Minimally, a down quark requires one moving vertex (of the down−left chirality). In these minimal quark configurations, a proton would have 5 moving vertices and a neutron would have 4. .... === Nucleons === [[File:Symmetrical_5-set_Venn_diagram.svg|thumb|[[W:Branko Grünbaum|Grünbaum's]] rotationally symmetrical 5-set Venn diagram, 1975. It is an illustration of the [[5-cell]] labeling each of its <math>2^5</math> elements. Think of it as an [[W:Nuclear magnetic resonance|NMR image]] of the 4-dimensional proton in projection to the plane.]] The proton is a very stable mass particle. Is there a stable orbit of 5 moving vertices in 4-dimensional Euclidean space? There are few known solutions to the 5-body problem, and fewer still to the [[W:n-body problem|{{mvar|n}}-body problem]], but one is known: the ''central configuration'' of {{mvar|n}} bodies in a space of dimension {{mvar|n}}-1. A [[W:Central configuration|central configuration]] is a system of [[W:Point particle|point masses]] with the property that each mass is pulled by the combined attractive force of the system directly towards the [[W:Center of mass|center of mass]], with acceleration proportional to its distance from the center. Placing three masses in an equilateral triangle, four at the vertices of a regular [[W:Tetrahedron|tetrahedron]], five at the vertices of a regular [[5-cell]], or more generally {{mvar|n}} masses at the vertices of a regular [[W:Simplex|simplex]] produces a central configuration [[W:Central configuration#Examples|even when the masses are not equal]]. In an isoclinic rotation, all the moving vertices orbit at the same radius and the same speed. Therefore if any 5 bodies are orbiting as an isoclinically rotating regular 5-cell (a rigid 4-simplex figure undergoing isoclinic rotation), they maintain a central configuration, describing 5 mutually stable orbits. Unlike the proton, the neutron is not always a stable particle; a free neutron will decay into a proton. A deficiency of the minimal configurations is that there is no way for this [[W:beta minus decay|beta minus decay]] to occur. The minimal neutron of 4 moving vertices described [[#Color|above]] cannot possibly decay into a proton by losing moving vertices, because it does not possess the four up+right moving vertices required in a proton. This deficiency could be remedied by giving the neutron configuration 8 moving vertices instead of 4: four down−left and four up+right moving vertices. Then by losing 3 down−left moving vertices the neutron could decay into the 5 vertex up-down-up proton configuration.{{Efn|Although protons are very stable, during [[W:stellar nucleosynthesis|stellar nucleosynthesis]] two H₁ protons are fused into an H₂ nucleus consisting of a proton and a neutron. This [[W:beta plus decay|beta plus "decay"]] of a proton into a neutron is actually the result of a rare high-energy collision between the two protons, in which a neutron is constructed. With respect to our nucleon configurations of moving vertices, it has to be explained as the conversion of two 5-point 5-cells into a 5-point 5-cell and an 8-point 16-cell, emitting two decay products of at least 1-point each. Thus it must involve the creation of moving vertices, by the conversion of kinetic energy to point-masses.}} A neutron configuration of 8 moving vertices could occur as the 8-point 16-cell, the second-smallest regular 4-polytope after the 5-point 5-cell (the hypothesized proton configuration). It is possible to double the neutron configuration in this way, without destroying the charge balance that defines the nucleons, by giving down quarks three moving vertices instead of just one: two −left vertices and one +right vertex. The net charge on the down quark remains −{{sfrac|1|3}}, but the down quark becomes heavier (at least in vertex count) than the up quark, as in fact its mass is measured to be. A nucleon's quark configuration is only a partial specification of its properties. There is much more to a nucleon than what is contained within its three quarks, which contribute only about 1% of the nucleon's energy. The additional 99% of the nucleon mass is said to be associated with the force that binds the three quarks together, rather than being intrinsic to the individual quarks separately. In the case of the proton, 5 moving vertices in the stable orbits of a central configuration (in one of the [[5-cell#Geodesics and rotations|isoclinic rotations characteristic of the regular 5-cell]]) might be sufficient to account for the stability of the proton, but not to account for most of the proton's energy. It is not the point-masses of the moving vertices themselves which constitute most of the mass of the nucleon; if mass is a consequence of geometry, we must look to the larger geometric elements of these polytopes as their major mass contributors. The quark configurations are thus incomplete specifications of the geometry of the nucleons, predictive of only some of the nucleon's properties, such as charge.{{Efn|Notice that by giving the down quark three moving vertices, we seem to have changed the quark model's prediction of the proton's number of moving vertices from 5 to 7, which would be incompatible with our theory that the proton configuration is a rotating regular 5-cell in a central configuration of 5 stable orbits. Fortunately, the actual quark model has nothing at all to say about moving vertices, so we may choose to regard that number as one of the geometric properties the quark model does not specify.}} In particular, they do not account for the forces binding the nucleon together. Moreover, if the rotating regular 5-cell is the proton configuration and the rotating regular 16-cell is the neutron configuration, then a nucleus is a complex of rotating 5-cells and 16-cells, and we must look to the geometric relationship between those two very different regular 4-polytopes for an understanding of the nuclear force binding them together. The most direct [[120-cell#Relationships among interior polytopes|geometric relationship among stationary regular 4-polytopes]] is the way they occupy a common 3-sphere together. Multiple 16-cells of equal radius can be compounded to form each of the larger regular 4-polytopes, the 8-cell, 24-cell, 600-cell, and 120-cell, but it is noteworthy that multiple regular 5-cells of equal radius cannot be compounded to form any of the other 4-polytopes except the largest, the 120-cell. The 120-cell is the unique intersection of the regular 5-cell and 16-cell: it is a compound of 120 regular 5-cells, and also a compound of 75 16-cells. All regular 4-polytopes except the 5-cell are compounds of 16-cells, but none of them except the largest, the 120-cell, contains any regular 5-cells. So in any compound of equal-radius 16-cells which also contains a regular 5-cell, whether that compound forms some single larger regular 4-polytope or does not, no two of the regular 5-cell's five vertices ever lie in the same 16-cell. So the geometric relationship between the regular 5-cell (our proton candidate) and the regular 16-cell (our neutron candidate) is quite a distant one: they are much more exclusive of each other's elements than they are distantly related, despite their complementary three-quark configurations and other similarities as nucleons. The relationship between a regular 5-cell and a regular 16-cell of equal radius is manifest only in the 120-cell, the most complex regular 4-polytope, which [[120-cell#Geometry|uniquely embodies all the containment relationships]] among all the regular 4-polytopes and their elements. If the nucleus is a complex of 5-cells (protons) and 16-cells (neutrons) rotating isoclinically around a common center, then its overall motion is a hybrid isoclinic rotation, because the 5-cell and the 16-cell have different characteristic isoclinic rotations, and they have no isoclinic rotation in common.{{Efn|The regular 5-cell does not occur inscribed in any other regular 4-polytope except one, the 600-vertex 120-cell. No two of the 5 vertices of a regular 5-cell can be vertices of the same 16-cell, 8-cell, 24-cell, or 600-cell. The isoclinic rotations characteristic of the regular 5-cell maintain the separation of its 5 moving vertices in 5 disjoint Clifford-parallel subspaces at all times. The [[16-cell#Rotations|isoclinic rotation characteristic of the 16-cell]] maintains the separation of its 8 moving vertices in 2 disjoint Clifford-parallel subspaces (completely orthogonal great square planes) at all times. Therefore, in any hybrid rotation of a concentric 5-cell and 16-cell, at most one 5-cell subspace (containing 1 vertex) might be synchronized with one 16-cell subspace (containing 4 vertices), such that the 1 + 4 vertices they jointly contain occupy the same moving subspace continually, forming a rigid 5-vertex polytope undergoing some kind of rotation. If in fact it existed, this 5-vertex rotating rigid polytope would not be [[5-cell#Geometry|not a 5-cell, since 4 of its vertices are coplanar]]; it is not a 4-polytope but merely a polyhedron, a [[W:square pyramid|square pyramid]].}} .... === Nuclides === ... === Quantum phenomena === The Bell-Kochen-Specker (BKS) theorem rules out the existence of deterministic noncontextual hidden variables theories. A proof of the theorem in a space of three or more dimensions can be given by exhibiting a finite set of lines through the origin that cannot each be colored black or white in such a way that (i) no two orthogonal lines are both black, and (ii) not all members of a set of ''d'' mutually orthogonal lines are white.{{Efn|"The Bell-Kochen-Specker theorem rules out the existence of deterministic noncontextual hidden variables theories. A proof of the theorem in a Hilbert space of dimension d ≥ 3 can be given by exhibiting a finite set of rays [9] that cannot each be assigned the value 0 or 1 in such a way that (i) no two orthogonal rays are both assigned the value 1, and (ii) not all members of a set of d mutually orthogonal rays are assigned the value 0."{{Sfn|Waegell & Aravind|2009|loc=2. The Bell-Kochen-Specker (BKS) theorem}}|name=BKS theorem}} .... === Motion === What does it mean to say that an object moves through space? Coxeter group theory provides precise answers to questions of this kind. A rigid object (polytope) moves by distinct transformations, changing itself in each discrete step into a congruent object in a different orientation and position.{{Efn|name=transformations}} .... === General relativity is just Galilean relativity in a general space of four orthogonal dimensions === .... Coxeter orthogonal dimensional relativity furnishes the geometric spatial definition of the inertial-gravitational force. It does so by limiting free motion (in the sense of free-falling) to one rotation and one translation, making double rotations accelerations which involve an energy transition. ...I now don't think that is quite right, rather: It does so by limiting free motion (in the sense of free-falling) to a double rotation (or one rotation and one translation) at conserved angles, making changes to the angles of rotation accelerations which involve an energy transition. === Physics === .... === Spherical relativity === We began our inquiry by wondering why physical space should be limited to just three dimensions (why ''three''). By visualizing the universe as a Euclidian space of four dimensions, we recognize that relativistic and quantum phenomena are natural consequences of symmetry group operations (including reflections and rotations) in four orthogonal dimensions. We are then led naturally to question whether the universe has only four dimensions. Physical space must bear as many dimensions as we need to ascribe to it, though the distinct phenomena for which we find a need to do so, in order to explain them, seem to be fewer and fewer as we consider higher and higher dimensions. To laws of physics generally, such as the principle of relativity in particular, we should always append the phrase "in Euclidean spaces of any number of dimensions". Laws of physics should operate in any flat Euclidean space <math>R^n</math> and in its corresponding spherical space <math>S^n</math>. The first and simplest sense in which we are forced to contemplate a fifth dimension is to accommodate our normal idea of time. Just as Einstein was forced to admit time as a dimension, in his four-dimensional spacetime of three spatial dimensions plus time, for some purposes we require a fifth time dimension to accompany our four spatial dimensions, when our purpose is orthogonal to (in the sense of independent of) the four spatial dimensions. For example, if we theorize that we observe a finite homogeneous universe, and that it is a Euclidean 4-space overall, we may prefer not to have to identify any distinct place within that 4-space as the center where the universe began in a big bang. To avoid having to pick a distinct place as the center of the universe, our model of it must be expanded, at least to be a ''spherical'' 4-dimensional space with the fifth radial dimension as time. Essentially, we require the fifth dimension in order to make our homogeneous 4-space finite, by wrapping it around into a 4-sphere. But perhaps we can still resist admitting the fifth radial dimension as a full-fledged Euclidean spatial dimension, at least so long as we have not observed how any naturally occurring object configurations are best described as 5-polytopes. One phenomenon which resists explanation in a space of just four dimensions is the propagation of light in a vacuum. The propagation of mass-carrying particles is explained as the consequence of their rotations in closed, curved spaces (3-spheres) of finite size, moving through four-dimensional Euclidean space at a universal constant speed, the speed of light. But an apparent paradox remains that light must seemingly propagate through four-dimensional Euclidean space at more than the speed of light. From an ''n''-dimensional viewpoint, this apparent paradox can be resolved, and in retrospect it is clear how massless particles can translate through four-dimensional space at twice the speed constant, since they are not simultaneously rotating. Another phenomenon justifying a five-dimensional view of space is the relation between the the 5-cell proton and the 16-cell neutron (the 4-simplex and 4-orthoplex polytopes). Their indirect relationship can be observed in the 4-600-point polytope (the 120-cell), and in its 11-cells,{{Sfn|Christie|2025|loc="A symmetrical arrangement of eleven 11-cells"}} but it is only directly observed (absent a 120-cell) in a five-dimensional reference frame. === Nuclear geometry === We have seen how isoclinic rotations (Clifford displacements) relate the orbits in the atomic nucleus to each other, just as they relate the regular convex 4-polytopes to each other, in a sequence of nested objects of increasing complexity. We have identified the proton as a 5-point, 5-cell 4-simplex 𝜶₄, the neutron as an 8-point, 16-cell 4-orthoplex 𝛽₄, and the shell of the atomic nucleus as a 24-point 24-cell. As Coxeter noted, that unique 24-point object stands quite alone in four dimensions, having no analogue above or below. === Atomic geometry === I'm on a plane flying to Eugene to visit Catalin, we'll talk after I arrive. I've been working on both my unpublished papers, the one going put for pre-publication review soon about 4D geometry, and the big one not going out soon about the 4D sun, 4D atoms, and 4D galaxies and n-D universe. I'vd just added the following paragraph to that big paper: Atomic geometry The force binding the protons and neutrons of the nucleus together into a distinct element is specifically an expression of the 11-cell 4-polytope, itself an expression of the pyritohedral symmetry, which binds the distinct 4-polytopes to each other, and relates the n-polytopes to their neighbors of different n by dimensional analogy. flying over mt shasta out my right-side window at the moment, that last text showing "not delivered" yet because there's no wifi on this plane, gazing at that great peak of the world and feeling as if i've just made the first ascent of it === Atomic elements === There are 120 5-cell protons in the 120-cell, and so there are 120 distinct atomic elements on the 3-sphere. === Molecular geometry === Molecules are 3-dimensional structures that live in the thin film of 3-membrane only one atom thick in most places that is our ordinary space, but since that is a significantly curved 3-dimensional space at the scale of a molecule, the way the molecule's covalent bonds form is influenced by the local curvature in 4-dimensions at that point. In the water molecule, there is a reason why the hydrogen atoms are attached to the oxygen atom at an angle of 104.45° in 3-dimensional space, and at root it must be the same symmetry that locates any two of the hydrogen proton's five vertices 104.45° apart on a great circle arc of its tiny 3-sphere. === Cosmology === ==== Solar systems ==== ===== Stars ===== ... ===== The Kepler problem ===== In the proper reference frame of some hypothetical observer in 4-dimensional Euclidean space, the Kepler problem has a solution in which all the planetary orbits are circles. This occurs because all bodies are always orbiting, each in some distinct isoclinic (equi-angled double) rotation. All atoms are rotating isoclinically at (double) the velocity <math>c</math>. In the observer's proper reference frame, the isoclinic double rotation of each of his own atoms is most conveniently represented as an equivalent screw displacement: a linear translation at velocity <math>c</math> (along the observer's proper time axis through 4-space), combined with a simple rotation at velocity <math>c</math> (of the atom internally). This representation, though proper, is subject to the over-simplification of special relativity, because it ignores the fact that the observer is himself moving on some circular orbit; he is not linearly translating through 4-space. When the observer's own orbit is also represented, the screw displacement becomes an elliptical double rotation, with a very slow, very large-radius simple circular orbit (almost a linear translation), combined with a very fast, very small-radius simple circular orbit of each atom internally. This is the observer's proper reference frame according to general relativity. An observer moving differently will perceive the motion of the observer and his atoms to be a different elliptical double rotation. We can find a proper reference frame, of some hypothetical observer moving hypothetically, in which the ratio of the radii of the two orthogonal rotations of the atom becomes any value we choose between 1 and <math>c</math>. In particular there exists, conceptually accessible to all observers independent of their motion, a distinct reference frame in which an atom is stationary, rotating isoclinically at (double) the velocity <math>c</math>. That is simply the stationary reference frame of the atom itself, such as the stationary reference frame of any observer and his own atoms. It is known as the observer's proper inertial reference frame. But in 4-dimensional physics, that is ''not'' the most convenient or sensible reference frame in which to consider objects moving differently, in different proper reference frames. In 4-dimensional physics, a reference frame in which an observer and his atoms are translating linearly at velocity <math>c</math> is known as the universal 4-coordinate reference frame, because it is the common reference frame of all observers (though it cannot be the proper reference frame of any observer). The universal 4-coordinate reference frame makes a great deal more sense for the purpose of considering the motion of objects moving differently in 4-space, just as a heliocentric solar system makes a great deal more sense than a geocentric solar system for the purpose of considering the motion of the planets. ... ==Appendix== {{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=2}} {{Regular convex 4-polytopes|columns=7|wiki=W:|radius=1}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|last=Mac Gregor|first=Malcolm H.|title=The energy packet generator 137 and the key energy equation E_w + E_z = E_top|date=2017|publisher=El Mac Books|place=Santa Cruz, CA|author-link=W:Fine-structure constant|ref={{Mac Gregor 137|2017}}}} {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} 0mitvc24fosogywh2bxux3mrxtbkinf 2842468 2842460 2026-10-24T23:55:04Z Dc.samizdat 2856930 /* The Kepler problem */ 2842468 wikitext text/x-wiki == Symmetry transformation group arising from the Laplace–Runge–Lenz vector == *{{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}} <blockquote>Through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which is well known in the literature. However, the physically relevant part of the LRL vector is its direction angle in the plane of motion (since its magnitude is just a function of energy and angular momentum). The ... infinitesimal dynamical symmetry corresponding to the direction part of the LRL vector ... when combined with the rotation symmetries ... is shown to be the semi-direct product of <math>SO(3)</math> and <math>\mathbb{R^3}</math>. This stands in contrast to the <math>SO(4)</math> symmetry group generated by the LRL symmetries and the rotations.</blockquote> In other words, bodies in Kepler orbits are translating through 4-space via <math>QT</math>, breaking their stationary <math>Q^2</math> symmetry. The LRL direction vector is their <math>T</math>, their proper time vector. The Kepler problem in Newtonian mechanics has been widely studied and encompasses both a particle moving under inverse-square force and two bodies moving under mutual gravitational attraction. One of its most interesting features is the existence of the conserved Laplace–Runge–Lenz (LRL) vector (see e.g.  and references therein). It lies in the plane of motion, which is orthogonal to the angular momentum vector, and points in the direction given by the periapsis in the case of inverse-square motion and the position of closest separation in the case of two-body motion. This conserved vector has a long history, going back over three centuries, and plays a key role in understanding the shape and orientation of astronomical orbits in the solar system. Its quantum mechanical analog was important historically in the initial derivation of energy levels in the hydrogen atom (see e.g.  and references therein). In contrast to the conservation of energy and angular momentum in the Kepler problem, which come directly from Noether’s theorem using the manifest time-translation and rotation symmetries of Newtonian space, the LRL vector does not arise from any kinematic symmetry principle. Nevertheless, Noether’s theorem in reverse  states that every conserved quantity is generated by some infinitesimal symmetry transformation that leaves invariant the action principle for the Kepler problem. Since the LRL vector comprises three independent conserved quantities, its symmetry generator consists of three independent infinitesimal transformations, which were first worked out in Ref.. They turn out to involve the velocity variable, unlike the infinitesimal time-translation and rotation transformations which involve only the time and position variables. In particular, the LRL infinitesimal symmetry transformations are primary examples of what are called ''dynamical symmetries'', in contradistinction to ''point symmetries'' which are typified by infinitesimal time-translations and rotations. The only additional point symmetry admitted by the Kepler problem consists of a scaling of the time and position variables, but this scaling transformation does not leave the action principle invariant. The five point symmetries together with the three LRL symmetries in the Kepler problem form a 6-dimensional Lie subalgebra when the LRL symmetries are normalized to have the same physical dimension as the rotations. This subalgebra’s structure turns out to be <math>\mathfrak{so}(4)</math> when the energy is negative, and <math>\mathfrak{so}(3,1)</math> when the energy is positive . These Lie algebras are isomorphic to the symmetry algebra of isometries of a 3-sphere <math>S^3</math> in <math>\mathbb{R^4}</math> and a 3-hyperboloid <math>H^3</math> in Minkowski space <math>\mathbb{R^{3+1}}</math>. One explanation for the appearance of four-dimensional spaces is provided by Moser’s observation  that, in the negative energy case, the Kepler problem is equivalent to non-affine geodesic motion on <math>S^3</math> in a four-dimensional auxiliary space under stereographic projection. A kinematical description of the projection is given by identifying the stereographic coordinates as the components of the Kepler velocity variable. The rotations in the auxiliary space correspond in the Kepler problem to changing the angular momentum while keeping the energy unchanged. These rotations constitute a <math>SO(4)</math> transformation group that combines the ordinary physical <math>SO(3)</math> rotations and the LRL symmetry transformations. A similar equivalence holds in the positive energy case by considering non-affine geodesic motion on <math>H(3)</math>. One remaining question is how does the group generated by the LRL infinitesimal symmetry transformations act just on the time, position, and velocity variables in the Kepler problem. It is hard to find an explicit description of this group in the literature (although a description in terms of the coordinates in the auxiliary 4-dimensional space is known ). The first main result of the present work will be to derive the explicit form of the LRL symmetry transformation group in terms of the kinematic physical variables. As a by-product, the action of the new infinitesimal symmetries on all of the conserved quantities is obtained. The results are given in terms of the physical kinematical variables in the Kepler problem, rather than in an enlarged auxiliary space in which the LRL symmetries are usually stated. Another remaining question, which is not commonly discussed in the literature, arises from the magnitude of LRL vector. This quantity is given by a scalar expression in terms of the energy and magnitude of the angular momentum. The corresponding infinitesimal symmetry transformation, coming from Noether’s theorem in reverse, is just a linear combination of the generators of time-translation and rotation in the plane of motion. Namely it is a point symmetry, involving only the time and position variables. On the one hand, this choice of normalization of the LRL vector is essential for the subalgebra generated by the LRL symmetries and rotations to be when energy is negative and when energy is positive. Yet on the other hand, it is a free choice, which raises the question of whether a different or perhaps simpler structure is possible for the resulting symmetry subalgebra if some other choice of normalization is made. A natural choice is to normalize it to be a unit vector, retaining only the physically relevant direction information contained in the LRL vector. This will be called the LRL direction vector. As a second main result, the components of the LRL direction vector will be shown to generate three infinitesimal symmetry transformations that commute with themselves. == 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 their Hubble redshifts, and by assuming that they are distributed in three dimensions of space, astronomers have plotted their locations in 3-space. If we abandon the last of these three assumptions, we can reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. To map the galaxy's stars in 4-coordinate space we would have to supply the missing fourth coordinate for each star, which corresponds to its angle above or below our 3-space hyperplane in our fourth dimension, the direction of our motion through 4-space at velocity <math>c</math>. If we assume that our galaxy and all its stars originated in the same big-bang, and that they still lie near the surface of its expanding 3-sphere (a domain which may or may not be our entire visible universe), we can interpret the redshift-determined distances of the galaxy's stars as chordal distances from us on the surface of that universal 3-sphere, and consequently as angles below our hyperplane of ordinary 3-space. Because our galaxy is only a very small patch on the universal 3-sphere, those angles will be small, but not zero. They appear to be zero to us in our 3-dimensional visual perspective from earth, because the 4-ball of space around us projects into a 3-ball of space in our hyperplane, where we lose the separation between stars in our fourth dimension. Near each point in the sky where we observe multiple objects at various distances from us, apparently directly behind each other, those objects are actually separated by an angular distance in our fourth dimension corresponding to their redshift chordal distance. That small separation might not make much difference in our view of the night sky, but their actual separation in the fourth dimension may be much greater, large enough to significantly transform our map of the heavens. That is because it is unlikely that the stars in the galaxy all lie exactly on the surface of the expanding universal 3-sphere, after millions of years of expansion....{{Efn| 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? That would be an expected consequence of the special rotational symmetry group of 4-space <math>SO(4)</math>, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits. Also of interest would be whether the central region of the galaxy is a 4-ball or a 4-ellipsoid.}} == Why the Equivalence Principle symmetry == How is the Weak Equivalence Principle (of Einstein) derived from Coxeter's group-theoretic formulation of the principle of relativity (his relation of congruent transformations in n-dimensional Euclidean space)? That is, what is the equivalence between rotations and translations, via their common basis in reflections? What exactly is the relative (i.e. symmetrical, interchangeable) relation between a translation and a rotation? == Light == === Polarization === How is polarized light oriented with respect to the proper time dimension? Is the magnetic field simply a fourth orthogonal orientation of the electric field? What is its orientation with respect to the proper time dimension? === Sails === On 3-sphere vs. 2-sphere - can you perhaps tack through 4-space on light pressure, if you can distinguish left and right hand radiation - relation to polarizations of light. Light pressure vs. particle pressure - <math>c\prime = 2c</math>. === Propagation kinematics through 4-space === 11-cell's role. Geodesics of truncated 5-cells. == 137 === * <math>137 = 5! + 4! - 3! - 2! + 1!</math> * a [[W:Pythagorean prime|Pythagorean prime]]: a prime number of the form <math>4n+1</math>, where <math>n=34</math>&nbsp;(<math>137=4\times 34+1</math>) or the sum of two squares <math>11^{2}+4^{2} = (121+16)</math>. * a combination of three terms <math>4^{3}+3^{4}-2^{3} = (64+81-8)</math>, cube of 4 + [[W:Triangular number|triangular number]] T4+T2 on each cube face (along 3 axes) - peaks (single 6th peak as free link) * The "73 calculator trick" involves picking any four-digit number, entering it twice to create an eight-digit number (e.g., 1234 becomes 12341234), then dividing by 137, and finally dividing that result by your original four-digit number to always get 73 as the answer. This works because repeating a four-digit decimal number N is the same as multiplying N by 10001, and 10001 is the product of 73 and 137 (73×137). == Galilean relativity in a space of four orthogonal dimensions == The paragraph ending in "That is all of physics" came to me on my mother's 101st birthday, just over a year after she died, like an example of rebirth in the spring. I say rebirth because it is probably not original, and even if it is, life must necessarily forget it someday, and just as necessarily rediscover it again someday afterwards. It seems to me to be as close an example to ''immortal life'' as I have encountered. That state, oxymoronic in nearly all senses I fear, may be said to exist where life discovers an immortal principle, and so (like the 11-cells) many lives distinct but not disjoint combine to give existence to an immortal object, where none could exist at all in singularity. [[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 19:44, 14 March 2024 (UTC) Writing this on the morning after my mother's 103rd birthday, after several related ideas finally assembled themselves in my mind in sleep last night, for the first time. This then is my first written observation of these several ideas that I have been pondering for years. The Higgs boson, the largest mediator of the fundamental force, has the mass/energy of the 120-cell. The photon as particle (as distinct from the free radiating photon as wave), the mediating boson of the electron, I think will turn out to have the mass/energy of the 11-cell. The 1/137 mass/energy relationship behind the ''e'' constant is due to the subsumption relationship between these two 4-polytopes and a third 4-polytope, the 4-simplex or 5-cell, which is the proton. Mass/energy is proportional to 4-content, and also a function of the isoclinic angle of rotation. [[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 03:57, 16 March 2026 (UTC) == Galaxies == The spacetime of general relativity is often illustrated as a projection to a curved 2D surface in which large gravitational objects make gravity wells or dimples in the surface. In the Euclidean 4D view of the universe the 3D surface of a large cosmic object such as a galaxy surrounds an empty 4D space, and large gravitational objects within the galaxy must make dimples in its surface. But should we see them as dimples exactly? Would they dimple inwards or outwards? In the spacetime illustrations they are naturally always shown as dimpling downwards, which is somewhat disingenuous, strongly suggesting to the viewer that the reason for gravity is that it flows downhill - the original tautology we are trying to surmount! In the Euclidean 4D galaxy the dimple, if it is one, must be either inward or outward, and which it is matters since the dimple is flying outward at velocity {{mvar|c}}. The universe is not collapsing inward. Is a large gravitational mass (such as a star) ''ahead'' of the smaller masses orbiting around it (such as its planets), or is it ''behind'' them, as they fly through 4-space on their Clifford parallel trajectories? The answer is ''both'' of course, because a star is not a dimple, it is a 4-ball, and it dimples the 3D surface both inwards and outwards. It is a thick place in the 3D surface. We should view it as having its gravitational center precisely at the surface of the expanding 3-sphere. What is a black hole? It is the hollow four-dimensional space that a galaxy is the three-dimensional surface of. When we view another galaxy, such as Andromeda, we are seeing that whole galaxy from a distance, the way the moon astronauts looked back at the whole earth. We see our own Milky Way galaxy from where we are on its surface, the way we see the earth from its surface, except that the earth is solid, but the galaxy is hollow and transparent. We can look across its empty center and see all the other stars also on its surface, including those opposite ours on the far side of its 3-sphere. The thicker band of stars we see in our night sky and identify as the Milky Way is not our whole galaxy; the majority of the other stars visible to the naked eye also lie in our galaxy. That dense band is not thicker and brighter than other parts of our galaxy because it lies toward a dense galactic center (our galaxy has an empty center), but for the opposite reason: those apparently more thickly clustered stars lie all around us on the galaxy's surface, in the nearest region of 4-space surrounding us. They appear to be densely packed only because we are looking at them "edge on". Actually, we are looking into this nearby apparently dense region ''face on'', not edge on, because it is a round sphere of nearby space surrounding us. In contrast, stars in our galaxy outside that bright band lie farther off from us, across the empty center of the galaxy, and we see them spread out as they actually are, instead of "edge on" where they appear to be more densely clustered. The "dense band" covers only an equatorial band of the night sky instead of all the sky, because when we look out into the four-dimensional space around us, we can see stars above and below our three-dimensional hyperplane in four-dimensional space. Everything in our solar system lies in our hyperplane, and the nearby stars around us in our galaxy are near our hyperplane (just slightly below it). All the other, more distant stars in our galaxy are also below our hyperplane. We can see objects outside our galaxy, such as other galaxies, both above and below our hyperplane, because we receive light signals from all directions in 4-space. We can see distant objects all around us above our hyperplane (looking up from the galactic surface into the fourth dimension), and all around us below our hyperplane (looking down through our transparent 4-dimensional galaxy and out the other side). But we do not observe (or measure) the angles of objects above or below our hyperplane, because we conceptualize the volume of space around us as a 3-ball when it is actually a 4-ball. We unknowingly conflate all the objects we receive light from, projecting them into a single 3-ball, whether they are above, below or in our hyperplane. == The dimpled fabric of spacetime == The spacetime of general relativity is often illustrated as a projection to a curved 2D surface in which large gravitational objects make gravity wells or dimples in the surface. In the Euclidean 4D view of the universe the 3D surface of a large cosmic object such as a galaxy surrounds an empty 4D space, and large gravitational objects within the galaxy must make dimples in its surface. But should we see them as dimples exactly? Would they dimple inwards or outwards? In the spacetime illustrations they are naturally always shown as dimpling downwards, which is somewhat disingenuous, strongly suggesting to the viewer that the reason for gravity is that it flows downhill - the original tautology we are trying to surmount! In the Euclidean 4D galaxy the dimple, if it is one, must be either inward or outward, and which it is matters. Is a large gravitational mass (such as a star) ''ahead'' of the smaller masses orbiting around it (such as its planets), or is it ''behind'' them, as they fly through 4-space on their Clifford parallel trajectories? The answer is ''both'' of course, because a star is not a dimple, it is a 4-ball, and it dimples the 3D surface both inwards and outwards. It is a thick place in the 3D surface. We should view it as having its gravitational center precisely at the surface of a 3-sphere expanding radially at velocity <math>c</math>. [[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 18:15, 10 November 2024 (UTC) == John Stewart Bell by Adam Becker == A roulette wheel has three orthogonal dimensions of binary parity: high/low (a finite integer set partitioned in two), even/odd (a different partitioning of the integer set), and a third dimension red/black (which belong to a distinct sequence, and I dare say a distinct group, than the other two parities). "Polarization is a property of light -- light is an electromagnetic wave, and the polarization is the direction that the wave is doing its waving in. Polarization is sort of like a little arrow that each photon carries with it that can point in different dirdctions. But ... we can't actually tell what direction a photon's polarization arrow is pointing. All we can do is measure a photon's polarization along one particular axis at a time in a somewhat indirect [binary] manner ...." [[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 21:36, 31 December 2024 (UTC) == Minkowski spacetime and Euclidean 4-space in relativity == ... Is there a configuration in Euclidean space where every observer, and every observed object, is moving at velocity <math>c</math>? Yes, there is one such configuration, in 4-dimensional Euclidean space. This configuration must be like the one described above, in our ''Abstract''. == The rate of atomic symmetry operations == ... == General relativity describes a curved 3-dimensional manifold embedded in Euclidean 4-dimensional space == ... == The geometry of the atomic nucleus == In [[W:Euclidean 4-space|Euclidean four dimensional space]], an [[W:atomic nucleus|atomic nucleus]] is a one or more concentric 4-polytopes of increasing radius. Each concentric shell is a single or compound [[24-cell]], the regular 4-polytope with [[W:Coxeter group#Symmetry groups of regular polytopes|𝔽₄ symmetry]]. Nuclear shells are concentric [[W:3-sphere|3-sphere]]s occupied (fully or partially) by the orbits of this 24-point [[#The 6 regular convex 4-polytopes|regular convex 4-polytope]]. An actual atomic nucleus is a rotating four dimensional object. It is not a ''rigid'' rotating object, it is a kinematic one, because the nucleus of an actual atom of a distinct [[W:nucleon number|nucleon number]] contains a distinct number of orbiting 24-cell vertices which may be in different isoclinic rotational orbits. These moving vertices never describe a compound of static 24-cells at any single instant in time, though their orbits do all the time. The physical configuration of a nucleus as concentric 24-cells can be reduced to the [[W:kinematics|kinematics]] of the orbits of its constituent protons and neutrons. The geometry of the atomic nucleus is therefore strictly [[W:Euclidean geometry#19th century|Euclidean]] in four dimensional space. === 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#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 (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 precise [[24-cell|detailed description]] enabling the reader to properly visualize it runs to many pages and illustrations, with many accompanying pages of explanatory notes on basic phenomena that arise only in 4-dimensional space: [[24-cell#Squares|completely orthogonal planes]], [[24-cell#Hexagons|Clifford parallelism]] and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Helical hexagrams and their isoclines|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 surprise. [[#The 6 regular convex 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 (generally), 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. [[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 is much harder for us to visualize, because the only way we can experience it is in our imaginations; we have no body of sensory experience in 4-dimensional space to draw upon. For that reason, 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. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than 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 any other at any time. That is still an example of a rigid object in a single distinct isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing the characteristic rotation of the 24-cell. But 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 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 shell without collisions? In adjacent 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 questions of this kind of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[W:kinetics|kinetics]]. In four dimensions, we find that space is more commodious than we thought it was, and 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 more symmetric packings. Sadoc studied twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack perfectly in 4-space without exhibiting any torsion, although their packing in 3-space was imperfect, "frustrated" by their torsion. <blockquote>The frustration, which arises when the molecular orientation is transported along the two [circular] AB paths of figure 1 [helix], is imposed by the very topological nature of the Euclidean space R³. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S³, 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]]....{{Efn|name=Petrie polygon of a honeycomb}} The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S³, 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_∞ 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 S³.{{Efn|name=dense fabric of pole-circles}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc & Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings ("frustrations") in 3-dimensional space.}}</blockquote> Of course we do not find molecules condensing to close-pack the 3-sphere in our experience. We find 3-spheres in the atomic realm (atoms are 4-polytopes), and in cosmologic realm (at the surface boundaries of stars, and perhaps at the surface masses of hollow galaxies). But in the molecular realm we find only objects which 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 wholly imaginary one, though no less illuminating for the symmetry it reveals not having a realization as a 3-sphere of actual molecules. And might it not have a realization in the atomic realm? We know that atoms have their own complex internal structure, which we are unable to model geometrically in flat 3-dimensional space. Suppose that the constituent particles of atoms actually close-pack the 3-sphere as helices, in the manner of Sadoc's imaginary twisted molecules, that is, as discrete Hopf fibrations of the atomic 4-polytope, where each discrete Hopf fibration corresponds to the distinct isoclinic rotation characteristic of a constituent particle. [[User:Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 19:28, 8 November 2025 (UTC) === Isospin === A [[W:Nucleon|nucleon]] is a [[W:proton|proton]] or a [[W:neutron|neutron]]. The proton carries a positive net [[W:Electric charge|charge]], and the neutron carries a zero net charge. The proton's [[W:Mass|mass]] is only about 0.13% less than the neutron's, and since they are observed to be identical in other respects, they can be viewed as two states of the same nucleon, together forming an isospin doublet ({{nowrap|''I'' {{=}} {{sfrac|1|2}}}}). In isospin space, neutrons can be transformed into protons and conversely by actions of the [[W:SU(2)|SU(2)]] symmetry group. In nature, protons are very stable (the most stable particle known); a proton and a neutron are a stable nuclide; but free neutrons decay into protons in about 10 or 15 seconds. According to the [[W:Noether theorem|Noether theorem]], [[W:Isospin|isospin]] is conserved with respect to the [[W:strong interaction|strong interaction]].<ref name=Griffiths2008>{{cite book |author=Griffiths, David J. |title=Introduction to Elementary Particles |edition=2nd revised |publisher=WILEY-VCH |year=2008 |isbn=978-3-527-40601-2}}</ref>{{rp|129–130}} Nucleons are acted upon equally by the strong interaction, which is invariant under rotation in isospin space. Isospin was introduced as a concept in 1932 by [[W:Werner Heisenberg|Werner Heisenberg]],<ref> {{cite journal |last=Heisenberg |first=W. |author-link=W:Werner Heisenberg |year=1932 |title=Über den Bau der Atomkerne |journal=[[W:Zeitschrift für Physik|Zeitschrift für Physik]] |volume=77 |issue=1–2 |pages=1–11 |doi=10.1007/BF01342433 |bibcode = 1932ZPhy...77....1H |s2cid=186218053 |language=de}}</ref> well before the 1960s development of the [[W:quark model|quark model]], to explain the symmetry of the proton and the then newly discovered neutron. Heisenberg introduced the concept of another conserved quantity that would cause the proton to turn into a neutron and vice versa. In 1937, [[W:Eugene Wigner|Eugene Wigner]] introduced the term "isospin" to indicate how the new quantity is similar to spin in behavior, but otherwise unrelated.<ref> {{cite journal |last=Wigner |first=E. |author-link=W:Eugene Wigner |year=1937 |title=On the Consequences of the Symmetry of the Nuclear Hamiltonian on the Spectroscopy of Nuclei |journal=[[W:Physical Review|Physical Review]] |volume=51 |pages=106–119 |doi=10.1103/PhysRev.51.106 |bibcode = 1937PhRv...51..106W |issue=2 }}</ref> Similar to a spin-1/2 particle, which has two states, protons and neutrons were said to be of isospin 1/2. The proton and neutron were then associated with different isospin projections ''I''₃&nbsp;=&nbsp;+1/2 and −1/2 respectively. Isospin is a different kind of rotation entirely than the ordinary spin which objects undergo when they rotate in three-dimensional space. Isospin does not correspond to a [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] in any space (of any number of dimensions). However, it does seem to correspond exactly to an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]] in a Euclidean space of four dimensions. Isospin space resembles the [[W:3-sphere|3-sphere]], the [[W:Elliptical space#Elliptic space (the 3D case)|curved 3-dimensional space]] that is the surface of a [[W:4-ball (mathematics)#In Euclidean space|4-dimensional ball]]. === Spinors === [[File:Spinor on the circle.png|thumb|upright=1.5|A spinor visualized as a vector pointing along the [[W:Möbius band|Möbius band]], exhibiting a sign inversion when the circle (the "physical system") is continuously rotated through a full turn of 360°.]][[W:Spinors|Spinors]] are [[W:representation of a Lie group|representations]] of a [[W:spin group|spin group]], which are [[W:Double covering group|double cover]]s of the [[W:special orthogonal group|special orthogonal groups]]. The spin group Spin(4) is the double cover of [[W:SO(4)|SO(4)]], the group of rotations in 4-dimensional Euclidean space. [[600-cell#Fibrations of isocline polygrams|Isoclines]], the helical geodesic paths followed by points under isoclinic rotation, correspond to spinors representing Spin(4). Spinors can be viewed as the "square roots" of [[W:Section (fiber bundle)|cross sections]] of [[W:vector bundle|vector bundle]]s; in this correspondence, a fiber bundle of isoclines (of a distinct isoclinic rotation) is a cross section (inverse bundle) of a fibration of great circles (in the invariant planes of that rotation). A spinor can be visualized as a moving vector on a Möbius strip which transforms to its negative when continuously rotated through 360°, just as [[24-cell#Helical hexagrams and their isoclines|an isocline can be visualized as a Möbius strip]] winding twice around the 3-sphere, during which [[24-cell#Isoclinic rotations|720° isoclinic rotation]] the rigid 4-polytope turns itself inside-out twice.{{Sfn|Goucher|2019|loc=Spin Groups}} Under isoclinic rotation, a rigid 4-polytope is an isospin-1/2 object with two states. === Isoclinic rotations in the nucleus === Isospin is regarded as a symmetry of the strong interaction under the [[W:Group action (mathematics)|action]] of the [[W:Lie group|Lie group]] [[W:SU(2)|SU(2)]], the two [[W:eigenstate|states]] being the [[W:Up quark|up flavour]] and [[W:Down quark|down flavour]]. A 360° isoclinic rotation of a rigid [[W:nuclide|nuclide]] would transform its protons into neutrons and vice versa, exchanging the up and down flavours of their constituent [[W:quarks|quarks]], by turning the nuclide and all its parts inside-out (or perhaps we should say upside-down). Because we never observe this, we know that the nucleus is not a ''rigid'' polytope undergoing isoclinic rotation. If the nucleus ''were'' a rigid object, nuclides that were isospin-rotated 360° would be isoclinic mirror images of each other, isospin +1/2 and isospin −1/2 states of the whole nucleus. We don't see whole nuclides rotating as a rigid object, but considering what would happen if they ''were'' rigid tells us something about the geometry we must expect inside the nucleons. One way that an isospin-rotated neutron could become a proton would be if the up quark and down quark were a left and right mirror-image pair of the same object; exchanging them in place would turn each down-down-up neutron into an up-up-down proton. But the case cannot be quite that simple, because the up quark and the down quark are not mirror-images of the same object: they have very different mass and other incongruities. Another way an isospin-rotated neutron could be a proton would be if the up and down quarks were asymmetrical kinematic polytopes (not indirectly congruent mirror-images, and not rigid polytopes), rotating within the nucleus in different ''hybrid'' orbits. By that we mean that they may have vertices orbiting in rotations characteristic of more than one 4-polytope, so they may change shape as they rotate. In that case their composites (protons and neutrons) could have a symmetry not manifest in their components, but emerging from their combination. .... === Hybrid isoclinic rotations === The 24-cell has [[24-cell#Isoclinic rotations|its own characteristic isoclinic rotations]] in 4 Clifford parallel hexagonal planes (each intersecting 6 vertices), and also inherits the [[16-cell#Rotations|characteristic isoclinic rotations of its 3 Clifford parallel constituent 16-cells]] in 6 Clifford parallel square planes (each intersecting 4 vertices). The twisted circular paths followed by vertices in these two different kinds of rotation have entirely different geometries. Vertices rotating in hexagonal invariant planes follow [[24-cell#Helical hexagrams and their isoclines|helical geodesic curves whose chords form hexagrams]], and vertices rotating in square invariant planes follow [[24-cell#Helical octagrams and their isoclines|helical geodesic curves whose chords form octagrams]]. In a rigid isoclinic rotation, ''all'' the [[24-cell#Geodesics|great circle polygons]] move, in any kind of rotation. What distinguishes the hexagonal and square isoclinic rotations is the invariant planes of rotation the vertices stay in. The rotation described [[#Rotations|above]] (of 8 vertices rotating in 4 Clifford parallel hexagonal planes) is a single hexagonal isoclinic rotation, not a kinematic or hybrid rotation. A ''kinematic'' isoclinic rotation in the 24-cell is any subset of the 24 vertices rotating through the same angle in the same time, but independently with respect to the choice of a Clifford parallel set of invariant planes of rotation and the chirality (left or right) of the rotation. A ''hybrid'' isoclinic rotation combines moving vertices from different kinds of isoclinic rotations, characteristic of different regular 4-polytopes. For example, if at least one vertex rotates in a square plane and at least one vertex rotates in a hexagonal plane, the kinematic rotation is a hybrid rotation, combining rotations characteristic of the 16-cell and characteristic of the 24-cell. As an example of the simplest hybrid isoclinic rotation, consider a 24-cell vertex rotating in a square plane, and a second vertex, initially one 24-cell edge-length distant, rotating in a hexagonal plane. Rotating isoclinically at the same rate, the two moving vertices will never collide where their paths intersect, so this is a ''valid'' hybrid rotation. To understand hybrid rotations in the 24-cell more generally, visualize the relationship between great squares and great hexagons. The [[24-cell#Squares|18 great squares]] occur as three sets of 6 orthogonal great squares,{{Efn|name=six orthogonal planes of the Cartesian basis}} each [[16-cell#Coordinates|forming a 16-cell]]. The three 16-cells are completely disjoint{{Efn|name=completely disjoint}} and [[24-cell#Clifford parallel polytopes|Clifford parallel]]: each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}).{{Efn|name=three isoclinic 16-cells}} The 18 square great circles are crossed by 16 hexagonal great circles; each [[24-cell#Hexagons|hexagon]] has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two [[24-cell#Triangles|great triangles]] inscribed in each great hexagon (occupying its alternate vertices, with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking three completely disjoint great squares, one from each of the three completely disjoint 16-cells''.{{Efn|There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}} Isoclinic rotations take the elements of the 4-polytope to congruent [[24-cell#Clifford parallel polytopes|Clifford parallel elements]] elsewhere in the 4-polytope. The square rotations do this ''locally'', confined within each 16-cell: for example, they take great squares to other great squares within the same 16-cell. The hexagonal rotations act ''globally'' within the entire 24-cell: for example, they take great squares to other great squares in ''different'' 16-cells. The [[16-cell#Helical construction|chords of the square rotations]] bind the 16-cells together internally, and the [[24-cell#Helical hexagrams and their isoclines|chords of the hexagonal rotations]] bind the three 16-cells together. .... === Charge === Positive and negative charge are [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#The perfection of Fuller's cyclic design|compression and tension]], respectively. ... === Color === When the existence of quarks was suspected in 1964, [[W:Oscar W. Greenberg|Greenberg]] introduced the notion of color charge to explain how quarks could coexist inside some [[W:hadron|hadron]]s in [[W:quark model#The discovery of color|otherwise identical quantum states]] without violating the [[W:Pauli exclusion principle|Pauli exclusion principle]]. The modern concept of [[W:color charge|color charge]] completely commuting with all other charges and providing the strong force charge was articulated in 1973, by [[W:William A. Bardeen|William Bardeen]], [[W:de:Harald Fritzsch|Harald Fritzsch]], and [[W:Murray Gell-Mann|Murray Gell-Mann]].<ref>{{cite conference |author1=Bardeen, W. |author2=Fritzsch, H. |author3=Gell-Mann, M. |year=1973 |title=Light cone current algebra, ''π''⁰ decay, and ''e''⁺ ''e''^&minus; annihilation |arxiv=hep-ph/0211388 |editor=Gatto, R. |book-title=Scale and conformal symmetry in hadron physics |page=[https://archive.org/details/scaleconformalsy0000unse/page/139 139] |publisher=[[W:John Wiley & Sons|John Wiley & Sons]] |isbn=0-471-29292-3 |bibcode=2002hep.ph...11388B |url-access=registration |url=https://archive.org/details/scaleconformalsy0000unse/page/139 }}</ref><ref>{{cite journal |title=Advantages of the color octet gluon picture |journal=[[W:Physics Letters B|Physics Letters B]] |volume=47 |issue=4 |page=365 |year=1973 |last1=Fritzsch |first1=H. |last2=Gell-Mann |first2=M. |last3=Leutwyler |first3=H. |doi=10.1016/0370-2693(73)90625-4 |bibcode=1973PhLB...47..365F |citeseerx=10.1.1.453.4712}}</ref> Color charge is not [[W:electric charge|electric charge]]; the whole point of it is that it is a quantum of something different. But it is related to electric charge, through the way in which the three different-colored quarks combine to contribute fractional quantities of electric charge to a nucleon. As we shall see, color is not really a separate kind of charge at all, but a partitioning of the electric charge into [[24-cell#Clifford parallel polytopes|Clifford parallel subspaces]]. The [[W:Color charge#Red, green, and blue|three different colors]] of quark charge might correspond to three different 16-cells, such as the three disjoint 16-cells inscribed in the 24-cell. Each color might be a disjoint domain in isospin space (the space of points on the 3-sphere).{{Efn|The 8 vertices of each disjoint 16-cell constitute an independent [[16-cell#Coordinates|orthonormal basis for a coordinate reference frame]].}} Alternatively, the three colors might correspond to three different fibrations of the same isospin space: three different ''sequences'' of the same total set of discrete points on the 3-sphere. These alternative possibilities constrain possible representations of the nuclides themselves, for example if we try to represent nuclides as particular rotating 4-polytopes. If the neutron is a (8-point) 16-cell, either of the two color possibilities might somehow make sense as far as the neutron is concerned. But if the proton is a (5-point) 5-cell, only the latter color possibility makes sense, because fibrations (which correspond to distinct isoclinic left-and-right rigid rotations) are the ''only'' thing the 5-cell has three of. Both the 5-cell and the 16-cell have three discrete rotational fibrations. Moreover, in the case of a rigid, isoclinically rotating 4-polytope, those three fibrations always come one-of-a-kind and two-of-a-kind, in at least two different ways. First, one fibration is the set of invariant planes currently being rotated through, and the other two are not. Second, when one considers the three fibrations of each of these 4-polytopes, in each fibration two isoclines carry the left and right rotations respectively, and the third isocline acts simply as a Petrie polygon, the difference between the fibrations being the role assigned to each isocline. If we associate each quark with one or more isoclinic rotations in which the moving vertices belong to different 16-cells of the 24-cell, and the sign (plus or minus) of the electric charge with the chirality (right or left) of isoclinic rotations generally, we can configure nucleons of three quarks, two performing rotations of one chirality and one performing rotations of the other chirality. The configuration will be a valid kinematic rotation because the completely disjoint 16-cells can rotate independently; their vertices would never collide even if the 16-cells were performing different rigid square isoclinic rotations (all 8 vertices rotating in unison). But we need not associate a quark with a [[16-cell#Rotations|rigidly rotating 16-cell]], or with a single distinct square rotation. Minimally, we must associate each quark with at least one moving vertex in each of three different 16-cells, following the twisted geodesic isocline of an isoclinic rotation. In the up quark, that could be the isocline of a right rotation; and in the down quark, the isocline of a left rotation. The chirality accounts for the sign of the electric charge (we have said conventionally as +right, −left), but we must also account for the quantity of charge: +{{sfrac|2|3}} in an up quark, and −{{sfrac|1|3}} in a down quark. One way to do that would be to give the three distinct quarks moving vertices of {{sfrac|1|3}} charge in different 16-cells, but provide up quarks with twice as many vertices moving on +right isoclines as down quarks have vertices moving on −left isoclines (assuming the correct chiral pairing is up+right, down−left). Minimally, an up quark requires two moving vertices (of the up+right chirality).{{Efn|Two moving vertices in one quark could belong to the same 16-cell. A 16-cell may have two vertices moving in the same isoclinic square (octagram) orbit, such as an antipodal pair (a rotating dipole), or two vertices moving in different square orbits of the same up+right chirality.{{Efn|There is only one [[16-cell#Helical construction|octagram orbit]] of each chirality in each fibration of the 16-cell, so two octagram orbits of the same chirality cannot be Clifford parallel (part of the same distinct rotation). Two vertices right-moving on different octagram isoclines in the same 16-cell is a combination of two distinct rotations, whose isoclines will intersect: a kinematic rotation. It can be a valid kinematic rotation if the moving vertices will never pass through a point of intersection at the same time. Octagram isoclines pass through all 8 vertices of the 16-cell, and all eight isoclines (the left and right isoclines of four different fibrations) intersect at ''every'' vertex.}} However, the theory of [[W:Color confinement|color confinement]] may not require that two moving vertices in one quark belong to the same 16-cell; like the moving vertices of different quarks, they could be drawn from the disjoint vertex sets of two different 16-cells.}} Minimally, a down quark requires one moving vertex (of the down−left chirality). In these minimal quark configurations, a proton would have 5 moving vertices and a neutron would have 4. .... === Nucleons === [[File:Symmetrical_5-set_Venn_diagram.svg|thumb|[[W:Branko Grünbaum|Grünbaum's]] rotationally symmetrical 5-set Venn diagram, 1975. It is an illustration of the [[5-cell]] labeling each of its <math>2^5</math> elements. Think of it as an [[W:Nuclear magnetic resonance|NMR image]] of the 4-dimensional proton in projection to the plane.]] The proton is a very stable mass particle. Is there a stable orbit of 5 moving vertices in 4-dimensional Euclidean space? There are few known solutions to the 5-body problem, and fewer still to the [[W:n-body problem|{{mvar|n}}-body problem]], but one is known: the ''central configuration'' of {{mvar|n}} bodies in a space of dimension {{mvar|n}}-1. A [[W:Central configuration|central configuration]] is a system of [[W:Point particle|point masses]] with the property that each mass is pulled by the combined attractive force of the system directly towards the [[W:Center of mass|center of mass]], with acceleration proportional to its distance from the center. Placing three masses in an equilateral triangle, four at the vertices of a regular [[W:Tetrahedron|tetrahedron]], five at the vertices of a regular [[5-cell]], or more generally {{mvar|n}} masses at the vertices of a regular [[W:Simplex|simplex]] produces a central configuration [[W:Central configuration#Examples|even when the masses are not equal]]. In an isoclinic rotation, all the moving vertices orbit at the same radius and the same speed. Therefore if any 5 bodies are orbiting as an isoclinically rotating regular 5-cell (a rigid 4-simplex figure undergoing isoclinic rotation), they maintain a central configuration, describing 5 mutually stable orbits. Unlike the proton, the neutron is not always a stable particle; a free neutron will decay into a proton. A deficiency of the minimal configurations is that there is no way for this [[W:beta minus decay|beta minus decay]] to occur. The minimal neutron of 4 moving vertices described [[#Color|above]] cannot possibly decay into a proton by losing moving vertices, because it does not possess the four up+right moving vertices required in a proton. This deficiency could be remedied by giving the neutron configuration 8 moving vertices instead of 4: four down−left and four up+right moving vertices. Then by losing 3 down−left moving vertices the neutron could decay into the 5 vertex up-down-up proton configuration.{{Efn|Although protons are very stable, during [[W:stellar nucleosynthesis|stellar nucleosynthesis]] two H₁ protons are fused into an H₂ nucleus consisting of a proton and a neutron. This [[W:beta plus decay|beta plus "decay"]] of a proton into a neutron is actually the result of a rare high-energy collision between the two protons, in which a neutron is constructed. With respect to our nucleon configurations of moving vertices, it has to be explained as the conversion of two 5-point 5-cells into a 5-point 5-cell and an 8-point 16-cell, emitting two decay products of at least 1-point each. Thus it must involve the creation of moving vertices, by the conversion of kinetic energy to point-masses.}} A neutron configuration of 8 moving vertices could occur as the 8-point 16-cell, the second-smallest regular 4-polytope after the 5-point 5-cell (the hypothesized proton configuration). It is possible to double the neutron configuration in this way, without destroying the charge balance that defines the nucleons, by giving down quarks three moving vertices instead of just one: two −left vertices and one +right vertex. The net charge on the down quark remains −{{sfrac|1|3}}, but the down quark becomes heavier (at least in vertex count) than the up quark, as in fact its mass is measured to be. A nucleon's quark configuration is only a partial specification of its properties. There is much more to a nucleon than what is contained within its three quarks, which contribute only about 1% of the nucleon's energy. The additional 99% of the nucleon mass is said to be associated with the force that binds the three quarks together, rather than being intrinsic to the individual quarks separately. In the case of the proton, 5 moving vertices in the stable orbits of a central configuration (in one of the [[5-cell#Geodesics and rotations|isoclinic rotations characteristic of the regular 5-cell]]) might be sufficient to account for the stability of the proton, but not to account for most of the proton's energy. It is not the point-masses of the moving vertices themselves which constitute most of the mass of the nucleon; if mass is a consequence of geometry, we must look to the larger geometric elements of these polytopes as their major mass contributors. The quark configurations are thus incomplete specifications of the geometry of the nucleons, predictive of only some of the nucleon's properties, such as charge.{{Efn|Notice that by giving the down quark three moving vertices, we seem to have changed the quark model's prediction of the proton's number of moving vertices from 5 to 7, which would be incompatible with our theory that the proton configuration is a rotating regular 5-cell in a central configuration of 5 stable orbits. Fortunately, the actual quark model has nothing at all to say about moving vertices, so we may choose to regard that number as one of the geometric properties the quark model does not specify.}} In particular, they do not account for the forces binding the nucleon together. Moreover, if the rotating regular 5-cell is the proton configuration and the rotating regular 16-cell is the neutron configuration, then a nucleus is a complex of rotating 5-cells and 16-cells, and we must look to the geometric relationship between those two very different regular 4-polytopes for an understanding of the nuclear force binding them together. The most direct [[120-cell#Relationships among interior polytopes|geometric relationship among stationary regular 4-polytopes]] is the way they occupy a common 3-sphere together. Multiple 16-cells of equal radius can be compounded to form each of the larger regular 4-polytopes, the 8-cell, 24-cell, 600-cell, and 120-cell, but it is noteworthy that multiple regular 5-cells of equal radius cannot be compounded to form any of the other 4-polytopes except the largest, the 120-cell. The 120-cell is the unique intersection of the regular 5-cell and 16-cell: it is a compound of 120 regular 5-cells, and also a compound of 75 16-cells. All regular 4-polytopes except the 5-cell are compounds of 16-cells, but none of them except the largest, the 120-cell, contains any regular 5-cells. So in any compound of equal-radius 16-cells which also contains a regular 5-cell, whether that compound forms some single larger regular 4-polytope or does not, no two of the regular 5-cell's five vertices ever lie in the same 16-cell. So the geometric relationship between the regular 5-cell (our proton candidate) and the regular 16-cell (our neutron candidate) is quite a distant one: they are much more exclusive of each other's elements than they are distantly related, despite their complementary three-quark configurations and other similarities as nucleons. The relationship between a regular 5-cell and a regular 16-cell of equal radius is manifest only in the 120-cell, the most complex regular 4-polytope, which [[120-cell#Geometry|uniquely embodies all the containment relationships]] among all the regular 4-polytopes and their elements. If the nucleus is a complex of 5-cells (protons) and 16-cells (neutrons) rotating isoclinically around a common center, then its overall motion is a hybrid isoclinic rotation, because the 5-cell and the 16-cell have different characteristic isoclinic rotations, and they have no isoclinic rotation in common.{{Efn|The regular 5-cell does not occur inscribed in any other regular 4-polytope except one, the 600-vertex 120-cell. No two of the 5 vertices of a regular 5-cell can be vertices of the same 16-cell, 8-cell, 24-cell, or 600-cell. The isoclinic rotations characteristic of the regular 5-cell maintain the separation of its 5 moving vertices in 5 disjoint Clifford-parallel subspaces at all times. The [[16-cell#Rotations|isoclinic rotation characteristic of the 16-cell]] maintains the separation of its 8 moving vertices in 2 disjoint Clifford-parallel subspaces (completely orthogonal great square planes) at all times. Therefore, in any hybrid rotation of a concentric 5-cell and 16-cell, at most one 5-cell subspace (containing 1 vertex) might be synchronized with one 16-cell subspace (containing 4 vertices), such that the 1 + 4 vertices they jointly contain occupy the same moving subspace continually, forming a rigid 5-vertex polytope undergoing some kind of rotation. If in fact it existed, this 5-vertex rotating rigid polytope would not be [[5-cell#Geometry|not a 5-cell, since 4 of its vertices are coplanar]]; it is not a 4-polytope but merely a polyhedron, a [[W:square pyramid|square pyramid]].}} .... === Nuclides === ... === Quantum phenomena === The Bell-Kochen-Specker (BKS) theorem rules out the existence of deterministic noncontextual hidden variables theories. A proof of the theorem in a space of three or more dimensions can be given by exhibiting a finite set of lines through the origin that cannot each be colored black or white in such a way that (i) no two orthogonal lines are both black, and (ii) not all members of a set of ''d'' mutually orthogonal lines are white.{{Efn|"The Bell-Kochen-Specker theorem rules out the existence of deterministic noncontextual hidden variables theories. A proof of the theorem in a Hilbert space of dimension d ≥ 3 can be given by exhibiting a finite set of rays [9] that cannot each be assigned the value 0 or 1 in such a way that (i) no two orthogonal rays are both assigned the value 1, and (ii) not all members of a set of d mutually orthogonal rays are assigned the value 0."{{Sfn|Waegell & Aravind|2009|loc=2. The Bell-Kochen-Specker (BKS) theorem}}|name=BKS theorem}} .... === Motion === What does it mean to say that an object moves through space? Coxeter group theory provides precise answers to questions of this kind. A rigid object (polytope) moves by distinct transformations, changing itself in each discrete step into a congruent object in a different orientation and position.{{Efn|name=transformations}} .... === General relativity is just Galilean relativity in a general space of four orthogonal dimensions === .... Coxeter orthogonal dimensional relativity furnishes the geometric spatial definition of the inertial-gravitational force. It does so by limiting free motion (in the sense of free-falling) to one rotation and one translation, making double rotations accelerations which involve an energy transition. ...I now don't think that is quite right, rather: It does so by limiting free motion (in the sense of free-falling) to a double rotation (or one rotation and one translation) at conserved angles, making changes to the angles of rotation accelerations which involve an energy transition. === Physics === .... === Spherical relativity === We began our inquiry by wondering why physical space should be limited to just three dimensions (why ''three''). By visualizing the universe as a Euclidian space of four dimensions, we recognize that relativistic and quantum phenomena are natural consequences of symmetry group operations (including reflections and rotations) in four orthogonal dimensions. We are then led naturally to question whether the universe has only four dimensions. Physical space must bear as many dimensions as we need to ascribe to it, though the distinct phenomena for which we find a need to do so, in order to explain them, seem to be fewer and fewer as we consider higher and higher dimensions. To laws of physics generally, such as the principle of relativity in particular, we should always append the phrase "in Euclidean spaces of any number of dimensions". Laws of physics should operate in any flat Euclidean space <math>R^n</math> and in its corresponding spherical space <math>S^n</math>. The first and simplest sense in which we are forced to contemplate a fifth dimension is to accommodate our normal idea of time. Just as Einstein was forced to admit time as a dimension, in his four-dimensional spacetime of three spatial dimensions plus time, for some purposes we require a fifth time dimension to accompany our four spatial dimensions, when our purpose is orthogonal to (in the sense of independent of) the four spatial dimensions. For example, if we theorize that we observe a finite homogeneous universe, and that it is a Euclidean 4-space overall, we may prefer not to have to identify any distinct place within that 4-space as the center where the universe began in a big bang. To avoid having to pick a distinct place as the center of the universe, our model of it must be expanded, at least to be a ''spherical'' 4-dimensional space with the fifth radial dimension as time. Essentially, we require the fifth dimension in order to make our homogeneous 4-space finite, by wrapping it around into a 4-sphere. But perhaps we can still resist admitting the fifth radial dimension as a full-fledged Euclidean spatial dimension, at least so long as we have not observed how any naturally occurring object configurations are best described as 5-polytopes. One phenomenon which resists explanation in a space of just four dimensions is the propagation of light in a vacuum. The propagation of mass-carrying particles is explained as the consequence of their rotations in closed, curved spaces (3-spheres) of finite size, moving through four-dimensional Euclidean space at a universal constant speed, the speed of light. But an apparent paradox remains that light must seemingly propagate through four-dimensional Euclidean space at more than the speed of light. From an ''n''-dimensional viewpoint, this apparent paradox can be resolved, and in retrospect it is clear how massless particles can translate through four-dimensional space at twice the speed constant, since they are not simultaneously rotating. Another phenomenon justifying a five-dimensional view of space is the relation between the the 5-cell proton and the 16-cell neutron (the 4-simplex and 4-orthoplex polytopes). Their indirect relationship can be observed in the 4-600-point polytope (the 120-cell), and in its 11-cells,{{Sfn|Christie|2025|loc="A symmetrical arrangement of eleven 11-cells"}} but it is only directly observed (absent a 120-cell) in a five-dimensional reference frame. === Nuclear geometry === We have seen how isoclinic rotations (Clifford displacements) relate the orbits in the atomic nucleus to each other, just as they relate the regular convex 4-polytopes to each other, in a sequence of nested objects of increasing complexity. We have identified the proton as a 5-point, 5-cell 4-simplex 𝜶₄, the neutron as an 8-point, 16-cell 4-orthoplex 𝛽₄, and the shell of the atomic nucleus as a 24-point 24-cell. As Coxeter noted, that unique 24-point object stands quite alone in four dimensions, having no analogue above or below. === Atomic geometry === I'm on a plane flying to Eugene to visit Catalin, we'll talk after I arrive. I've been working on both my unpublished papers, the one going put for pre-publication review soon about 4D geometry, and the big one not going out soon about the 4D sun, 4D atoms, and 4D galaxies and n-D universe. I'vd just added the following paragraph to that big paper: Atomic geometry The force binding the protons and neutrons of the nucleus together into a distinct element is specifically an expression of the 11-cell 4-polytope, itself an expression of the pyritohedral symmetry, which binds the distinct 4-polytopes to each other, and relates the n-polytopes to their neighbors of different n by dimensional analogy. flying over mt shasta out my right-side window at the moment, that last text showing "not delivered" yet because there's no wifi on this plane, gazing at that great peak of the world and feeling as if i've just made the first ascent of it === Atomic elements === There are 120 5-cell protons in the 120-cell, and so there are 120 distinct atomic elements on the 3-sphere. === Molecular geometry === Molecules are 3-dimensional structures that live in the thin film of 3-membrane only one atom thick in most places that is our ordinary space, but since that is a significantly curved 3-dimensional space at the scale of a molecule, the way the molecule's covalent bonds form is influenced by the local curvature in 4-dimensions at that point. In the water molecule, there is a reason why the hydrogen atoms are attached to the oxygen atom at an angle of 104.45° in 3-dimensional space, and at root it must be the same symmetry that locates any two of the hydrogen proton's five vertices 104.45° apart on a great circle arc of its tiny 3-sphere. === Cosmology === ==== Solar systems ==== ===== Stars ===== ... ===== The Kepler problem ===== In the proper reference frame of some hypothetical observer in 4-dimensional Euclidean space, the Kepler problem has a solution in which all the planetary orbits are circles. This occurs because all bodies are always orbiting, each in some distinct isoclinic (equi-angled double) rotation. All atoms are rotating isoclinically (in two completely orthogonal planes) at velocity <math>c</math>. In the observer's proper reference frame, the isoclinic double rotation of each of his own atoms is most conveniently represented as an equivalent screw displacement: a linear translation at velocity <math>c</math> (along the observer's proper time axis through 4-space), combined with a simple rotation at velocity <math>c</math> (of the atom internally). This representation, though proper, is subject to the over-simplification of special relativity, because it ignores the fact that the observer is himself moving on some circular orbit; he is not linearly translating through 4-space. When the observer's own orbit is also represented, the screw displacement becomes an elliptical double rotation, with a very slow, very large-radius simple circular orbit (almost a linear translation), combined with a very fast, very small-radius simple circular orbit of each atom internally. This is the observer's proper reference frame according to general relativity. An observer moving differently will perceive the motion of the observer and his atoms to be a different elliptical double rotation. We can find a proper reference frame, of some hypothetical observer moving hypothetically, in which the ratio of the radii of the two orthogonal rotations of the atom becomes any value we choose between 1 and <math>c</math>. In particular there exists, conceptually accessible to all observers independent of their motion, a distinct reference frame in which an atom is stationary, rotating isoclinically (in two completely orthogonal planes) at the velocity <math>c</math>. That is simply the stationary reference frame of the atom itself, such as the stationary reference frame of any observer and his own atoms. It is known as the observer's proper inertial reference frame. But in 4-dimensional physics, that is no the most convenient or useful reference frame in which to consider objects moving differently, in different proper reference frames. In 4-dimensional physics, a reference frame in which every observer and his atoms are translating linearly at velocity <math>c</math> is known as the universal 4-coordinate reference frame, because it is the common reference frame of all observers (though it cannot be the proper reference frame of any observer). The universal 4-coordinate reference frame makes a great deal more sense for the purpose of considering the motion of objects moving differently in 4-space, just as a heliocentric solar system makes a great deal more sense than a geocentric solar system for the purpose of considering the motion of the planets. ... ==Appendix== {{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=2}} {{Regular convex 4-polytopes|columns=7|wiki=W:|radius=1}} ==Notes== {{Regular convex 4-polytopes Notelist|wiki=W:}} ==Citations== {{Regular convex 4-polytopes Reflist|wiki=W:}} ==References== {{Refbegin}} * {{Cite book|last=Mac Gregor|first=Malcolm H.|title=The energy packet generator 137 and the key energy equation E_w + E_z = E_top|date=2017|publisher=El Mac Books|place=Santa Cruz, CA|author-link=W:Fine-structure constant|ref={{Mac Gregor 137|2017}}}} {{Regular convex 4-polytopes Refs|wiki=W:}} {{Refend}} 5si7v1wlvyab4do82axcwf7ikexejf1 5 330057 2842466 2836599 2026-10-24T22:52:59Z Codename Noreste 2969951 /* Approval */ new topic ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]]) 2842466 wikitext text/x-wiki == Notice to colloquium == What is the sence of noticing community about that? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:10, 8 June 2026 (UTC) : A notice would be posted at the inactive SSM's user talk page, and a separate notification at the village pump listing the inactive support staff member(s). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:46, 25 June 2026 (UTC) ::And the reasoning behind why whole community should know, there is inactive staff? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:08, 2 July 2026 (UTC) ::: This is standard practice as the stewards have done this similar procedure per [[m:Admin activity review]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:06, 2 July 2026 (UTC) : I think this practice helps towards openness and transparency. For example, someone else in the community may become aware and comment about reasons why the SSM may be inactive. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:43, 21 July 2026 (UTC) ::I concur that the community should be made aware of, and given an opportunity to comment on, decisions that have an effect on our site. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 22:11, 27 August 2026 (UTC) == Inactive curator template == Just a note if this policy is agreet the template should be fixed. No it counts with 2 years. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:12, 8 June 2026 (UTC) : I'm not sure what you are trying to explain. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:47, 25 June 2026 (UTC) ::I am saying that if the inactive period is changed, the {{tl|Inactive curator}} template text ''"no edits or no logged actions for 2 years"'' should be changed to the appropriate one. This is just a notice, not to forget to do so. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:10, 2 July 2026 (UTC) ::: I see, thank you for explaining. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:05, 2 July 2026 (UTC) == Communication with the SSM and deadlines == A notification on the user's user page is a decent way to communicate with support staff. If they don't respond, it's clear that there's no point in waiting any longer and their rights have been revoked. On the contrary, if they respond, they suspect that they should start working on Wikiversity, but it may happen that they won't, i.e. SSM will respond, but they will continue to be inactive, so they will have another year of "peace". I would probably reduce the inactivity time to '''8 months''' (i.e. 6 months + 2 months, which may take to creat a custodian), but I would leave the response time at a '''month or more'''. I assume that sometimes the reason for inactivity is health problems or personal problems, and in such situations a person is usually not very reactive - i.e. they don't manage to respond quickly to all the requests that come to them. Another reason may be the busy work schedule of university teachers, who, for example, are on the job for 4 months during exams. This means, yes, you have been inactive for a while for some reason and then someone invites you to return to activity, but you are sick, or you are writing a scientific article, grant report, etc. and you don't have much time right now. Here, it would perhaps require standardized posts for all SSM roles, where a notice would be written that according to the policy, a SSM cannot be inactive for a given period. ''Then a question whether they will resume activity within 2 months.'' Yes - rights retained, no/no answer - rights removed within a month. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:29, 8 June 2026 (UTC) : I still think we should leave the timeframe as one year to maintain consistency with some other projects. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:40, 25 June 2026 (UTC) ::Well, why not. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:25, 25 June 2026 (UTC) : I think 12 months is a simple, clear, appropriate timeframe for inactivity. : I agree that one month to reply is more reasonable than 2 weeks. It is quite common for people to be on annual leave e.g., for 4 weeks. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:47, 21 July 2026 (UTC) == wording == This may sound pendantic... the prhase "are considered inactive if they have made no edits and logged actions within one year." Does this mean that any edit [[w:Logical conjunction|AND]] a log action are required in one year? Does it mean that either one is required in that time frame, but not necessarily both? [[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 04:39, 10 July 2026 (UTC) : This means an edit and a logged action, together. I am not sure what is the point of desysopping a custodian who makes edits but otherwise does not use their permission(s). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:45, 10 July 2026 (UTC) ::I updated the draft language to remove any ambiguity. [[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 17:18, 16 July 2026 (UTC) == 'crat actions == For reference, please see the en.wikipedia section at [[meta:Admin activity review/Local inactivity policies]]. <blockquote>Bureaucrats: same as the Inactive administrators rule above, plus if a bureaucrat does not participate in bureaucrat activity for over three years, their rights may also be removed.</blockquote> As someone who uses this tool, I can tell you that these rights are infrequently used. If more than 12 months passes with no need to flip a custodian bit (a not particularity rare occurrence) it would result in every 'crat here having the bit simultaneously removed. Comments? [[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 17:25, 16 July 2026 (UTC) : Should this be a part of this policy, a notice would be recommended explaining that some bureaucrat haven't used their bit involving bureaucrat actions, and a week should be given whether they would like to retain it or not, or otherwise resign at [[m:SRP]]. [[User:Codename Noreste|<span style="color: blue">Codename Noreste</span>]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:56, 16 July 2026 (UTC) : Pinging bureaucrats [[User:Atcovi|Atcovi]], [[User:Dave Braunschweig|Dave Braunschweig]], [[User:Jtneill|Jtneill]] and [[User:Koavf|Koavf]] to this discussion. [[User:Codename Noreste|<span style="color: blue">Codename Noreste</span>]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:57, 16 July 2026 (UTC) ::I am fine with CN's proposal. I think a more sensible policy would be to collapse bureaucrat and sysop rights together, so if someone hasn't changed users rights (bureaucrat) <em>or</em> blocked a user, deleted a page, deleted a rev, imported a page, etc. (sysop rights), then that person's advanced user rights would fall under the inactivity policy. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 23:26, 16 July 2026 (UTC) :::I agree with Justin. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 00:07, 19 July 2026 (UTC) ::: Yes, I agree with Koavf's suggestion; it resolves Mu301's concerns that Bureaucrat-specific actions may not be used in a 12 month period even though the user is active. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:50, 21 July 2026 (UTC) :::: Koavf's suggestion adequately addresses my concerns. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 22:15, 27 August 2026 (UTC) :: The current draft refers to administrative actions in a broad sense, not bureaucrat-specific actions. So, I think it is reasonable to interpret this as meaning that a bureaucrat would only be considered inactive if they haven't used any administrative action and haven't edited in 12 months. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 06:31, 21 July 2026 (UTC) ::: I concur with that interpretation. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 05:44, 7 October 2026 (UTC) == Process == *For item #2 of process I would recommend requiring both talk page notification and also "email this user." If someone is inactive they might not notice the talk page edit. *For #3 we should make clear that a steward request can only be made after a community discussion has concluded. The draft policy seems to skip over the step of review by the community, how long the review should remain open, and other important due process details. :--[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 22:24, 16 July 2026 (UTC) : For number 3, the discussion should last about the same as the inactivity response timeframe, but I don't think there should be voting, as this is about inactivity. [[User:Codename Noreste|<span style="color: blue">Codename Noreste</span>]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:20, 18 July 2026 (UTC) : In that case, go ahead. I am not against these changes. [[User:Codename Noreste|<span style="color: blue">Codename Noreste</span>]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:22, 18 July 2026 (UTC) :: Pinging @[[User:Mu301|Mu301]] a second time–in case you did not see my response(s). [[User:Codename Noreste|<span style="color: blue">Codename Noreste</span>]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:03, 23 August 2026 (UTC) ==Vested interest== Which admins could be affected by this proposal? I think that we should ping them to solicit input on a proposed policy before finalizing the terms. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 01:43, 28 August 2026 (UTC) : User:Evolution and evolvability. [[User:Codename Noreste|<span style="color: blue">Codename Noreste</span>]] /// [[User talk:Codename Noreste|⭐️✨]] 18:46, 7 October 2026 (UTC) == Approval == Is this policy good enough, or does it need additional changes? I await feedback here. [[User:Codename Noreste|<span style="color: blue">Codename Noreste</span>]] /// [[User talk:Codename Noreste|⭐️✨]] 22:52, 24 October 2026 (UTC) 4nd18nlt7bq9221ge8s26gzv92telgs 0 330075 2842556 2836266 2026-10-25T11:10:07Z Isa Murphy 3105869 2842556 wikitext text/x-wiki {{title|Future orientation and criminal behaviour:<br>How does future orientation influence the risk of criminal activity?}} ==Overview== {{RoundBoxTop|theme=7}} [[File:Exams during Covid-19 - France (50651621763).jpg|thumb|200px|'''Figure 1'''. Exams during Covid-19.]] ; Acar and Kaya (2024) study Amongst 23 adolescent participants, psychological drivers, social and environmental influences, and cognitive and planning capacities were identified as primary categories influencing adolescent future orientation, alongside family, peers, educational access, and cultural norms contributing to the facilitation or hindering of future aspirations (Acar & Kaya, 2024). '''Psychological drivers''': 'hope and optimism, fear of failure, self-identity, motivation and ambition, and emotional regulation' (Acar & Kaya, 2024). '''Social and environmental''': 'parental support, peer influence, school environment, socioeconomic constraints, and community expectations' (Acar & Kaya, 2024). '''Cognitive and planning capacities''': 'goal-setting skills, time perspective, decision-making ability, problem-solving skills, information-seeking behavior [''sic''], and academic self-efficacy' (Acar & Kaya, 2024). {{RoundBoxBottom}} When focusing on adolescents, future orientation is a vital component of their education (Johnson et al., 2014). Conversely, adolescent crime rates have been steadily decreasing (Australian Bureau of Statistics [ABS], 2026), yet they remain a major concern (Hinchcliff et al., 2026; Australian Human Rights Commission [AHRC], 2024; World Health Organisation [WHO], 2026). {{RoundBoxTop|theme=2}} '''Focus questions''' {{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}} '''1.''' Why is Future Orientation Important? '''2.''' Can Future Orientation Mitigate Criminal Behaviour? '''3.''' Are Adolescents More Susceptible to Aversive Effects of Low Future Orientation? '''4.''' Does Criminal Behaviour Depend Upon Future-Oriented Cognition? {{RoundBoxBottom}} = Future Orientation = {{ic|Include at least an introductory paragraph before branching into sub-headings}} ==What is Future Orientation== According to House et al. (2004), future orientation is the tendency for a person to engage in thoughts, actions, and the delaying of immediate gratification in favour of future goals. Referring to adolescents, many disciplines have ratified the importance of future-oriented behaviour in promoting positive health and development (Johnson et al., 2016). Trommsdorff (9179) proposed that future orientation included eight dimensions: extension, detail, domain, affect, motivation, control, sequence of events, and number of cognitions. {{center top}} {| border=1 cellspacing=0 cellpadding=6 |- ! Dimension ! Definition |- | extension | bar |- | detail | bar |- | domain | bar |- | affect | bar |- | motivation | bar |- | control | bar |- | sequence of events | bar |- | number of cognitions | far |} {{center bottom}} A study by Beal (2011) revealed that Trommsdorff’s (1979) conceptualisation indicates that different influences can be indicators of future orientation and the eight dimensions. Findings consistently supported previous research that future orientation is an important predictor of adolescent well-being and career adaptability and revealed that ‘stronger future orientation and future-focused time perspectives are consistently associated with higher adolescent well-being, health-promoting habits (e.g., physical activity), and lower engagement in health-risk behaviours (e.g., substance use, risky sexual behaviour)’ (Kovács et al., 2026). Lindstrom Johnson et al. (2014) examines future orientation as a mediator or moderator of problem behaviour, determining that while conclusions should be interpreted cautiously, findings indicated future-oriented thinking as a mediator can be expressed as reasoning, but not the sole, for behavioural outcomes (Weems, 2025). Kovács et al. (2026) conducted a systematic review on the effects and different forms future orientation has on adolescents, focusing on future time perspective and future orientation. Findings on future time perspective revealed an interesting trend: that positivity-endorsing programmes on past behaviours, future-oriented planning, actively developing time perspective, and contextual health behaviour–focused prevention programmes greatly improve adolescent health and well-being (Kovács et al., 2026; Kuan & Zhang, 2022). Thus, the following article will inform adolescent future orientation as a preventative measure for criminal behaviour before it occurs. It should be noted that additional factors, such as drug-oriented perspectives, can explain insignificant positive effects in future-oriented programmes. Considering the major transitional stage that is adolescence (Khurana et al., 2015), the enticing and control-eliciting nature of drugs and alcohol, and social and environmental influences on various behaviours (Barnett et al., 2013; Zhang et al., 2021), potential negative effects resulting from illicit drug and alcohol use are plausible (Finan et al., 2022; Peters et al., 2005; Jones et al., 2021) and should be recognised when discussing and assessing adolescent future-oriented thinking (Geier et al., 2013; Stoddard et al., 2018). ==Influences on Future Orientation== Multidisciplinary findings have indicated that a variety of influences contribute to future orientation. Much psychological research has focused on adolescents and young adults, so findings are skewed, though still extensive and reliable. ===The BioPsychoSocial Model=== In 1977, George Engel proposed an extension to the biomedical model that incorporated the psychological and social factors influencing health and wellbeing, known as the BioPsychoSocial Model (BPSM). Vögele (2015) defines BPSM as ‘a general model positing that biological, psychological (which includes thoughts, emotions, and behaviors [sic]), and social (e.g., socioeconomical, socioenvironmental, and cultural) factors, all play a significant role in health. …[and] are best understood in terms of a combination…rather than purely in biological terms.’ It is important to recognise that biological, psychological, and social influences are individual yet equally important additions to health and wellbeing with a variety of crossover (Dfarhud et al., 2014; Chu et al., 2024; Candeias et al., 2024; Kahn et al., 2024). For example, Subramanyam et al. (2024) studied the effects of neurodevelopment, epigenetics, parenting, communication skills, career-focused future orientation, and the environment on child and adolescent wellbeing, with findings indicating each component held a myriad of interplaying causes and effects. Regarding future orientation, multidisciplinary findings have indicated that a variety of influences contribute to future orientation. It is important to note that much of the psychological research has targeted adolescents and young adults, and therefore findings are skewed, yet extensive and reliable. ====Psychological Influences==== McCue et al. (2019) researched and measured the effects of future orientation on adolescent decision-making. Previously, connectedness to future-self and episodic future thinking demonstrates future-oriented cognition, and while their correlation is undetermined, individual measures have shown significant effects in minimising hazardous decision-making in both adolescents and adults (Grekin et al., 2025; Goger et al., 2024; Arora, 2023). Future-connectedness is defined by behaviours and cognition involving the active imagination, verbal and physical communication, and other related engagements towards a person’s future in all forms. Disconnect from the future self has been shown to increase irrational, short-sighted, and risky behaviours deemed beneficial in the present (Hershfield, 2011). Episodic future thinking is defined as the mental stimulation experienced towards possible future occurrences, and supports future-orientation and highlights the importance of delaying rewards and countering impulsive behaviours (Gilbert & Wilson, 2007; Benoit et al., 2011; Bar, 2009). This is particularly important as future-oriented thinking has historically shown its profound necessity amongst adolescents (Nurmi, 1991), and a lower predilection for impulsive and risky behaviour in favour of future rewards (Steinberg, 2004; Madden & Bickel, 2012). McCue et al. (2019) found a significant positive correlation between future-connectedness and episodic future thinking, claiming to be one of, if not the first, study to identify a beneficial relationship and independent effects within adolescents. Giollabhui et al. (2018) found that increased future-oriented behaviours and cognition contribute to reduced adolescent hopelessness. During adolescence, hopelessness is categorised by adamant negative future expectations and vulnerability (Casey et al., 2008; Lester, 2015). The severity of hopelessness in adolescence, and its relation to increased engagement in violent and/or high-risk behaviours, severe depression and anxiety, and unsafe sexual conduct (Becker-Weidman et al., 2009; Bolland, 2003; James et al., 2017; Giollabhui et al., 2018), indicates the importance of minimising adolescent hopelessness. Suicide has been a primary cause of death amongst adolescents for many years, skyrocketing after COVID-19 (Kim et al., 2024), and highlighting the worrisome trend. Considering the relationship between hopelessness, future-orientation, and adolescent depression and suicidal behaviours, increasing positive cognitions actively declines time-limited rates of hopelessness across confounds, including socio-economic status and stressors (Giollabhui et al., 2018; Faura-Garcia et al., 2024). ====Biological Influences==== Biological aspects of future orientation are currently under-researched, likely due to difficulty in quantifying results. Gan et al. (2024) investigated effects of the Colony-Stimulating Factor 3 Receptor (CSF3R) polymorphism and early stress on future orientation amongst 14,675 Chinese participants. Participants self-reported personal occurrences of maltreatment throughout childhood. Results indicated the CSF3R gene, which is closely related to chronic psychological stress responses, greatly interacted with early trauma’s influence on future orientation, specifically as levels of maltreatment increased (Gan et al., 2024). Of great importance to this chapter, childhood maltreatment demonstrated higher levels of stress in comparison to other stressors. This can either deplete opportunities to grow or encourage stress-related improvement in response to triggering situations (Gan et al., 2024), particularly for those with certain genotypes of the CSF3R (Drury et al., 2010). Zheng et al. (2022) explored the heritability of future orientation, identifying that genetic influence and mutual enviornments significantly affected future orientation development amongst adolescents. Findings revealed a negative effect between age and genetic influence, and a positive effect between age and individual environments on future orientation, as well as no effect between male and females (Zheng et al., 2022). Conversely, Greene et al. (2004) found that historical gender roles and expectations influenced future orientation, with females maintaining interpersonal goals, whereas males had further extension of goals. Importantly, the astronomical effects of environments on adolescent future orientation development and health and wellbeing, unrestricted from socio-cultural contexts, such as race, gender, and religion (Greene et al., 2004). ====Social And Environmental Influences==== Social influence can be argued as the largest influence on future orientation due to the proximity of environmental situations throughout childhood and adolescence (Laursen & Faur, 2022). Research has shown socioeconomic determinants positively correlated to the future orientation of adolescents (Nurmi, 1991), particularly higher parental and personal education (Borowsky et al., 2009; Nguyen et al., 2012), whereas neighbourhood poverty and racial minorities demonstrate weaker future-oriented thinking (Kerpelman et al., 2004; Guthrie et al., 2009). Johnson et al. (2016) explored how parenting style, including acceptance and involvement, attachment security, and autonomy granting, has consistently demonstrated positive effects on adolescent future-oriented thinking. Interestingly, Kai and Tienda (1998) found that adolescent future orientation as represented through college aspirations did not rely on race as it did for other influencing factors. Chen and Vazsonyi (2013) conducted a multilevel study on the effects of schooling on future-oriented thought processes amongst nine thousand 9th to 12th graders. Findings showed adolescent problem behaviours negatively impacted future-oriented thinking, especially those in larger cohorts and schools with higher positivity towards future orientation (Chen & Vazsonyi, 2013). Interestingly, students with a low sense of future orientation experienced opposite effects than those with problem behaviours, displaying a significant effect from positive future orientation attitudes at school (Chen & Vazsonyi, 2013). Rydell and Brocki (2024) found similar effects, with problem behaviours and inattention around mid-adolescence negatively affecting future orientation. However, high levels of hyperactivity and positive social interactions and relationships had compensatory properties, improving future orientation. Continuing research on adolescence, Skinner et al. (2022) investigated the effects of parental authority on future influence throughout COVID-19. Parental control is known to have an influence on adolescent wellbeing and future orientation (Wang et al., 2024; Zhu et al., 2026; Roman et al., 2015). For instance, psychological control is characterised by ‘intrusive and manipulative behaviors [sic] aimed at children or adolescents’ thoughts and feelings, through which adults exert power by controlling their children’s psychological world’ (Pérez et al., 2021), and can hinder adolescent development and wellbeing (Mandara & Pikes, 2008; Xu and Ren, 2025), and is connected to the development of depressive symptoms (Skinner et al., 2022). Data collected from adolescents and their parents across three years from nine countries, accounting for confounding factors, demonstrated the importance of positive normative standards for future orientation through parent-child interactions, and an educated, targeted approach to overcome adverse effects of childhood (Merrick et al., 2017; Kerpelman et al. 2008), such as an ecological systems approach (Rosanbalm et al. 2020). Wang et al. (2024) reiterated these points, finding that parental control significantly influences future orientation and positively affects perceived significance of life, future orientation, and personal growth initiative. Interestingly, psychological control had the only negative effect on future orientation and no effect on perceived significance of life. Given these findings, the social influences on future orientation are of the utmost importance and should be diligently recognised for positive effects to occur (Rydell et al., 2024; Crespo et al., 2013), particularly in adolescents and the objective to minimise bad behaviour (Chen & Vazsonyi, 2011; Voisin et al., 2020) and decrease hopelessness (Giollabhui et al., 2018). It should be noted that social influences on adolescent health and wellbeing are connected to psychological wellbeing and perceived significance of life, and therefore can be argued as the component of the BPSM that takes priority, but not as most important (Long et al., 2021; Webster et al., 2021; World Health Organization [WHO], 2025). ===All Three Components=== The BPSM aims to put the ‘nature vs nurture’ debate to rest by highlighting primary considerations as equal contributors to behaviour. Within the ‘Nature’ conversation, particularly neuroscience, researchers dictate that ‘the capacity of the brain to change its connections. …must be the consequence of having molecular mechanisms that mediate those changes’. (United States Institute of Medicine [USIM]. Beauchaine et al. (2009) provided ten reasons to consider biological processes in psychological prevention and intervention in research, including biomarkers and endophenotypes in identifying priority prevention methods, how biological vulnerabilities moderate environmental experiences and differences, and interactions between environmental and neural systems sensitivity across lifespans. Similarly, Marder and Miller (2024) discuss how biological psychology is the focal point of psychophysiology, investigating the effects of biological markers in behaviour, accompanied by the influences of social trends creates the foundation of individual disposition. Eriksson et al. (2025) corroborated the multilevel consequences on behaviour and wellbeing, stating ‘It is essential to move beyond focusing solely on individual psychological and cognitive factors to an understanding of the complex processes involved in health behaviour change.’ Findings illustrated variance between health behaviours for different health identities amongst middle-aged men and women, guiding future socio-ecological interventions promoting healthier behaviours (Eriksson et al., 2025). Regarding the BPSM within future orientation and criminal behaviour, each element attracts differing levels of authority from many sources. For example, the majority of research in future orientation revolves around environmental influences (Wittmann & Sircova, 2018), compared to psychological and biological components. Conversely, criminal behaviour prioritises biological and psychological influences (Miller, 2014; Focquaert, 2019). ===Why is this particularly important for adolescents?=== Kovács et al. (2026) detail that future orientation is “not a singular construct but a multifaceted phenomenon operationalized [sic] through cognitive, affective, and contextual lenses”. This is particularly important for adolescents, as biological, social, and psychological factors interact dynamically, with each influencing the others and shaping development (Lindstrom Johnson et al., 2014). The BPSM is particularly relevant as stronger future orientation is associated with adolescent wellbeing and healthier behaviours, including greater engagement in health-promoting activities and reduced risk-taking, although these relationships vary across contexts (Kovács et al., 2026). Future orientation may support these outcomes by strengthening adolescents’ capacity to establish meaningful goals, anticipate the consequences of their decisions, and regulate present behaviour in pursuit of longer-term outcomes (Bandura, 2001; Nurmi, 1991). However, the extent to which future-oriented thinking translates into behaviour depends on individual motivation and adolescents’ perceived opportunities, social relationships, and environmental circumstances (Lindstrom Johnson et al., 2014). Future orientation should not be considered as universally protective, as an intense focus on future outcomes may also be associated with anxiety or heightened pressure to achieve in certain contexts (Kovács et al., 2026). This indicates the importance of the BPSM in future orientation, implying that no singular factor is the sole cause of development, and interventions should aim at promoting adolescent wellbeing beyond encouraging long-term goal setting to include the development of self-regulatory skills and supportive environments that enable adolescents to pursue their aspirations (Nurmi, 1991; Lindstrom Johnson et al., 2014). === Future Orientation’s Relevance to Criminal Activity === Relevant to criminal behaviour, higher levels of future-oriented cognition indicate further consideration of long-term consequences of offending (Kwon et al., 2025); in contrast, limited future orientation correlates with increased criminal behaviour by encouraging immediate gratification and reducing consideration of delayed consequences (Kübel et al., 2023). Clinkinbeard (2014) found that high future orientation and self-control predict adolescent criminal behaviour; particularly, higher future-oriented cognition mitigates negative effects of low self-control and lessens impulsivity, risky behaviour engagement, criminal cognition, and adolescent crime rates (Khetarpal et al., 2021; Squillaro & Bixter, 2025; Simmons et al., 2019). Importantly, Walters (2019) identified a gradual decrease in criminal behaviour with maturation, associated with increased future orientation and decreased adolescent hopelessness. Similarly, Wood et al. (2017) recognised that cognitive immaturity relates to engagement in aversive behaviours, and thereby suffering future negative life effects; promoting future orientation and positive childhood effects can mitigate these negative life effects and promote sustainability (Kübel et al., 2023; Cauffman et al., 2023; Mueller et al., 2023). Given the findings on future orientation, it is now important to consider the role it plays in criminal behaviour, and therefore how research can facilitate future orientation to prevent, mitigate or rehabilitate criminal behaviour. ;Quiz Choose your answers and click "Submit": <quiz display=simple> {A variety of influences contribute to future orientation: |type="(+)"} + True - False {Future-connectedness is defined by behaviours and cognition involving the engagement towards a person’s future: |type="(+)"} + True - False {Disconnect from the future self has been shown to have no aversive risks: |type="(+)"} + True - False {Hopelessness is a persistent and serious outcome of adolescence: |type="(+)"} + True - False {Future orientation is ''the'' primary indicator for engagement with criminal behaviour: |type="(-)"} + True - False </quiz> ==Key Points== * Future orientation is the tendency for a person to engage in thoughts and actions in favour of future goals. *Future-connectedness is defined by behaviours and cognition involving engagement towards a person’s future in all forms. *The severity of hopelessness in adolescence, and its high relation to adverse behaviours, indicates the importance of minimising adolescent hopelessness. * Higher levels of future-oriented cognition indicate further consideration of long-term consequences of offending. * Limited future orientation correlates with increased criminal behaviour. =Criminal Behaviour= Dent and Ward (2022) conceptualised criminal behaviour as ‘an action or set of actions... understood by the actor to be adaptive in relation to their functional needs, but...make it more difficult to function adaptively...to other sociocultural needs.’ Criminal behaviour can include minor crimes, such as ‘theft or traffic violations’, and major crimes, such as ‘physical abuse, sexual offense [sic], or murder’ (Prent et al., 2023), and is considered a ‘major social problem with complex causes... [and involves] a myriad of environmental, social, and psychological factors are associated with increased risk of convictions for this type of criminality’ (Prent et al., 2023; Levitt, 2012; Swaab & Meynen, 2023). ==Influences on Criminal Behaviour== Reiterating findings from Prent et al. (2023), criminal behaviour is shaped by many complex, interconnected facets, including social, psychological, biological, and environmental factors, each with influential components such as geographical location and socio-economic status. Importantly, future orientation has shown positive effects in minimising negative behaviour across different contributing factors (Giollabhui et al., 2018). ==Psychological Influences== In media and history, psychology has often been treated as the primary explanation for criminal acts; however, crime is not limited to a single discipline (South Australia News, 2026). While this is well established, it is often a misconception that one factor is more influential than the other. A longstanding theory explaining human behaviour, cognition, and reasoning is Nature v Nurture. Nurture, as a psychological construct, argues that a person’s circumstances stem from upbringing and environment rather than other factors. Hadsell (2026) stated that the ‘connection between parents and children has two pathways: biological, covering both genes and hormones, and environmental.’ ==Biological Influences== When discussing the Nature concept, it is theorised that an individual's biological makeup is the sole predictor of behaviour (Alper, 1995). Baschetti (2008) argued that genetic evidence proved Nature over Nurture. Drawing on Charles Darwin’s theories of evolution and biology, current and overwhelming research indicates that criminality is heritable, yet persistent (Baschetti, 2008; Pizzi, 2004). It should be noted that Nature can refer to the genetic contribution within mental illness. Rao (2007) argued that all humans have a ‘dark side that loves crime and violence... [while] we may all deny it, but the contrary is true...regardless of our age, caste, social status, region, religion or education.’ [''APA style: Direct quotes should use double quotation marks and include page numbers''] Relevant to crime rates, it can be reasoned that modern-day society is the ‘most violent culture in history in the number of crimes and...the nature of brutality.’ Ling et al. (2019) explored how psychophysiology, brain mechanisms, and genetics stimulate criminal behaviour. Particularly, disrupted prefrontal cortex-amygdala connectivity and changes in autonomic functioning damaging the hypothalamic-pituitary-adrenal axis correlate with increased criminal and anti-social tendencies (Gillespie et al., 2018; Critchley, 2005; Brzozowski & Mitchell, 2018) [''APA style: Arrange multiple citations in alphabetical order'']. Baschetti (2008) signified the importance of systematic changes acknowledging the existence of a biological influence to encourage appropriate reforms and minimise negative outcomes. ==Relationship Between Criminal Behaviour and Future Orientation== {{expand}} ==Does Future Orientation Influence the Risk of Criminal Activity?== Future orientation is an incredibly [''Avoid overly emotive language''] important skill, particularly for adolescents or those experiencing hopelessness. A lack of future-oriented cognition can exacerbate impulsive and risky behaviour and may lead to criminal behaviour (Squillaro & Bixter, 2025; Mueller et al., 2023). Given adolescents' vulnerability to criminal behaviours (Bui & Deakin, 2021) and hopelessness (Bolland, 2003; Tonkus et al., 2022), encouraging future orientation is highly relevant. Reiterating that future orientation positively relates to curbing adolescent criminal activity and juvenile incarceration (Walters, 2019; Simmons et al., 2019) reinforces future orientation as an imperative protective factor across the lifespan (Yang et al., 2026; Clinkinbeard, 2014). ==Book Chapter Direction== * Environmental influences in criminal activity. *Social influences in criminal activity. *Understanding the relationship between future orientation and criminal activity. *Protective factors within future orientation *Risk factors within future orientation *Future research directions ==Conclusion== ''''' Include key sentence on the importance of understanding future orientation in criminal behaviour''.''' Currently, there is substantial research into the effects and relationships of future orientation and adolescent criminal activity. However, conclusive findings on the relevance of using future orientation to mitigate criminal behaviour remain limited. Given the strong positive correlations between future-oriented thinking and reduced disturbances in life quality, it would be unwise to ignore how future orientation can help prevent rising crime rates among adolescents ''and'' adults. ==See also== {{expand}} ==References== {{Hanging indent|1= Alper, J. S. (1995). Biological influences on criminal behaviour: How good is the evidence? *BMJ, 310*(6975), 272–273. https://doi.org/10.1136/bmj.310.6975.272 Arora, S. (2023). *“It’s a future-me problem”: How future-self continuity and message framing affect climate change risk perceptions* [Doctoral dissertation, University of Guelph]. Atrium. https://atrium.lib.uoguelph.ca/items/4d97286e-5635-45b6-a1c3-e9be464b1e7c Bar, M. (2009). The proactive brain: Memory for predictions. *Philosophical Transactions of the Royal Society B: Biological Sciences, 364*(1521), 1235–1243. https://doi.org/10.1098/rstb.2008.0310 Baschetti, R. (2008). Genetic evidence that Darwin was right about criminality: Nature, not nurture. *Medical Hypotheses, 70*(6), 1092–1102. https://doi.org/10.1016/j.mehy.2008.01.005 Becker-Weidman, E. G., Jacobs, R. H., Reinecke, M. A., Silva, S. G., & March, J. S. (2010). Social problem-solving among adolescents treated for depression. *Behaviour Research and Therapy, 48*(1), 11–18. https://doi.org/10.1016/j.brat.2009.08.006 Benoit, R. G., Gilbert, S. J., & Burgess, P. W. (2011). A neural mechanism mediating the impact of episodic prospection on farsighted decisions. *The Journal of Neuroscience, 31*(18), 6771–6779. https://doi.org/10.1523/JNEUROSCI.6559-10.2011 Bickel, W. K., Athamneh, L. N., Basso, J. C., Mellis, A. M., DeHart, W. B., Craft, W. H., & Pope, D. (2019). Excessive discounting of delayed reinforcers as a trans-disease process: Update on the state of the science. *Current Opinion in Psychology, 30*, 59–64. https://doi.org/10.1016/j.copsyc.2019.01.005 Bolland, J. M. (2003). Hopelessness and risk behaviour among adolescents living in high-poverty inner-city neighbourhoods. *Journal of Adolescence, 26*, 145–158. https://doi.org/10.1016/S0140-1971(02)00136-7 Bui, L., & Deakin, J. (2021). What we talk about when we talk about vulnerability and youth crime: A narrative review. *Aggression and Violent Behavior, 58*, 101605. https://doi.org/10.1016/j.avb.2021.101605 Cauffman, E., Gillespie, M. L., Beardslee, J., Davis, F., Hernandez, M., & Williams, T. (2023). Adolescent contact, lasting impact? Lessons learned from two longitudinal studies spanning 20 years of developmental science research with justice-system-involved youths. *Psychological Science in the Public Interest, 24*(3), 133–161. https://doi.org/10.1177/15291006231205173 Chadwick, R. (n.d.). Genetics and crime. In M. Levitt (Ed.), *Encyclopedia of applied ethics* (2nd ed., pp. 462–469). Academic Press. https://doi.org/10.1016/B978-0-12-373932-2.000375 Clinkinbeard, S. S. (2014). What lies ahead: An exploration of future orientation, self-control, and delinquency. *Criminal Justice Review, 39*(1), 19–36. https://doi.org/10.1177/0734016813501193 Critchley, H. D. (2005). Neural mechanisms of autonomic, affective, and cognitive integration. *The Journal of Comparative Neurology, 493*(1), 154–166. https://doi.org/10.1002/cne.20749 Dent, H., & Ward, T. (2022). Emotion and correctional practice: Theoretical foundations of an enactive practice framework. *Aggression and Violent Behavior, 63*, 101672. https://doi.org/10.1016/j.avb.2021.101672 Faura-Garcia, J., Calvete, E., & Orue, I. (2024). Longitudinal associations between nonsuicidal self-injury, depressive symptoms, hopelessness, and emotional dysregulation in adolescents. *Archives of Suicide Research, 28*(3), 800–814. https://doi.org/10.1080/13811118.2023.2237075 Gillespie, S. M., Brzozowski, A., & Mitchell, I. J. (2018). Self-regulation and aggressive antisocial behaviour: Insights from amygdala-prefrontal and heart-brain interactions. *Psychology, Crime & Law, 24*(3), 243–257. https://doi.org/10.1080/1068316X.2017.1414816 Gilbert, D. T., & Wilson, T. D. (2007). Prospection: Experiencing the future. *Science, 317*(5843), 1351–1354. https://doi.org/10.1126/science.1144161 Goger, P., Nam, R. J., Lowry, N., Bell, K. A., Parvez, N., Pollak, O. H., Robinaugh, D. J., Schacter, D. L., & Cha, C. B. (2024). Testing the modifiability of episodic future thinking and episodic memory among suicidal and nonsuicidal adolescents. *JCPP Advances, 4*(3), e12236. https://doi.org/10.1002/jcv2.12236 Grekin, E. R., Thomas, H. A., Souweidane, M. A., & Stidham, J. L. (2025). The effects of future self-continuity interventions on behavioral outcomes in adults: A systematic review of the literature. *Personality Science, 6*. https://doi.org/10.1177/27000710251391610 Hadsell, L. (2026). Parents: Nature and nurture. In *Social economies*. Palgrave Macmillan. https://doi.org/10.1007/978-3-032-21916-9_18 Hershfield, H. E. (2011). Future self-continuity: How conceptions of the future self transform intertemporal choice. *Annals of the New York Academy of Sciences, 1235*, 30–43. https://doi.org/10.1111/j.1749-6632.2011.06201.x James, S., Reddy, S. P., Ellahebokus, A., Sewpaul, R., & Naidoo, P. (2017). The association between adolescent risk behaviours and feelings of sadness or hopelessness: A cross-sectional survey of South African secondary school learners. *Psychology, Health & Medicine, 22*(7), 778–789. https://doi.org/10.1080/13548506.2017.1300669 Johnson, S. R., Blum, R. W., & Cheng, T. L. (2014). Future orientation: A construct with implications for adolescent health and wellbeing. *International Journal of Adolescent Medicine and Health, 26*(4), 459–468. https://doi.org/10.1515/ijamh-2013-0333 Khetarpal, S. K., Szoko, N., Ragavan, M. I., & Culyba, A. J. (2021). Future orientation as a cross-cutting protective factor against multiple forms of violence. *The Journal of Pediatrics, 235*, 288–291. https://doi.org/10.1016/j.jpeds.2021.05.019 Kübel, L. S., Deitzer, R. J., Frankenhuis, E. W., Ribeaud, D., Eisner, P. M., & van Gelder, J.-L. (2023). The shortsighted victim: Short-term mindsets mediate the link between victimization and later offending. *Journal of Criminal Justice, 86*, 102062. https://doi.org/10.1016/j.jcrimjus.2023.102062 Kwon, K. Y., Sutin, A. R., & Kim, J. (2025). Juvenile delinquency and cognitive function in adulthood: Differentiating violent and nonviolent behaviors and exploring multiple mechanisms. *Journal of Research on Adolescence, 35*(4), e70095. https://doi.org/10.1111/jora.70095 Lester, D. (2015). Hopelessness in adolescents. *Journal of Affective Disorders, 173*, 221–225. https://doi.org/10.1016/j.jad.2014.10.048 Ling, S., Umbach, R., & Raine, A. (2019). Biological explanations of criminal behavior. *Psychology, Crime & Law, 25*(6), 626–640. https://doi.org/10.1080/1068316X.2019.1572753 Mac Giollabhui, N., Hamilton, J. L., Nielsen, J., Connolly, S. L., Stange, J. P., Varga, S., Burdette, E., Olino, T. M., Abramson, L. Y., & Alloy, L. B. (2018). Negative cognitive style interacts with negative life events to predict first onset of a major depressive episode in adolescence via hopelessness. *Journal of Abnormal Psychology, 127*(1), 1–11. https://doi.org/10.1037/abn0000301 McCue, R., McCormack, T., McElnay, J., Alto, A., & Feeney, A. (2019). The future and me: Imagining the future and the future self in adolescent decision making. *Cognitive Development, 50*, 142–156. https://doi.org/10.1016/j.cogdev.2019.04.001 Mueller, K. C., Cavitt, J., & Carey, M. T. (2023). Future orientation in justice-involved youth: The effects of adverse and positive childhood experiences, and being dual-status. *Child Abuse & Neglect, 141*, 106230. https://doi.org/10.1016/j.chiabu.2023.106230 Nurmi, E. J. (1991). How do adolescents see their future? A review of the development of future orientation and planning. *Developmental Review, 11*(1), 1–59. https://doi.org/10.1016/0273-2297(91)90002-6 Peterson, M. F. (2004). [Review of the book *Culture, leadership and organizations: The GLOBE study of 62 societies*, by R. J. House, M. Javidan, P. W. Dorfman, V. Gupta, & P. J. Hanges]. *Administrative Science Quarterly, 49*(4), 641–647. https://www.jstor.org/stable/4131493 Pizzi, W. J. (2004). Nature via nurture: Genes, experience, and what makes us human. *Journal of Undergraduate Neuroscience Education, 2*(2), R10–R11. Prent, N., Jonker, A. F., Schouws, N. T. M. S., & Jonker, C. (2023). The risk of criminal behavior in the elderly and patients with neurodegenerative disease. In H. Swaab & G. Meynen (Eds.), *Handbook of clinical neurology* (Vol. 197, pp. 181–196). Elsevier. https://doi.org/10.1016/B978-0-12-821375-9.00004-9 Sathyanarayana Rao, T. S. (2007). Psychiatrist and the science of criminology: Sociological, psychological and psychiatric analysis of the dark side. *Indian Journal of Psychiatry, 49*(1), 3–5. https://doi.org/10.4103/0019-5545.31511 Simmons, C., Rowan, Z., Knowles, A., Steinberg, L., Frick, P. J., & Cauffman, E. (2019). A life history approach to understanding juvenile offending and aggression. *Aggression and Violent Behavior, 49*, 101317. https://doi.org/10.1016/j.avb.2019.07.012 South Australia News. (2026, April 25). *Psychology of a criminal: Why do people commit crimes?* https://sanews.in/psychology-of-a-criminal/ Squillaro, D., & Bixter, M. T. (2025). The relationship between four indicators of future time orientation, criminal thinking style, and illegal behaviors. *Journal of Forensic Sciences, 70*(1), 200–214. https://doi.org/10.1111/1556-4029.15640 Steinberg, L. (2004). Risk taking in adolescence: What changes, and why? *Annals of the New York Academy of Sciences, 1021*, 51–58. https://doi.org/10.1196/annals.1308.005 Swaab, H., & Meynen, G. (2023). Introduction: On brain and crime. In H. Swaab & G. Meynen (Eds.), *Handbook of clinical neurology* (Vol. 197, pp. 3–9). Elsevier. https://doi.org/10.1016/B978-0-12-821375-9.00018-9 Turton, H., Berry, K., Danquah, A., & Pratt, D. (2021). The relationship between emotion dysregulation and suicide ideation and behaviour: A systematic review. *Journal of Affective Disorders Reports, 5*, 100136. https://doi.org/10.1016/j.jadr.2021.100136 Walters, G. D. (2019). Maturing out of delinquency: Unpacking the effects of identity achievement and future orientation on crime desistance. *Self and Identity, 18*(3), 267–283. https://doi.org/10.1080/15298868.2018.1439848 Wager, T. D., Waugh, C. E., Lindquist, M., Noll, D. C., Fredrickson, B. L., & Taylor, S. F. (2009). Brain mediators of cardiovascular responses to social threat: Part I: Reciprocal dorsal and ventral sub-regions of the medial prefrontal cortex and heart-rate reactivity. *NeuroImage, 47*(3), 821–835. https://doi.org/10.1016/j.neuroimage.2009.05.043 Wood, D., Crapnell, T., Lau, L., et al. (2017). Emerging adulthood as a critical stage in the life course. In N. Halfon, C. B. Forrest, R. M. Lerner, et al. (Eds.), *Handbook of life course health development*. Springer. https://www.ncbi.nlm.nih.gov/books/NBK543712/ Yang, E. S., Kim, S., Oh, S.-K., Kang, H., & Chung, I.-J. (2026). Disrupting the link between exposure to violence and adolescent delinquency: The protective role of expectations about the future. *Journal of Child and Family Studies*. https://doi.org/10.1007/s10826-026-03311-3 }} ==External links== {{expand}} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Cognitive]] [[Category:Motivation and emotion/Book/Forensic]] scyaj7te6ui5v1b4z4j6xjv17wxwsyo 2842557 2842556 2026-10-25T11:10:30Z Isa Murphy 3105869 /* Social And Environmental Influences */ 2842557 wikitext text/x-wiki {{title|Future orientation and criminal behaviour:<br>How does future orientation influence the risk of criminal activity?}} ==Overview== {{RoundBoxTop|theme=7}} [[File:Exams during Covid-19 - France (50651621763).jpg|thumb|200px|'''Figure 1'''. Exams during Covid-19.]] ; Acar and Kaya (2024) study Amongst 23 adolescent participants, psychological drivers, social and environmental influences, and cognitive and planning capacities were identified as primary categories influencing adolescent future orientation, alongside family, peers, educational access, and cultural norms contributing to the facilitation or hindering of future aspirations (Acar & Kaya, 2024). '''Psychological drivers''': 'hope and optimism, fear of failure, self-identity, motivation and ambition, and emotional regulation' (Acar & Kaya, 2024). '''Social and environmental''': 'parental support, peer influence, school environment, socioeconomic constraints, and community expectations' (Acar & Kaya, 2024). '''Cognitive and planning capacities''': 'goal-setting skills, time perspective, decision-making ability, problem-solving skills, information-seeking behavior [''sic''], and academic self-efficacy' (Acar & Kaya, 2024). {{RoundBoxBottom}} When focusing on adolescents, future orientation is a vital component of their education (Johnson et al., 2014). Conversely, adolescent crime rates have been steadily decreasing (Australian Bureau of Statistics [ABS], 2026), yet they remain a major concern (Hinchcliff et al., 2026; Australian Human Rights Commission [AHRC], 2024; World Health Organisation [WHO], 2026). {{RoundBoxTop|theme=2}} '''Focus questions''' {{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}} '''1.''' Why is Future Orientation Important? '''2.''' Can Future Orientation Mitigate Criminal Behaviour? '''3.''' Are Adolescents More Susceptible to Aversive Effects of Low Future Orientation? '''4.''' Does Criminal Behaviour Depend Upon Future-Oriented Cognition? {{RoundBoxBottom}} = Future Orientation = {{ic|Include at least an introductory paragraph before branching into sub-headings}} ==What is Future Orientation== According to House et al. (2004), future orientation is the tendency for a person to engage in thoughts, actions, and the delaying of immediate gratification in favour of future goals. Referring to adolescents, many disciplines have ratified the importance of future-oriented behaviour in promoting positive health and development (Johnson et al., 2016). Trommsdorff (9179) proposed that future orientation included eight dimensions: extension, detail, domain, affect, motivation, control, sequence of events, and number of cognitions. {{center top}} {| border=1 cellspacing=0 cellpadding=6 |- ! Dimension ! Definition |- | extension | bar |- | detail | bar |- | domain | bar |- | affect | bar |- | motivation | bar |- | control | bar |- | sequence of events | bar |- | number of cognitions | far |} {{center bottom}} A study by Beal (2011) revealed that Trommsdorff’s (1979) conceptualisation indicates that different influences can be indicators of future orientation and the eight dimensions. Findings consistently supported previous research that future orientation is an important predictor of adolescent well-being and career adaptability and revealed that ‘stronger future orientation and future-focused time perspectives are consistently associated with higher adolescent well-being, health-promoting habits (e.g., physical activity), and lower engagement in health-risk behaviours (e.g., substance use, risky sexual behaviour)’ (Kovács et al., 2026). Lindstrom Johnson et al. (2014) examines future orientation as a mediator or moderator of problem behaviour, determining that while conclusions should be interpreted cautiously, findings indicated future-oriented thinking as a mediator can be expressed as reasoning, but not the sole, for behavioural outcomes (Weems, 2025). Kovács et al. (2026) conducted a systematic review on the effects and different forms future orientation has on adolescents, focusing on future time perspective and future orientation. Findings on future time perspective revealed an interesting trend: that positivity-endorsing programmes on past behaviours, future-oriented planning, actively developing time perspective, and contextual health behaviour–focused prevention programmes greatly improve adolescent health and well-being (Kovács et al., 2026; Kuan & Zhang, 2022). Thus, the following article will inform adolescent future orientation as a preventative measure for criminal behaviour before it occurs. It should be noted that additional factors, such as drug-oriented perspectives, can explain insignificant positive effects in future-oriented programmes. Considering the major transitional stage that is adolescence (Khurana et al., 2015), the enticing and control-eliciting nature of drugs and alcohol, and social and environmental influences on various behaviours (Barnett et al., 2013; Zhang et al., 2021), potential negative effects resulting from illicit drug and alcohol use are plausible (Finan et al., 2022; Peters et al., 2005; Jones et al., 2021) and should be recognised when discussing and assessing adolescent future-oriented thinking (Geier et al., 2013; Stoddard et al., 2018). ==Influences on Future Orientation== Multidisciplinary findings have indicated that a variety of influences contribute to future orientation. Much psychological research has focused on adolescents and young adults, so findings are skewed, though still extensive and reliable. ===The BioPsychoSocial Model=== In 1977, George Engel proposed an extension to the biomedical model that incorporated the psychological and social factors influencing health and wellbeing, known as the BioPsychoSocial Model (BPSM). Vögele (2015) defines BPSM as ‘a general model positing that biological, psychological (which includes thoughts, emotions, and behaviors [sic]), and social (e.g., socioeconomical, socioenvironmental, and cultural) factors, all play a significant role in health. …[and] are best understood in terms of a combination…rather than purely in biological terms.’ It is important to recognise that biological, psychological, and social influences are individual yet equally important additions to health and wellbeing with a variety of crossover (Dfarhud et al., 2014; Chu et al., 2024; Candeias et al., 2024; Kahn et al., 2024). For example, Subramanyam et al. (2024) studied the effects of neurodevelopment, epigenetics, parenting, communication skills, career-focused future orientation, and the environment on child and adolescent wellbeing, with findings indicating each component held a myriad of interplaying causes and effects. Regarding future orientation, multidisciplinary findings have indicated that a variety of influences contribute to future orientation. It is important to note that much of the psychological research has targeted adolescents and young adults, and therefore findings are skewed, yet extensive and reliable. ====Psychological Influences==== McCue et al. (2019) researched and measured the effects of future orientation on adolescent decision-making. Previously, connectedness to future-self and episodic future thinking demonstrates future-oriented cognition, and while their correlation is undetermined, individual measures have shown significant effects in minimising hazardous decision-making in both adolescents and adults (Grekin et al., 2025; Goger et al., 2024; Arora, 2023). Future-connectedness is defined by behaviours and cognition involving the active imagination, verbal and physical communication, and other related engagements towards a person’s future in all forms. Disconnect from the future self has been shown to increase irrational, short-sighted, and risky behaviours deemed beneficial in the present (Hershfield, 2011). Episodic future thinking is defined as the mental stimulation experienced towards possible future occurrences, and supports future-orientation and highlights the importance of delaying rewards and countering impulsive behaviours (Gilbert & Wilson, 2007; Benoit et al., 2011; Bar, 2009). This is particularly important as future-oriented thinking has historically shown its profound necessity amongst adolescents (Nurmi, 1991), and a lower predilection for impulsive and risky behaviour in favour of future rewards (Steinberg, 2004; Madden & Bickel, 2012). McCue et al. (2019) found a significant positive correlation between future-connectedness and episodic future thinking, claiming to be one of, if not the first, study to identify a beneficial relationship and independent effects within adolescents. Giollabhui et al. (2018) found that increased future-oriented behaviours and cognition contribute to reduced adolescent hopelessness. During adolescence, hopelessness is categorised by adamant negative future expectations and vulnerability (Casey et al., 2008; Lester, 2015). The severity of hopelessness in adolescence, and its relation to increased engagement in violent and/or high-risk behaviours, severe depression and anxiety, and unsafe sexual conduct (Becker-Weidman et al., 2009; Bolland, 2003; James et al., 2017; Giollabhui et al., 2018), indicates the importance of minimising adolescent hopelessness. Suicide has been a primary cause of death amongst adolescents for many years, skyrocketing after COVID-19 (Kim et al., 2024), and highlighting the worrisome trend. Considering the relationship between hopelessness, future-orientation, and adolescent depression and suicidal behaviours, increasing positive cognitions actively declines time-limited rates of hopelessness across confounds, including socio-economic status and stressors (Giollabhui et al., 2018; Faura-Garcia et al., 2024). ====Biological Influences==== Biological aspects of future orientation are currently under-researched, likely due to difficulty in quantifying results. Gan et al. (2024) investigated effects of the Colony-Stimulating Factor 3 Receptor (CSF3R) polymorphism and early stress on future orientation amongst 14,675 Chinese participants. Participants self-reported personal occurrences of maltreatment throughout childhood. Results indicated the CSF3R gene, which is closely related to chronic psychological stress responses, greatly interacted with early trauma’s influence on future orientation, specifically as levels of maltreatment increased (Gan et al., 2024). Of great importance to this chapter, childhood maltreatment demonstrated higher levels of stress in comparison to other stressors. This can either deplete opportunities to grow or encourage stress-related improvement in response to triggering situations (Gan et al., 2024), particularly for those with certain genotypes of the CSF3R (Drury et al., 2010). Zheng et al. (2022) explored the heritability of future orientation, identifying that genetic influence and mutual enviornments significantly affected future orientation development amongst adolescents. Findings revealed a negative effect between age and genetic influence, and a positive effect between age and individual environments on future orientation, as well as no effect between male and females (Zheng et al., 2022). Conversely, Greene et al. (2004) found that historical gender roles and expectations influenced future orientation, with females maintaining interpersonal goals, whereas males had further extension of goals. Importantly, the astronomical effects of environments on adolescent future orientation development and health and wellbeing, unrestricted from socio-cultural contexts, such as race, gender, and religion (Greene et al., 2004). ====Social And Environmental Influences==== Social influence can be argued as the largest influence on future orientation due to the proximity of environmental situations throughout childhood and adolescence (Laursen & Faur, 2022). Research has shown socioeconomic determinants positively correlated to the future orientation of adolescents (Nurmi, 1991), particularly higher parental and personal education (Borowsky et al., 2009; Nguyen et al., 2012), whereas neighbourhood poverty and racial minorities demonstrate weaker future-oriented thinking (Kerpelman et al., 2004; Guthrie et al., 2009). Johnson et al. (2016) explored how parenting style, including acceptance and involvement, attachment security, and autonomy granting, has consistently demonstrated positive effects on adolescent future-oriented thinking. Interestingly, Kai and Tienda (1998) found that adolescent future orientation as represented through college aspirations did not rely on race as it did for other influencing factors. Chen and Vazsonyi (2013) conducted a multilevel study on the effects of schooling on future-oriented thought processes amongst nine thousand 9th to 12th graders. Findings showed adolescent problem behaviours negatively impacted future-oriented thinking, especially those in larger cohorts and schools with higher positivity towards future orientation (Chen & Vazsonyi, 2013). Interestingly, students with a low sense of future orientation experienced opposite effects than those with problem behaviours, displaying a significant effect from positive future orientation attitudes at school (Chen & Vazsonyi, 2013). Rydell and Brocki (2024) found similar effects, with problem behaviours and inattention around mid-adolescence negatively affecting future orientation. However, high levels of hyperactivity and positive social interactions and relationships had compensatory properties, improving future orientation. Continuing research on adolescence, Skinner et al. (2022) investigated the effects of parental authority on future influence throughout COVID-19. Parental control is known to have an influence on adolescent wellbeing and future orientation (Wang et al., 2024; Zhu et al., 2026; Roman et al., 2015). For instance, psychological control is characterised by ‘intrusive and manipulative behaviors [sic] aimed at children or adolescents’ thoughts and feelings, through which adults exert power by controlling their children’s psychological world’ (Pérez et al., 2021), and can hinder adolescent development and wellbeing (Mandara & Pikes, 2008; Xu and Ren, 2025), and is connected to the development of depressive symptoms (Skinner et al., 2022). Data collected from adolescents and their parents across three years from nine countries, accounting for confounding factors, demonstrated the importance of positive normative standards for future orientation through parent-child interactions, and an educated, targeted approach to overcome adverse effects of childhood (Merrick et al., 2017; Kerpelman et al. 2008), such as an ecological systems approach (Rosanbalm et al. 2020). Wang et al. (2024) reiterated these points, finding that parental control significantly influences future orientation and positively affects perceived significance of life, future orientation, and personal growth initiative. Interestingly, psychological control had the only negative effect on future orientation and no effect on perceived significance of life. Given these findings, the social influences on future orientation are of the utmost importance and should be diligently recognised for positive effects to occur (Rydell et al., 2024; Crespo et al., 2013), particularly in adolescents and the objective to minimise bad behaviour (Chen & Vazsonyi, 2011; Voisin et al., 2020) and decrease hopelessness (Giollabhui et al., 2018). It should be noted that social influences on adolescent health and wellbeing are connected to psychological wellbeing and perceived significance of life, and therefore can be argued as the component of the BPSM that takes priority, but not as most important (Long et al., 2021; Webster et al., 2021; World Health Organization [WHO], 2025). ===All Three Components=== The BPSM aims to put the ‘nature vs nurture’ debate to rest by highlighting primary considerations as equal contributors to behaviour. Within the ‘Nature’ conversation, particularly neuroscience, researchers dictate that ‘the capacity of the brain to change its connections. …must be the consequence of having molecular mechanisms that mediate those changes’. (United States Institute of Medicine [USIM]. Beauchaine et al. (2009) provided ten reasons to consider biological processes in psychological prevention and intervention in research, including biomarkers and endophenotypes in identifying priority prevention methods, how biological vulnerabilities moderate environmental experiences and differences, and interactions between environmental and neural systems sensitivity across lifespans. Similarly, Marder and Miller (2024) discuss how biological psychology is the focal point of psychophysiology, investigating the effects of biological markers in behaviour, accompanied by the influences of social trends creates the foundation of individual disposition. Eriksson et al. (2025) corroborated the multilevel consequences on behaviour and wellbeing, stating ‘It is essential to move beyond focusing solely on individual psychological and cognitive factors to an understanding of the complex processes involved in health behaviour change.’ Findings illustrated variance between health behaviours for different health identities amongst middle-aged men and women, guiding future socio-ecological interventions promoting healthier behaviours (Eriksson et al., 2025). Regarding the BPSM within future orientation and criminal behaviour, each element attracts differing levels of authority from many sources. For example, the majority of research in future orientation revolves around environmental influences (Wittmann & Sircova, 2018), compared to psychological and biological components. Conversely, criminal behaviour prioritises biological and psychological influences (Miller, 2014; Focquaert, 2019). ===Why is this particularly important for adolescents?=== Kovács et al. (2026) detail that future orientation is “not a singular construct but a multifaceted phenomenon operationalized [sic] through cognitive, affective, and contextual lenses”. This is particularly important for adolescents, as biological, social, and psychological factors interact dynamically, with each influencing the others and shaping development (Lindstrom Johnson et al., 2014). The BPSM is particularly relevant as stronger future orientation is associated with adolescent wellbeing and healthier behaviours, including greater engagement in health-promoting activities and reduced risk-taking, although these relationships vary across contexts (Kovács et al., 2026). Future orientation may support these outcomes by strengthening adolescents’ capacity to establish meaningful goals, anticipate the consequences of their decisions, and regulate present behaviour in pursuit of longer-term outcomes (Bandura, 2001; Nurmi, 1991). However, the extent to which future-oriented thinking translates into behaviour depends on individual motivation and adolescents’ perceived opportunities, social relationships, and environmental circumstances (Lindstrom Johnson et al., 2014). Future orientation should not be considered as universally protective, as an intense focus on future outcomes may also be associated with anxiety or heightened pressure to achieve in certain contexts (Kovács et al., 2026). This indicates the importance of the BPSM in future orientation, implying that no singular factor is the sole cause of development, and interventions should aim at promoting adolescent wellbeing beyond encouraging long-term goal setting to include the development of self-regulatory skills and supportive environments that enable adolescents to pursue their aspirations (Nurmi, 1991; Lindstrom Johnson et al., 2014). === Future Orientation’s Relevance to Criminal Activity === Relevant to criminal behaviour, higher levels of future-oriented cognition indicate further consideration of long-term consequences of offending (Kwon et al., 2025); in contrast, limited future orientation correlates with increased criminal behaviour by encouraging immediate gratification and reducing consideration of delayed consequences (Kübel et al., 2023). Clinkinbeard (2014) found that high future orientation and self-control predict adolescent criminal behaviour; particularly, higher future-oriented cognition mitigates negative effects of low self-control and lessens impulsivity, risky behaviour engagement, criminal cognition, and adolescent crime rates (Khetarpal et al., 2021; Squillaro & Bixter, 2025; Simmons et al., 2019). Importantly, Walters (2019) identified a gradual decrease in criminal behaviour with maturation, associated with increased future orientation and decreased adolescent hopelessness. Similarly, Wood et al. (2017) recognised that cognitive immaturity relates to engagement in aversive behaviours, and thereby suffering future negative life effects; promoting future orientation and positive childhood effects can mitigate these negative life effects and promote sustainability (Kübel et al., 2023; Cauffman et al., 2023; Mueller et al., 2023). Given the findings on future orientation, it is now important to consider the role it plays in criminal behaviour, and therefore how research can facilitate future orientation to prevent, mitigate or rehabilitate criminal behaviour. ;Quiz Choose your answers and click "Submit": <quiz display=simple> {A variety of influences contribute to future orientation: |type="(+)"} + True - False {Future-connectedness is defined by behaviours and cognition involving the engagement towards a person’s future: |type="(+)"} + True - False {Disconnect from the future self has been shown to have no aversive risks: |type="(+)"} + True - False {Hopelessness is a persistent and serious outcome of adolescence: |type="(+)"} + True - False {Future orientation is ''the'' primary indicator for engagement with criminal behaviour: |type="(-)"} + True - False </quiz> ==Key Points== * Future orientation is the tendency for a person to engage in thoughts and actions in favour of future goals. *Future-connectedness is defined by behaviours and cognition involving engagement towards a person’s future in all forms. *The severity of hopelessness in adolescence, and its high relation to adverse behaviours, indicates the importance of minimising adolescent hopelessness. * Higher levels of future-oriented cognition indicate further consideration of long-term consequences of offending. * Limited future orientation correlates with increased criminal behaviour. =Criminal Behaviour= Dent and Ward (2022) conceptualised criminal behaviour as ‘an action or set of actions... understood by the actor to be adaptive in relation to their functional needs, but...make it more difficult to function adaptively...to other sociocultural needs.’ Criminal behaviour can include minor crimes, such as ‘theft or traffic violations’, and major crimes, such as ‘physical abuse, sexual offense [sic], or murder’ (Prent et al., 2023), and is considered a ‘major social problem with complex causes... [and involves] a myriad of environmental, social, and psychological factors are associated with increased risk of convictions for this type of criminality’ (Prent et al., 2023; Levitt, 2012; Swaab & Meynen, 2023). ==Influences on Criminal Behaviour== Reiterating findings from Prent et al. (2023), criminal behaviour is shaped by many complex, interconnected facets, including social, psychological, biological, and environmental factors, each with influential components such as geographical location and socio-economic status. Importantly, future orientation has shown positive effects in minimising negative behaviour across different contributing factors (Giollabhui et al., 2018). ==Psychological Influences== In media and history, psychology has often been treated as the primary explanation for criminal acts; however, crime is not limited to a single discipline (South Australia News, 2026). While this is well established, it is often a misconception that one factor is more influential than the other. A longstanding theory explaining human behaviour, cognition, and reasoning is Nature v Nurture. Nurture, as a psychological construct, argues that a person’s circumstances stem from upbringing and environment rather than other factors. Hadsell (2026) stated that the ‘connection between parents and children has two pathways: biological, covering both genes and hormones, and environmental.’ ==Biological Influences== When discussing the Nature concept, it is theorised that an individual's biological makeup is the sole predictor of behaviour (Alper, 1995). Baschetti (2008) argued that genetic evidence proved Nature over Nurture. Drawing on Charles Darwin’s theories of evolution and biology, current and overwhelming research indicates that criminality is heritable, yet persistent (Baschetti, 2008; Pizzi, 2004). It should be noted that Nature can refer to the genetic contribution within mental illness. Rao (2007) argued that all humans have a ‘dark side that loves crime and violence... [while] we may all deny it, but the contrary is true...regardless of our age, caste, social status, region, religion or education.’ [''APA style: Direct quotes should use double quotation marks and include page numbers''] Relevant to crime rates, it can be reasoned that modern-day society is the ‘most violent culture in history in the number of crimes and...the nature of brutality.’ Ling et al. (2019) explored how psychophysiology, brain mechanisms, and genetics stimulate criminal behaviour. Particularly, disrupted prefrontal cortex-amygdala connectivity and changes in autonomic functioning damaging the hypothalamic-pituitary-adrenal axis correlate with increased criminal and anti-social tendencies (Gillespie et al., 2018; Critchley, 2005; Brzozowski & Mitchell, 2018) [''APA style: Arrange multiple citations in alphabetical order'']. Baschetti (2008) signified the importance of systematic changes acknowledging the existence of a biological influence to encourage appropriate reforms and minimise negative outcomes. ==Relationship Between Criminal Behaviour and Future Orientation== {{expand}} ==Does Future Orientation Influence the Risk of Criminal Activity?== Future orientation is an incredibly [''Avoid overly emotive language''] important skill, particularly for adolescents or those experiencing hopelessness. A lack of future-oriented cognition can exacerbate impulsive and risky behaviour and may lead to criminal behaviour (Squillaro & Bixter, 2025; Mueller et al., 2023). Given adolescents' vulnerability to criminal behaviours (Bui & Deakin, 2021) and hopelessness (Bolland, 2003; Tonkus et al., 2022), encouraging future orientation is highly relevant. Reiterating that future orientation positively relates to curbing adolescent criminal activity and juvenile incarceration (Walters, 2019; Simmons et al., 2019) reinforces future orientation as an imperative protective factor across the lifespan (Yang et al., 2026; Clinkinbeard, 2014). ==Book Chapter Direction== * Environmental influences in criminal activity. *Social influences in criminal activity. *Understanding the relationship between future orientation and criminal activity. *Protective factors within future orientation *Risk factors within future orientation *Future research directions ==Conclusion== ''''' Include key sentence on the importance of understanding future orientation in criminal behaviour''.''' Currently, there is substantial research into the effects and relationships of future orientation and adolescent criminal activity. However, conclusive findings on the relevance of using future orientation to mitigate criminal behaviour remain limited. Given the strong positive correlations between future-oriented thinking and reduced disturbances in life quality, it would be unwise to ignore how future orientation can help prevent rising crime rates among adolescents ''and'' adults. ==See also== {{expand}} ==References== {{Hanging indent|1= Alper, J. S. (1995). Biological influences on criminal behaviour: How good is the evidence? *BMJ, 310*(6975), 272–273. https://doi.org/10.1136/bmj.310.6975.272 Arora, S. (2023). *“It’s a future-me problem”: How future-self continuity and message framing affect climate change risk perceptions* [Doctoral dissertation, University of Guelph]. Atrium. https://atrium.lib.uoguelph.ca/items/4d97286e-5635-45b6-a1c3-e9be464b1e7c Bar, M. (2009). The proactive brain: Memory for predictions. *Philosophical Transactions of the Royal Society B: Biological Sciences, 364*(1521), 1235–1243. https://doi.org/10.1098/rstb.2008.0310 Baschetti, R. (2008). Genetic evidence that Darwin was right about criminality: Nature, not nurture. *Medical Hypotheses, 70*(6), 1092–1102. https://doi.org/10.1016/j.mehy.2008.01.005 Becker-Weidman, E. G., Jacobs, R. H., Reinecke, M. A., Silva, S. G., & March, J. S. (2010). Social problem-solving among adolescents treated for depression. *Behaviour Research and Therapy, 48*(1), 11–18. https://doi.org/10.1016/j.brat.2009.08.006 Benoit, R. G., Gilbert, S. J., & Burgess, P. W. (2011). A neural mechanism mediating the impact of episodic prospection on farsighted decisions. *The Journal of Neuroscience, 31*(18), 6771–6779. https://doi.org/10.1523/JNEUROSCI.6559-10.2011 Bickel, W. K., Athamneh, L. N., Basso, J. C., Mellis, A. M., DeHart, W. B., Craft, W. H., & Pope, D. (2019). Excessive discounting of delayed reinforcers as a trans-disease process: Update on the state of the science. *Current Opinion in Psychology, 30*, 59–64. https://doi.org/10.1016/j.copsyc.2019.01.005 Bolland, J. M. (2003). Hopelessness and risk behaviour among adolescents living in high-poverty inner-city neighbourhoods. *Journal of Adolescence, 26*, 145–158. https://doi.org/10.1016/S0140-1971(02)00136-7 Bui, L., & Deakin, J. (2021). What we talk about when we talk about vulnerability and youth crime: A narrative review. *Aggression and Violent Behavior, 58*, 101605. https://doi.org/10.1016/j.avb.2021.101605 Cauffman, E., Gillespie, M. L., Beardslee, J., Davis, F., Hernandez, M., & Williams, T. (2023). Adolescent contact, lasting impact? Lessons learned from two longitudinal studies spanning 20 years of developmental science research with justice-system-involved youths. *Psychological Science in the Public Interest, 24*(3), 133–161. https://doi.org/10.1177/15291006231205173 Chadwick, R. (n.d.). Genetics and crime. In M. Levitt (Ed.), *Encyclopedia of applied ethics* (2nd ed., pp. 462–469). Academic Press. https://doi.org/10.1016/B978-0-12-373932-2.000375 Clinkinbeard, S. S. (2014). What lies ahead: An exploration of future orientation, self-control, and delinquency. *Criminal Justice Review, 39*(1), 19–36. https://doi.org/10.1177/0734016813501193 Critchley, H. D. (2005). Neural mechanisms of autonomic, affective, and cognitive integration. *The Journal of Comparative Neurology, 493*(1), 154–166. https://doi.org/10.1002/cne.20749 Dent, H., & Ward, T. (2022). Emotion and correctional practice: Theoretical foundations of an enactive practice framework. *Aggression and Violent Behavior, 63*, 101672. https://doi.org/10.1016/j.avb.2021.101672 Faura-Garcia, J., Calvete, E., & Orue, I. (2024). Longitudinal associations between nonsuicidal self-injury, depressive symptoms, hopelessness, and emotional dysregulation in adolescents. *Archives of Suicide Research, 28*(3), 800–814. https://doi.org/10.1080/13811118.2023.2237075 Gillespie, S. M., Brzozowski, A., & Mitchell, I. J. (2018). Self-regulation and aggressive antisocial behaviour: Insights from amygdala-prefrontal and heart-brain interactions. *Psychology, Crime & Law, 24*(3), 243–257. https://doi.org/10.1080/1068316X.2017.1414816 Gilbert, D. T., & Wilson, T. D. (2007). Prospection: Experiencing the future. *Science, 317*(5843), 1351–1354. https://doi.org/10.1126/science.1144161 Goger, P., Nam, R. J., Lowry, N., Bell, K. A., Parvez, N., Pollak, O. H., Robinaugh, D. J., Schacter, D. L., & Cha, C. B. (2024). Testing the modifiability of episodic future thinking and episodic memory among suicidal and nonsuicidal adolescents. *JCPP Advances, 4*(3), e12236. https://doi.org/10.1002/jcv2.12236 Grekin, E. R., Thomas, H. A., Souweidane, M. A., & Stidham, J. L. (2025). The effects of future self-continuity interventions on behavioral outcomes in adults: A systematic review of the literature. *Personality Science, 6*. https://doi.org/10.1177/27000710251391610 Hadsell, L. (2026). Parents: Nature and nurture. In *Social economies*. Palgrave Macmillan. https://doi.org/10.1007/978-3-032-21916-9_18 Hershfield, H. E. (2011). Future self-continuity: How conceptions of the future self transform intertemporal choice. *Annals of the New York Academy of Sciences, 1235*, 30–43. https://doi.org/10.1111/j.1749-6632.2011.06201.x James, S., Reddy, S. P., Ellahebokus, A., Sewpaul, R., & Naidoo, P. (2017). The association between adolescent risk behaviours and feelings of sadness or hopelessness: A cross-sectional survey of South African secondary school learners. *Psychology, Health & Medicine, 22*(7), 778–789. https://doi.org/10.1080/13548506.2017.1300669 Johnson, S. R., Blum, R. W., & Cheng, T. L. (2014). Future orientation: A construct with implications for adolescent health and wellbeing. *International Journal of Adolescent Medicine and Health, 26*(4), 459–468. https://doi.org/10.1515/ijamh-2013-0333 Khetarpal, S. K., Szoko, N., Ragavan, M. I., & Culyba, A. J. (2021). Future orientation as a cross-cutting protective factor against multiple forms of violence. *The Journal of Pediatrics, 235*, 288–291. https://doi.org/10.1016/j.jpeds.2021.05.019 Kübel, L. S., Deitzer, R. J., Frankenhuis, E. W., Ribeaud, D., Eisner, P. M., & van Gelder, J.-L. (2023). The shortsighted victim: Short-term mindsets mediate the link between victimization and later offending. *Journal of Criminal Justice, 86*, 102062. https://doi.org/10.1016/j.jcrimjus.2023.102062 Kwon, K. Y., Sutin, A. R., & Kim, J. (2025). Juvenile delinquency and cognitive function in adulthood: Differentiating violent and nonviolent behaviors and exploring multiple mechanisms. *Journal of Research on Adolescence, 35*(4), e70095. https://doi.org/10.1111/jora.70095 Lester, D. (2015). Hopelessness in adolescents. *Journal of Affective Disorders, 173*, 221–225. https://doi.org/10.1016/j.jad.2014.10.048 Ling, S., Umbach, R., & Raine, A. (2019). Biological explanations of criminal behavior. *Psychology, Crime & Law, 25*(6), 626–640. https://doi.org/10.1080/1068316X.2019.1572753 Mac Giollabhui, N., Hamilton, J. L., Nielsen, J., Connolly, S. L., Stange, J. P., Varga, S., Burdette, E., Olino, T. M., Abramson, L. Y., & Alloy, L. B. (2018). Negative cognitive style interacts with negative life events to predict first onset of a major depressive episode in adolescence via hopelessness. *Journal of Abnormal Psychology, 127*(1), 1–11. https://doi.org/10.1037/abn0000301 McCue, R., McCormack, T., McElnay, J., Alto, A., & Feeney, A. (2019). The future and me: Imagining the future and the future self in adolescent decision making. *Cognitive Development, 50*, 142–156. https://doi.org/10.1016/j.cogdev.2019.04.001 Mueller, K. C., Cavitt, J., & Carey, M. T. (2023). Future orientation in justice-involved youth: The effects of adverse and positive childhood experiences, and being dual-status. *Child Abuse & Neglect, 141*, 106230. https://doi.org/10.1016/j.chiabu.2023.106230 Nurmi, E. J. (1991). How do adolescents see their future? A review of the development of future orientation and planning. *Developmental Review, 11*(1), 1–59. https://doi.org/10.1016/0273-2297(91)90002-6 Peterson, M. F. (2004). [Review of the book *Culture, leadership and organizations: The GLOBE study of 62 societies*, by R. J. House, M. Javidan, P. W. Dorfman, V. Gupta, & P. J. Hanges]. *Administrative Science Quarterly, 49*(4), 641–647. https://www.jstor.org/stable/4131493 Pizzi, W. J. (2004). Nature via nurture: Genes, experience, and what makes us human. *Journal of Undergraduate Neuroscience Education, 2*(2), R10–R11. Prent, N., Jonker, A. F., Schouws, N. T. M. S., & Jonker, C. (2023). The risk of criminal behavior in the elderly and patients with neurodegenerative disease. In H. Swaab & G. Meynen (Eds.), *Handbook of clinical neurology* (Vol. 197, pp. 181–196). Elsevier. https://doi.org/10.1016/B978-0-12-821375-9.00004-9 Sathyanarayana Rao, T. S. (2007). Psychiatrist and the science of criminology: Sociological, psychological and psychiatric analysis of the dark side. *Indian Journal of Psychiatry, 49*(1), 3–5. https://doi.org/10.4103/0019-5545.31511 Simmons, C., Rowan, Z., Knowles, A., Steinberg, L., Frick, P. J., & Cauffman, E. (2019). A life history approach to understanding juvenile offending and aggression. *Aggression and Violent Behavior, 49*, 101317. https://doi.org/10.1016/j.avb.2019.07.012 South Australia News. (2026, April 25). *Psychology of a criminal: Why do people commit crimes?* https://sanews.in/psychology-of-a-criminal/ Squillaro, D., & Bixter, M. T. (2025). The relationship between four indicators of future time orientation, criminal thinking style, and illegal behaviors. *Journal of Forensic Sciences, 70*(1), 200–214. https://doi.org/10.1111/1556-4029.15640 Steinberg, L. (2004). Risk taking in adolescence: What changes, and why? *Annals of the New York Academy of Sciences, 1021*, 51–58. https://doi.org/10.1196/annals.1308.005 Swaab, H., & Meynen, G. (2023). Introduction: On brain and crime. In H. Swaab & G. Meynen (Eds.), *Handbook of clinical neurology* (Vol. 197, pp. 3–9). Elsevier. https://doi.org/10.1016/B978-0-12-821375-9.00018-9 Turton, H., Berry, K., Danquah, A., & Pratt, D. (2021). The relationship between emotion dysregulation and suicide ideation and behaviour: A systematic review. *Journal of Affective Disorders Reports, 5*, 100136. https://doi.org/10.1016/j.jadr.2021.100136 Walters, G. D. (2019). Maturing out of delinquency: Unpacking the effects of identity achievement and future orientation on crime desistance. *Self and Identity, 18*(3), 267–283. https://doi.org/10.1080/15298868.2018.1439848 Wager, T. D., Waugh, C. E., Lindquist, M., Noll, D. C., Fredrickson, B. L., & Taylor, S. F. (2009). Brain mediators of cardiovascular responses to social threat: Part I: Reciprocal dorsal and ventral sub-regions of the medial prefrontal cortex and heart-rate reactivity. *NeuroImage, 47*(3), 821–835. https://doi.org/10.1016/j.neuroimage.2009.05.043 Wood, D., Crapnell, T., Lau, L., et al. (2017). Emerging adulthood as a critical stage in the life course. In N. Halfon, C. B. Forrest, R. M. Lerner, et al. (Eds.), *Handbook of life course health development*. Springer. https://www.ncbi.nlm.nih.gov/books/NBK543712/ Yang, E. S., Kim, S., Oh, S.-K., Kang, H., & Chung, I.-J. (2026). Disrupting the link between exposure to violence and adolescent delinquency: The protective role of expectations about the future. *Journal of Child and Family Studies*. https://doi.org/10.1007/s10826-026-03311-3 }} ==External links== {{expand}} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Cognitive]] [[Category:Motivation and emotion/Book/Forensic]] lwofullnyjb6sb8ocwni17px3gk9dpv 2842558 2842557 2026-10-25T11:10:44Z Isa Murphy 3105869 2842558 wikitext text/x-wiki {{title|Future orientation and criminal behaviour:<br>How does future orientation influence the risk of criminal activity?}} ==Overview== {{RoundBoxTop|theme=7}} [[File:Exams during Covid-19 - France (50651621763).jpg|thumb|200px|'''Figure 1'''. Exams during Covid-19.]] ; Acar and Kaya (2024) study Amongst 23 adolescent participants, psychological drivers, social and environmental influences, and cognitive and planning capacities were identified as primary categories influencing adolescent future orientation, alongside family, peers, educational access, and cultural norms contributing to the facilitation or hindering of future aspirations (Acar & Kaya, 2024). '''Psychological drivers''': 'hope and optimism, fear of failure, self-identity, motivation and ambition, and emotional regulation' (Acar & Kaya, 2024). '''Social and environmental''': 'parental support, peer influence, school environment, socioeconomic constraints, and community expectations' (Acar & Kaya, 2024). '''Cognitive and planning capacities''': 'goal-setting skills, time perspective, decision-making ability, problem-solving skills, information-seeking behavior [''sic''], and academic self-efficacy' (Acar & Kaya, 2024). {{RoundBoxBottom}} When focusing on adolescents, future orientation is a vital component of their education (Johnson et al., 2014). Conversely, adolescent crime rates have been steadily decreasing (Australian Bureau of Statistics [ABS], 2026), yet they remain a major concern (Hinchcliff et al., 2026; Australian Human Rights Commission [AHRC], 2024; World Health Organisation [WHO], 2026). {{RoundBoxTop|theme=2}} '''Focus questions''' {{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}} '''1.''' Why is Future Orientation Important? '''2.''' Can Future Orientation Mitigate Criminal Behaviour? '''3.''' Are Adolescents More Susceptible to Aversive Effects of Low Future Orientation? '''4.''' Does Criminal Behaviour Depend Upon Future-Oriented Cognition? {{RoundBoxBottom}} = Future Orientation = {{ic|Include at least an introductory paragraph before branching into sub-headings}} ==What is Future Orientation== According to House et al. (2004), future orientation is the tendency for a person to engage in thoughts, actions, and the delaying of immediate gratification in favour of future goals. Referring to adolescents, many disciplines have ratified the importance of future-oriented behaviour in promoting positive health and development (Johnson et al., 2016). Trommsdorff (9179) proposed that future orientation included eight dimensions: extension, detail, domain, affect, motivation, control, sequence of events, and number of cognitions. {{center top}} {| border=1 cellspacing=0 cellpadding=6 |- ! Dimension ! Definition |- | extension | bar |- | detail | bar |- | domain | bar |- | affect | bar |- | motivation | bar |- | control | bar |- | sequence of events | bar |- | number of cognitions | far |} {{center bottom}} A study by Beal (2011) revealed that Trommsdorff’s (1979) conceptualisation indicates that different influences can be indicators of future orientation and the eight dimensions. Findings consistently supported previous research that future orientation is an important predictor of adolescent well-being and career adaptability and revealed that ‘stronger future orientation and future-focused time perspectives are consistently associated with higher adolescent well-being, health-promoting habits (e.g., physical activity), and lower engagement in health-risk behaviours (e.g., substance use, risky sexual behaviour)’ (Kovács et al., 2026). Lindstrom Johnson et al. (2014) examines future orientation as a mediator or moderator of problem behaviour, determining that while conclusions should be interpreted cautiously, findings indicated future-oriented thinking as a mediator can be expressed as reasoning, but not the sole, for behavioural outcomes (Weems, 2025). Kovács et al. (2026) conducted a systematic review on the effects and different forms future orientation has on adolescents, focusing on future time perspective and future orientation. Findings on future time perspective revealed an interesting trend: that positivity-endorsing programmes on past behaviours, future-oriented planning, actively developing time perspective, and contextual health behaviour–focused prevention programmes greatly improve adolescent health and well-being (Kovács et al., 2026; Kuan & Zhang, 2022). Thus, the following article will inform adolescent future orientation as a preventative measure for criminal behaviour before it occurs. It should be noted that additional factors, such as drug-oriented perspectives, can explain insignificant positive effects in future-oriented programmes. Considering the major transitional stage that is adolescence (Khurana et al., 2015), the enticing and control-eliciting nature of drugs and alcohol, and social and environmental influences on various behaviours (Barnett et al., 2013; Zhang et al., 2021), potential negative effects resulting from illicit drug and alcohol use are plausible (Finan et al., 2022; Peters et al., 2005; Jones et al., 2021) and should be recognised when discussing and assessing adolescent future-oriented thinking (Geier et al., 2013; Stoddard et al., 2018). ==Influences on Future Orientation== Multidisciplinary findings have indicated that a variety of influences contribute to future orientation. Much psychological research has focused on adolescents and young adults, so findings are skewed, though still extensive and reliable. ===The BioPsychoSocial Model=== In 1977, George Engel proposed an extension to the biomedical model that incorporated the psychological and social factors influencing health and wellbeing, known as the BioPsychoSocial Model (BPSM). Vögele (2015) defines BPSM as ‘a general model positing that biological, psychological (which includes thoughts, emotions, and behaviors [sic]), and social (e.g., socioeconomical, socioenvironmental, and cultural) factors, all play a significant role in health. …[and] are best understood in terms of a combination…rather than purely in biological terms.’ It is important to recognise that biological, psychological, and social influences are individual yet equally important additions to health and wellbeing with a variety of crossover (Dfarhud et al., 2014; Chu et al., 2024; Candeias et al., 2024; Kahn et al., 2024). For example, Subramanyam et al. (2024) studied the effects of neurodevelopment, epigenetics, parenting, communication skills, career-focused future orientation, and the environment on child and adolescent wellbeing, with findings indicating each component held a myriad of interplaying causes and effects. Regarding future orientation, multidisciplinary findings have indicated that a variety of influences contribute to future orientation. It is important to note that much of the psychological research has targeted adolescents and young adults, and therefore findings are skewed, yet extensive and reliable. ====Psychological Influences==== McCue et al. (2019) researched and measured the effects of future orientation on adolescent decision-making. Previously, connectedness to future-self and episodic future thinking demonstrates future-oriented cognition, and while their correlation is undetermined, individual measures have shown significant effects in minimising hazardous decision-making in both adolescents and adults (Grekin et al., 2025; Goger et al., 2024; Arora, 2023). Future-connectedness is defined by behaviours and cognition involving the active imagination, verbal and physical communication, and other related engagements towards a person’s future in all forms. Disconnect from the future self has been shown to increase irrational, short-sighted, and risky behaviours deemed beneficial in the present (Hershfield, 2011). Episodic future thinking is defined as the mental stimulation experienced towards possible future occurrences, and supports future-orientation and highlights the importance of delaying rewards and countering impulsive behaviours (Gilbert & Wilson, 2007; Benoit et al., 2011; Bar, 2009). This is particularly important as future-oriented thinking has historically shown its profound necessity amongst adolescents (Nurmi, 1991), and a lower predilection for impulsive and risky behaviour in favour of future rewards (Steinberg, 2004; Madden & Bickel, 2012). McCue et al. (2019) found a significant positive correlation between future-connectedness and episodic future thinking, claiming to be one of, if not the first, study to identify a beneficial relationship and independent effects within adolescents. Giollabhui et al. (2018) found that increased future-oriented behaviours and cognition contribute to reduced adolescent hopelessness. During adolescence, hopelessness is categorised by adamant negative future expectations and vulnerability (Casey et al., 2008; Lester, 2015). The severity of hopelessness in adolescence, and its relation to increased engagement in violent and/or high-risk behaviours, severe depression and anxiety, and unsafe sexual conduct (Becker-Weidman et al., 2009; Bolland, 2003; James et al., 2017; Giollabhui et al., 2018), indicates the importance of minimising adolescent hopelessness. Suicide has been a primary cause of death amongst adolescents for many years, skyrocketing after COVID-19 (Kim et al., 2024), and highlighting the worrisome trend. Considering the relationship between hopelessness, future-orientation, and adolescent depression and suicidal behaviours, increasing positive cognitions actively declines time-limited rates of hopelessness across confounds, including socio-economic status and stressors (Giollabhui et al., 2018; Faura-Garcia et al., 2024). ====Biological Influences==== Biological aspects of future orientation are currently under-researched, likely due to difficulty in quantifying results. Gan et al. (2024) investigated effects of the Colony-Stimulating Factor 3 Receptor (CSF3R) polymorphism and early stress on future orientation amongst 14,675 Chinese participants. Participants self-reported personal occurrences of maltreatment throughout childhood. Results indicated the CSF3R gene, which is closely related to chronic psychological stress responses, greatly interacted with early trauma’s influence on future orientation, specifically as levels of maltreatment increased (Gan et al., 2024). Of great importance to this chapter, childhood maltreatment demonstrated higher levels of stress in comparison to other stressors. This can either deplete opportunities to grow or encourage stress-related improvement in response to triggering situations (Gan et al., 2024), particularly for those with certain genotypes of the CSF3R (Drury et al., 2010). Zheng et al. (2022) explored the heritability of future orientation, identifying that genetic influence and mutual enviornments significantly affected future orientation development amongst adolescents. Findings revealed a negative effect between age and genetic influence, and a positive effect between age and individual environments on future orientation, as well as no effect between male and females (Zheng et al., 2022). Conversely, Greene et al. (2004) found that historical gender roles and expectations influenced future orientation, with females maintaining interpersonal goals, whereas males had further extension of goals. Importantly, the astronomical effects of environments on adolescent future orientation development and health and wellbeing, unrestricted from socio-cultural contexts, such as race, gender, and religion (Greene et al., 2004). ====Social And Environmental Influences==== Social influence can be argued as the largest influence on future orientation due to the proximity of environmental situations throughout childhood and adolescence (Laursen & Faur, 2022). Research has shown socioeconomic determinants positively correlated to the future orientation of adolescents (Nurmi, 1991), particularly higher parental and personal education (Borowsky et al., 2009; Nguyen et al., 2012), whereas neighbourhood poverty and racial minorities demonstrate weaker future-oriented thinking (Kerpelman et al., 2004; Guthrie et al., 2009). Johnson et al. (2016) explored how parenting style, including acceptance and involvement, attachment security, and autonomy granting, has consistently demonstrated positive effects on adolescent future-oriented thinking. Interestingly, Kai and Tienda (1998) found that adolescent future orientation as represented through college aspirations did not rely on race as it did for other influencing factors. Chen and Vazsonyi (2013) conducted a multilevel study on the effects of schooling on future-oriented thought processes amongst nine thousand 9th to 12th graders. Findings showed adolescent problem behaviours negatively impacted future-oriented thinking, especially those in larger cohorts and schools with higher positivity towards future orientation (Chen & Vazsonyi, 2013). Interestingly, students with a low sense of future orientation experienced opposite effects than those with problem behaviours, displaying a significant effect from positive future orientation attitudes at school (Chen & Vazsonyi, 2013). Rydell and Brocki (2024) found similar effects, with problem behaviours and inattention around mid-adolescence negatively affecting future orientation. However, high levels of hyperactivity and positive social interactions and relationships had compensatory properties, improving future orientation. Continuing research on adolescence, Skinner et al. (2022) investigated the effects of parental authority on future influence throughout COVID-19. Parental control is known to have an influence on adolescent wellbeing and future orientation (Wang et al., 2024; Zhu et al., 2026; Roman et al., 2015). For instance, psychological control is characterised by ‘intrusive and manipulative behaviors [sic] aimed at children or adolescents’ thoughts and feelings, through which adults exert power by controlling their children’s psychological world’ (Pérez et al., 2021), and can hinder adolescent development and wellbeing (Mandara & Pikes, 2008; Xu and Ren, 2025), and is connected to the development of depressive symptoms (Skinner et al., 2022). Data collected from adolescents and their parents across three years from nine countries, accounting for confounding factors, demonstrated the importance of positive normative standards for future orientation through parent-child interactions, and an educated, targeted approach to overcome adverse effects of childhood (Merrick et al., 2017; Kerpelman et al. 2008), such as an ecological systems approach (Rosanbalm et al. 2020). Wang et al. (2024) reiterated these points, finding that parental control significantly influences future orientation and positively affects perceived significance of life, future orientation, and personal growth initiative. Interestingly, psychological control had the only negative effect on future orientation and no effect on perceived significance of life. Given these findings, the social influences on future orientation are of the utmost importance and should be diligently recognised for positive effects to occur (Rydell et al., 2024; Crespo et al., 2013), particularly in adolescents and the objective to minimise bad behaviour (Chen & Vazsonyi, 2011; Voisin et al., 2020) and decrease hopelessness (Giollabhui et al., 2018). It should be noted that social influences on adolescent health and wellbeing are connected to psychological wellbeing and perceived significance of life, and therefore can be argued as the component of the BPSM that takes priority, but not as most important (Long et al., 2021; Webster et al., 2021; World Health Organization [WHO], 2025). ===All Three Components=== The BPSM aims to put the ‘nature vs nurture’ debate to rest by highlighting primary considerations as equal contributors to behaviour. Within the ‘Nature’ conversation, particularly neuroscience, researchers dictate that ‘the capacity of the brain to change its connections. …must be the consequence of having molecular mechanisms that mediate those changes’. (United States Institute of Medicine [USIM]. Beauchaine et al. (2009) provided ten reasons to consider biological processes in psychological prevention and intervention in research, including biomarkers and endophenotypes in identifying priority prevention methods, how biological vulnerabilities moderate environmental experiences and differences, and interactions between environmental and neural systems sensitivity across lifespans. Similarly, Marder and Miller (2024) discuss how biological psychology is the focal point of psychophysiology, investigating the effects of biological markers in behaviour, accompanied by the influences of social trends creates the foundation of individual disposition. Eriksson et al. (2025) corroborated the multilevel consequences on behaviour and wellbeing, stating ‘It is essential to move beyond focusing solely on individual psychological and cognitive factors to an understanding of the complex processes involved in health behaviour change.’ Findings illustrated variance between health behaviours for different health identities amongst middle-aged men and women, guiding future socio-ecological interventions promoting healthier behaviours (Eriksson et al., 2025). Regarding the BPSM within future orientation and criminal behaviour, each element attracts differing levels of authority from many sources. For example, the majority of research in future orientation revolves around environmental influences (Wittmann & Sircova, 2018), compared to psychological and biological components. Conversely, criminal behaviour prioritises biological and psychological influences (Miller, 2014; Focquaert, 2019). ===Why is this particularly important for adolescents?=== Kovács et al. (2026) detail that future orientation is “not a singular construct but a multifaceted phenomenon operationalized [sic] through cognitive, affective, and contextual lenses”. This is particularly important for adolescents, as biological, social, and psychological factors interact dynamically, with each influencing the others and shaping development (Lindstrom Johnson et al., 2014). The BPSM is particularly relevant as stronger future orientation is associated with adolescent wellbeing and healthier behaviours, including greater engagement in health-promoting activities and reduced risk-taking, although these relationships vary across contexts (Kovács et al., 2026). Future orientation may support these outcomes by strengthening adolescents’ capacity to establish meaningful goals, anticipate the consequences of their decisions, and regulate present behaviour in pursuit of longer-term outcomes (Bandura, 2001; Nurmi, 1991). However, the extent to which future-oriented thinking translates into behaviour depends on individual motivation and adolescents’ perceived opportunities, social relationships, and environmental circumstances (Lindstrom Johnson et al., 2014). Future orientation should not be considered as universally protective, as an intense focus on future outcomes may also be associated with anxiety or heightened pressure to achieve in certain contexts (Kovács et al., 2026). This indicates the importance of the BPSM in future orientation, implying that no singular factor is the sole cause of development, and interventions should aim at promoting adolescent wellbeing beyond encouraging long-term goal setting to include the development of self-regulatory skills and supportive environments that enable adolescents to pursue their aspirations (Nurmi, 1991; Lindstrom Johnson et al., 2014). === Future Orientation’s Relevance to Criminal Activity === Relevant to criminal behaviour, higher levels of future-oriented cognition indicate further consideration of long-term consequences of offending (Kwon et al., 2025); in contrast, limited future orientation correlates with increased criminal behaviour by encouraging immediate gratification and reducing consideration of delayed consequences (Kübel et al., 2023). Clinkinbeard (2014) found that high future orientation and self-control predict adolescent criminal behaviour; particularly, higher future-oriented cognition mitigates negative effects of low self-control and lessens impulsivity, risky behaviour engagement, criminal cognition, and adolescent crime rates (Khetarpal et al., 2021; Squillaro & Bixter, 2025; Simmons et al., 2019). Importantly, Walters (2019) identified a gradual decrease in criminal behaviour with maturation, associated with increased future orientation and decreased adolescent hopelessness. Similarly, Wood et al. (2017) recognised that cognitive immaturity relates to engagement in aversive behaviours, and thereby suffering future negative life effects; promoting future orientation and positive childhood effects can mitigate these negative life effects and promote sustainability (Kübel et al., 2023; Cauffman et al., 2023; Mueller et al., 2023). Given the findings on future orientation, it is now important to consider the role it plays in criminal behaviour, and therefore how research can facilitate future orientation to prevent, mitigate or rehabilitate criminal behaviour. ;Quiz Choose your answers and click "Submit": <quiz display=simple> {A variety of influences contribute to future orientation: |type="(+)"} + True - False {Future-connectedness is defined by behaviours and cognition involving the engagement towards a person’s future: |type="(+)"} + True - False {Disconnect from the future self has been shown to have no aversive risks: |type="(+)"} + True - False {Hopelessness is a persistent and serious outcome of adolescence: |type="(+)"} + True - False {Future orientation is ''the'' primary indicator for engagement with criminal behaviour: |type="(-)"} + True - False </quiz> ==Key Points== * Future orientation is the tendency for a person to engage in thoughts and actions in favour of future goals. *Future-connectedness is defined by behaviours and cognition involving engagement towards a person’s future in all forms. *The severity of hopelessness in adolescence, and its high relation to adverse behaviours, indicates the importance of minimising adolescent hopelessness. * Higher levels of future-oriented cognition indicate further consideration of long-term consequences of offending. * Limited future orientation correlates with increased criminal behaviour. =Criminal Behaviour= Dent and Ward (2022) conceptualised criminal behaviour as ‘an action or set of actions... understood by the actor to be adaptive in relation to their functional needs, but...make it more difficult to function adaptively...to other sociocultural needs.’ Criminal behaviour can include minor crimes, such as ‘theft or traffic violations’, and major crimes, such as ‘physical abuse, sexual offense [sic], or murder’ (Prent et al., 2023), and is considered a ‘major social problem with complex causes... [and involves] a myriad of environmental, social, and psychological factors are associated with increased risk of convictions for this type of criminality’ (Prent et al., 2023; Levitt, 2012; Swaab & Meynen, 2023). ==Influences on Criminal Behaviour== Reiterating findings from Prent et al. (2023), criminal behaviour is shaped by many complex, interconnected facets, including social, psychological, biological, and environmental factors, each with influential components such as geographical location and socio-economic status. Importantly, future orientation has shown positive effects in minimising negative behaviour across different contributing factors (Giollabhui et al., 2018). ==Psychological Influences== In media and history, psychology has often been treated as the primary explanation for criminal acts; however, crime is not limited to a single discipline (South Australia News, 2026). While this is well established, it is often a misconception that one factor is more influential than the other. A longstanding theory explaining human behaviour, cognition, and reasoning is Nature v Nurture. Nurture, as a psychological construct, argues that a person’s circumstances stem from upbringing and environment rather than other factors. Hadsell (2026) stated that the ‘connection between parents and children has two pathways: biological, covering both genes and hormones, and environmental.’ ==Biological Influences== When discussing the Nature concept, it is theorised that an individual's biological makeup is the sole predictor of behaviour (Alper, 1995). Baschetti (2008) argued that genetic evidence proved Nature over Nurture. Drawing on Charles Darwin’s theories of evolution and biology, current and overwhelming research indicates that criminality is heritable, yet persistent (Baschetti, 2008; Pizzi, 2004). It should be noted that Nature can refer to the genetic contribution within mental illness. Rao (2007) argued that all humans have a ‘dark side that loves crime and violence... [while] we may all deny it, but the contrary is true...regardless of our age, caste, social status, region, religion or education.’ [''APA style: Direct quotes should use double quotation marks and include page numbers''] Relevant to crime rates, it can be reasoned that modern-day society is the ‘most violent culture in history in the number of crimes and...the nature of brutality.’ Ling et al. (2019) explored how psychophysiology, brain mechanisms, and genetics stimulate criminal behaviour. Particularly, disrupted prefrontal cortex-amygdala connectivity and changes in autonomic functioning damaging the hypothalamic-pituitary-adrenal axis correlate with increased criminal and anti-social tendencies (Gillespie et al., 2018; Critchley, 2005; Brzozowski & Mitchell, 2018) [''APA style: Arrange multiple citations in alphabetical order'']. Baschetti (2008) signified the importance of systematic changes acknowledging the existence of a biological influence to encourage appropriate reforms and minimise negative outcomes. ==Relationship Between Criminal Behaviour and Future Orientation== {{expand}} ==Does Future Orientation Influence the Risk of Criminal Activity?== Future orientation is an incredibly [''Avoid overly emotive language''] important skill, particularly for adolescents or those experiencing hopelessness. A lack of future-oriented cognition can exacerbate impulsive and risky behaviour and may lead to criminal behaviour (Squillaro & Bixter, 2025; Mueller et al., 2023). Given adolescents' vulnerability to criminal behaviours (Bui & Deakin, 2021) and hopelessness (Bolland, 2003; Tonkus et al., 2022), encouraging future orientation is highly relevant. Reiterating that future orientation positively relates to curbing adolescent criminal activity and juvenile incarceration (Walters, 2019; Simmons et al., 2019) reinforces future orientation as an imperative protective factor across the lifespan (Yang et al., 2026; Clinkinbeard, 2014). ==Book Chapter Direction== * Environmental influences in criminal activity. *Social influences in criminal activity. *Understanding the relationship between future orientation and criminal activity. *Protective factors within future orientation *Risk factors within future orientation *Future research directions ==Conclusion== ''''' Include key sentence on the importance of understanding future orientation in criminal behaviour''.''' Currently, there is substantial research into the effects and relationships of future orientation and adolescent criminal activity. However, conclusive findings on the relevance of using future orientation to mitigate criminal behaviour remain limited. Given the strong positive correlations between future-oriented thinking and reduced disturbances in life quality, it would be unwise to ignore how future orientation can help prevent rising crime rates among adolescents ''and'' adults. ==See also== {{expand}} ==References== {{Hanging indent|1= Alper, J. S. (1995). Biological influences on criminal behaviour: How good is the evidence? *BMJ, 310*(6975), 272–273. https://doi.org/10.1136/bmj.310.6975.272 Arora, S. (2023). *“It’s a future-me problem”: How future-self continuity and message framing affect climate change risk perceptions* [Doctoral dissertation, University of Guelph]. Atrium. https://atrium.lib.uoguelph.ca/items/4d97286e-5635-45b6-a1c3-e9be464b1e7c Bar, M. (2009). The proactive brain: Memory for predictions. *Philosophical Transactions of the Royal Society B: Biological Sciences, 364*(1521), 1235–1243. https://doi.org/10.1098/rstb.2008.0310 Baschetti, R. (2008). Genetic evidence that Darwin was right about criminality: Nature, not nurture. *Medical Hypotheses, 70*(6), 1092–1102. https://doi.org/10.1016/j.mehy.2008.01.005 Becker-Weidman, E. G., Jacobs, R. H., Reinecke, M. A., Silva, S. G., & March, J. S. (2010). Social problem-solving among adolescents treated for depression. *Behaviour Research and Therapy, 48*(1), 11–18. https://doi.org/10.1016/j.brat.2009.08.006 Benoit, R. G., Gilbert, S. J., & Burgess, P. W. (2011). A neural mechanism mediating the impact of episodic prospection on farsighted decisions. *The Journal of Neuroscience, 31*(18), 6771–6779. https://doi.org/10.1523/JNEUROSCI.6559-10.2011 Bickel, W. K., Athamneh, L. N., Basso, J. C., Mellis, A. M., DeHart, W. B., Craft, W. H., & Pope, D. (2019). Excessive discounting of delayed reinforcers as a trans-disease process: Update on the state of the science. *Current Opinion in Psychology, 30*, 59–64. https://doi.org/10.1016/j.copsyc.2019.01.005 Bolland, J. M. (2003). Hopelessness and risk behaviour among adolescents living in high-poverty inner-city neighbourhoods. *Journal of Adolescence, 26*, 145–158. https://doi.org/10.1016/S0140-1971(02)00136-7 Bui, L., & Deakin, J. (2021). What we talk about when we talk about vulnerability and youth crime: A narrative review. *Aggression and Violent Behavior, 58*, 101605. https://doi.org/10.1016/j.avb.2021.101605 Cauffman, E., Gillespie, M. L., Beardslee, J., Davis, F., Hernandez, M., & Williams, T. (2023). Adolescent contact, lasting impact? Lessons learned from two longitudinal studies spanning 20 years of developmental science research with justice-system-involved youths. *Psychological Science in the Public Interest, 24*(3), 133–161. https://doi.org/10.1177/15291006231205173 Chadwick, R. (n.d.). Genetics and crime. In M. Levitt (Ed.), *Encyclopedia of applied ethics* (2nd ed., pp. 462–469). Academic Press. https://doi.org/10.1016/B978-0-12-373932-2.000375 Clinkinbeard, S. S. (2014). What lies ahead: An exploration of future orientation, self-control, and delinquency. *Criminal Justice Review, 39*(1), 19–36. https://doi.org/10.1177/0734016813501193 Critchley, H. D. (2005). Neural mechanisms of autonomic, affective, and cognitive integration. *The Journal of Comparative Neurology, 493*(1), 154–166. https://doi.org/10.1002/cne.20749 Dent, H., & Ward, T. (2022). Emotion and correctional practice: Theoretical foundations of an enactive practice framework. *Aggression and Violent Behavior, 63*, 101672. https://doi.org/10.1016/j.avb.2021.101672 Faura-Garcia, J., Calvete, E., & Orue, I. (2024). Longitudinal associations between nonsuicidal self-injury, depressive symptoms, hopelessness, and emotional dysregulation in adolescents. *Archives of Suicide Research, 28*(3), 800–814. https://doi.org/10.1080/13811118.2023.2237075 Gillespie, S. M., Brzozowski, A., & Mitchell, I. J. (2018). Self-regulation and aggressive antisocial behaviour: Insights from amygdala-prefrontal and heart-brain interactions. *Psychology, Crime & Law, 24*(3), 243–257. https://doi.org/10.1080/1068316X.2017.1414816 Gilbert, D. T., & Wilson, T. D. (2007). Prospection: Experiencing the future. *Science, 317*(5843), 1351–1354. https://doi.org/10.1126/science.1144161 Goger, P., Nam, R. J., Lowry, N., Bell, K. A., Parvez, N., Pollak, O. H., Robinaugh, D. J., Schacter, D. L., & Cha, C. B. (2024). Testing the modifiability of episodic future thinking and episodic memory among suicidal and nonsuicidal adolescents. *JCPP Advances, 4*(3), e12236. https://doi.org/10.1002/jcv2.12236 Grekin, E. R., Thomas, H. A., Souweidane, M. A., & Stidham, J. L. (2025). The effects of future self-continuity interventions on behavioral outcomes in adults: A systematic review of the literature. *Personality Science, 6*. https://doi.org/10.1177/27000710251391610 Hadsell, L. (2026). Parents: Nature and nurture. In *Social economies*. Palgrave Macmillan. https://doi.org/10.1007/978-3-032-21916-9_18 Hershfield, H. E. (2011). Future self-continuity: How conceptions of the future self transform intertemporal choice. *Annals of the New York Academy of Sciences, 1235*, 30–43. https://doi.org/10.1111/j.1749-6632.2011.06201.x James, S., Reddy, S. P., Ellahebokus, A., Sewpaul, R., & Naidoo, P. (2017). The association between adolescent risk behaviours and feelings of sadness or hopelessness: A cross-sectional survey of South African secondary school learners. *Psychology, Health & Medicine, 22*(7), 778–789. https://doi.org/10.1080/13548506.2017.1300669 Johnson, S. R., Blum, R. W., & Cheng, T. L. (2014). Future orientation: A construct with implications for adolescent health and wellbeing. *International Journal of Adolescent Medicine and Health, 26*(4), 459–468. https://doi.org/10.1515/ijamh-2013-0333 Khetarpal, S. K., Szoko, N., Ragavan, M. I., & Culyba, A. J. (2021). Future orientation as a cross-cutting protective factor against multiple forms of violence. *The Journal of Pediatrics, 235*, 288–291. https://doi.org/10.1016/j.jpeds.2021.05.019 Kübel, L. S., Deitzer, R. J., Frankenhuis, E. W., Ribeaud, D., Eisner, P. M., & van Gelder, J.-L. (2023). The shortsighted victim: Short-term mindsets mediate the link between victimization and later offending. *Journal of Criminal Justice, 86*, 102062. https://doi.org/10.1016/j.jcrimjus.2023.102062 Kwon, K. Y., Sutin, A. R., & Kim, J. (2025). Juvenile delinquency and cognitive function in adulthood: Differentiating violent and nonviolent behaviors and exploring multiple mechanisms. *Journal of Research on Adolescence, 35*(4), e70095. https://doi.org/10.1111/jora.70095 Lester, D. (2015). Hopelessness in adolescents. *Journal of Affective Disorders, 173*, 221–225. https://doi.org/10.1016/j.jad.2014.10.048 Ling, S., Umbach, R., & Raine, A. (2019). Biological explanations of criminal behavior. *Psychology, Crime & Law, 25*(6), 626–640. https://doi.org/10.1080/1068316X.2019.1572753 Mac Giollabhui, N., Hamilton, J. L., Nielsen, J., Connolly, S. L., Stange, J. P., Varga, S., Burdette, E., Olino, T. M., Abramson, L. Y., & Alloy, L. B. (2018). Negative cognitive style interacts with negative life events to predict first onset of a major depressive episode in adolescence via hopelessness. *Journal of Abnormal Psychology, 127*(1), 1–11. https://doi.org/10.1037/abn0000301 McCue, R., McCormack, T., McElnay, J., Alto, A., & Feeney, A. (2019). The future and me: Imagining the future and the future self in adolescent decision making. *Cognitive Development, 50*, 142–156. https://doi.org/10.1016/j.cogdev.2019.04.001 Mueller, K. C., Cavitt, J., & Carey, M. T. (2023). Future orientation in justice-involved youth: The effects of adverse and positive childhood experiences, and being dual-status. *Child Abuse & Neglect, 141*, 106230. https://doi.org/10.1016/j.chiabu.2023.106230 Nurmi, E. J. (1991). How do adolescents see their future? A review of the development of future orientation and planning. *Developmental Review, 11*(1), 1–59. https://doi.org/10.1016/0273-2297(91)90002-6 Peterson, M. F. (2004). [Review of the book *Culture, leadership and organizations: The GLOBE study of 62 societies*, by R. J. House, M. Javidan, P. W. Dorfman, V. Gupta, & P. J. Hanges]. *Administrative Science Quarterly, 49*(4), 641–647. https://www.jstor.org/stable/4131493 Pizzi, W. J. (2004). Nature via nurture: Genes, experience, and what makes us human. *Journal of Undergraduate Neuroscience Education, 2*(2), R10–R11. Prent, N., Jonker, A. F., Schouws, N. T. M. S., & Jonker, C. (2023). The risk of criminal behavior in the elderly and patients with neurodegenerative disease. In H. Swaab & G. Meynen (Eds.), *Handbook of clinical neurology* (Vol. 197, pp. 181–196). Elsevier. https://doi.org/10.1016/B978-0-12-821375-9.00004-9 Sathyanarayana Rao, T. S. (2007). Psychiatrist and the science of criminology: Sociological, psychological and psychiatric analysis of the dark side. *Indian Journal of Psychiatry, 49*(1), 3–5. https://doi.org/10.4103/0019-5545.31511 Simmons, C., Rowan, Z., Knowles, A., Steinberg, L., Frick, P. J., & Cauffman, E. (2019). A life history approach to understanding juvenile offending and aggression. *Aggression and Violent Behavior, 49*, 101317. https://doi.org/10.1016/j.avb.2019.07.012 South Australia News. (2026, April 25). *Psychology of a criminal: Why do people commit crimes?* https://sanews.in/psychology-of-a-criminal/ Squillaro, D., & Bixter, M. T. (2025). The relationship between four indicators of future time orientation, criminal thinking style, and illegal behaviors. *Journal of Forensic Sciences, 70*(1), 200–214. https://doi.org/10.1111/1556-4029.15640 Steinberg, L. (2004). Risk taking in adolescence: What changes, and why? *Annals of the New York Academy of Sciences, 1021*, 51–58. https://doi.org/10.1196/annals.1308.005 Swaab, H., & Meynen, G. (2023). Introduction: On brain and crime. In H. Swaab & G. Meynen (Eds.), *Handbook of clinical neurology* (Vol. 197, pp. 3–9). Elsevier. https://doi.org/10.1016/B978-0-12-821375-9.00018-9 Turton, H., Berry, K., Danquah, A., & Pratt, D. (2021). The relationship between emotion dysregulation and suicide ideation and behaviour: A systematic review. *Journal of Affective Disorders Reports, 5*, 100136. https://doi.org/10.1016/j.jadr.2021.100136 Walters, G. D. (2019). Maturing out of delinquency: Unpacking the effects of identity achievement and future orientation on crime desistance. *Self and Identity, 18*(3), 267–283. https://doi.org/10.1080/15298868.2018.1439848 Wager, T. D., Waugh, C. E., Lindquist, M., Noll, D. C., Fredrickson, B. L., & Taylor, S. F. (2009). Brain mediators of cardiovascular responses to social threat: Part I: Reciprocal dorsal and ventral sub-regions of the medial prefrontal cortex and heart-rate reactivity. *NeuroImage, 47*(3), 821–835. https://doi.org/10.1016/j.neuroimage.2009.05.043 Wood, D., Crapnell, T., Lau, L., et al. (2017). Emerging adulthood as a critical stage in the life course. In N. Halfon, C. B. Forrest, R. M. Lerner, et al. (Eds.), *Handbook of life course health development*. Springer. https://www.ncbi.nlm.nih.gov/books/NBK543712/ Yang, E. S., Kim, S., Oh, S.-K., Kang, H., & Chung, I.-J. (2026). Disrupting the link between exposure to violence and adolescent delinquency: The protective role of expectations about the future. *Journal of Child and Family Studies*. https://doi.org/10.1007/s10826-026-03311-3 }} ==External links== {{expand}} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Cognitive]] [[Category:Motivation and emotion/Book/Forensic]] ha0cl6qt4036w2qgn8jh7d2i32v4oe5 2842559 2842558 2026-10-25T11:11:24Z Isa Murphy 3105869 2842559 wikitext text/x-wiki {{title|Future orientation and criminal behaviour:<br>How does future orientation influence the risk of criminal activity?}} ==Overview== {{RoundBoxTop|theme=7}} [[File:Exams during Covid-19 - France (50651621763).jpg|thumb|200px|'''Figure 1'''. Exams during Covid-19.]] ; Acar and Kaya (2024) study Amongst 23 adolescent participants, psychological drivers, social and environmental influences, and cognitive and planning capacities were identified as primary categories influencing adolescent future orientation, alongside family, peers, educational access, and cultural norms contributing to the facilitation or hindering of future aspirations (Acar & Kaya, 2024). '''Psychological drivers''': 'hope and optimism, fear of failure, self-identity, motivation and ambition, and emotional regulation' (Acar & Kaya, 2024). '''Social and environmental''': 'parental support, peer influence, school environment, socioeconomic constraints, and community expectations' (Acar & Kaya, 2024). '''Cognitive and planning capacities''': 'goal-setting skills, time perspective, decision-making ability, problem-solving skills, information-seeking behavior [''sic''], and academic self-efficacy' (Acar & Kaya, 2024). {{RoundBoxBottom}} When focusing on adolescents, future orientation is a vital component of their education (Johnson et al., 2014). Conversely, adolescent crime rates have been steadily decreasing (Australian Bureau of Statistics [ABS], 2026), yet they remain a major concern (Hinchcliff et al., 2026; Australian Human Rights Commission [AHRC], 2024; World Health Organisation [WHO], 2026). {{RoundBoxTop|theme=2}} '''Focus questions''' {{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}} '''1.''' Why is Future Orientation Important? '''2.''' Can Future Orientation Mitigate Criminal Behaviour? '''3.''' Are Adolescents More Susceptible to Aversive Effects of Low Future Orientation? '''4.''' Does Criminal Behaviour Depend Upon Future-Oriented Cognition? {{RoundBoxBottom}} = Future Orientation = {{ic|Include at least an introductory paragraph before branching into sub-headings}} ==What is Future Orientation== According to House et al. (2004), future orientation is the tendency for a person to engage in thoughts, actions, and the delaying of immediate gratification in favour of future goals. Referring to adolescents, many disciplines have ratified the importance of future-oriented behaviour in promoting positive health and development (Johnson et al., 2016). Trommsdorff (9179) proposed that future orientation included eight dimensions: extension, detail, domain, affect, motivation, control, sequence of events, and number of cognitions. {{center top}} {| border=1 cellspacing=0 cellpadding=6 |- ! Dimension ! Definition |- | extension | bar |- | detail | bar |- | domain | bar |- | affect | bar |- | motivation | bar |- | control | bar |- | sequence of events | bar |- | number of cognitions | far |} {{center bottom}} A study by Beal (2011) revealed that Trommsdorff’s (1979) conceptualisation indicates that different influences can be indicators of future orientation and the eight dimensions. Findings consistently supported previous research that future orientation is an important predictor of adolescent well-being and career adaptability and revealed that ‘stronger future orientation and future-focused time perspectives are consistently associated with higher adolescent well-being, health-promoting habits (e.g., physical activity), and lower engagement in health-risk behaviours (e.g., substance use, risky sexual behaviour)’ (Kovács et al., 2026). Lindstrom Johnson et al. (2014) examines future orientation as a mediator or moderator of problem behaviour, determining that while conclusions should be interpreted cautiously, findings indicated future-oriented thinking as a mediator can be expressed as reasoning, but not the sole, for behavioural outcomes (Weems, 2025). Kovács et al. (2026) conducted a systematic review on the effects and different forms future orientation has on adolescents, focusing on future time perspective and future orientation. Findings on future time perspective revealed an interesting trend: that positivity-endorsing programmes on past behaviours, future-oriented planning, actively developing time perspective, and contextual health behaviour–focused prevention programmes greatly improve adolescent health and well-being (Kovács et al., 2026; Kuan & Zhang, 2022). Thus, the following article will inform adolescent future orientation as a preventative measure for criminal behaviour before it occurs. It should be noted that additional factors, such as drug-oriented perspectives, can explain insignificant positive effects in future-oriented programmes. Considering the major transitional stage that is adolescence (Khurana et al., 2015), the enticing and control-eliciting nature of drugs and alcohol, and social and environmental influences on various behaviours (Barnett et al., 2013; Zhang et al., 2021), potential negative effects resulting from illicit drug and alcohol use are plausible (Finan et al., 2022; Peters et al., 2005; Jones et al., 2021) and should be recognised when discussing and assessing adolescent future-oriented thinking (Geier et al., 2013; Stoddard et al., 2018). ==Influences on Future Orientation== Multidisciplinary findings have indicated that a variety of influences contribute to future orientation. Much psychological research has focused on adolescents and young adults, so findings are skewed, though still extensive and reliable. ===The BioPsychoSocial Model=== In 1977, George Engel proposed an extension to the biomedical model that incorporated the psychological and social factors influencing health and wellbeing, known as the BioPsychoSocial Model (BPSM). Vögele (2015) defines BPSM as ‘a general model positing that biological, psychological (which includes thoughts, emotions, and behaviors [sic]), and social (e.g., socioeconomical, socioenvironmental, and cultural) factors, all play a significant role in health. …[and] are best understood in terms of a combination…rather than purely in biological terms.’ It is important to recognise that biological, psychological, and social influences are individual yet equally important additions to health and wellbeing with a variety of crossover (Dfarhud et al., 2014; Chu et al., 2024; Candeias et al., 2024; Kahn et al., 2024). For example, Subramanyam et al. (2024) studied the effects of neurodevelopment, epigenetics, parenting, communication skills, career-focused future orientation, and the environment on child and adolescent wellbeing, with findings indicating each component held a myriad of interplaying causes and effects. Regarding future orientation, multidisciplinary findings have indicated that a variety of influences contribute to future orientation. It is important to note that much of the psychological research has targeted adolescents and young adults, and therefore findings are skewed, yet extensive and reliable. ====Psychological Influences==== McCue et al. (2019) researched and measured the effects of future orientation on adolescent decision-making. Previously, connectedness to future-self and episodic future thinking demonstrates future-oriented cognition, and while their correlation is undetermined, individual measures have shown significant effects in minimising hazardous decision-making in both adolescents and adults (Grekin et al., 2025; Goger et al., 2024; Arora, 2023). Future-connectedness is defined by behaviours and cognition involving the active imagination, verbal and physical communication, and other related engagements towards a person’s future in all forms. Disconnect from the future self has been shown to increase irrational, short-sighted, and risky behaviours deemed beneficial in the present (Hershfield, 2011). Episodic future thinking is defined as the mental stimulation experienced towards possible future occurrences, and supports future-orientation and highlights the importance of delaying rewards and countering impulsive behaviours (Gilbert & Wilson, 2007; Benoit et al., 2011; Bar, 2009). This is particularly important as future-oriented thinking has historically shown its profound necessity amongst adolescents (Nurmi, 1991), and a lower predilection for impulsive and risky behaviour in favour of future rewards (Steinberg, 2004; Madden & Bickel, 2012). McCue et al. (2019) found a significant positive correlation between future-connectedness and episodic future thinking, claiming to be one of, if not the first, study to identify a beneficial relationship and independent effects within adolescents. Giollabhui et al. (2018) found that increased future-oriented behaviours and cognition contribute to reduced adolescent hopelessness. During adolescence, hopelessness is categorised by adamant negative future expectations and vulnerability (Casey et al., 2008; Lester, 2015). The severity of hopelessness in adolescence, and its relation to increased engagement in violent and/or high-risk behaviours, severe depression and anxiety, and unsafe sexual conduct (Becker-Weidman et al., 2009; Bolland, 2003; James et al., 2017; Giollabhui et al., 2018), indicates the importance of minimising adolescent hopelessness. Suicide has been a primary cause of death amongst adolescents for many years, skyrocketing after COVID-19 (Kim et al., 2024), and highlighting the worrisome trend. Considering the relationship between hopelessness, future-orientation, and adolescent depression and suicidal behaviours, increasing positive cognitions actively declines time-limited rates of hopelessness across confounds, including socio-economic status and stressors (Giollabhui et al., 2018; Faura-Garcia et al., 2024). ====Biological Influences==== Biological aspects of future orientation are currently under-researched, likely due to difficulty in quantifying results. Gan et al. (2024) investigated effects of the Colony-Stimulating Factor 3 Receptor (CSF3R) polymorphism and early stress on future orientation amongst 14,675 Chinese participants. Participants self-reported personal occurrences of maltreatment throughout childhood. Results indicated the CSF3R gene, which is closely related to chronic psychological stress responses, greatly interacted with early trauma’s influence on future orientation, specifically as levels of maltreatment increased (Gan et al., 2024). Of great importance to this chapter, childhood maltreatment demonstrated higher levels of stress in comparison to other stressors. This can either deplete opportunities to grow or encourage stress-related improvement in response to triggering situations (Gan et al., 2024), particularly for those with certain genotypes of the CSF3R (Drury et al., 2010). Zheng et al. (2022) explored the heritability of future orientation, identifying that genetic influence and mutual enviornments significantly affected future orientation development amongst adolescents. Findings revealed a negative effect between age and genetic influence, and a positive effect between age and individual environments on future orientation, as well as no effect between male and females (Zheng et al., 2022). Conversely, Greene et al. (2004) found that historical gender roles and expectations influenced future orientation, with females maintaining interpersonal goals, whereas males had further extension of goals. Importantly, the astronomical effects of environments on adolescent future orientation development and health and wellbeing, unrestricted from socio-cultural contexts, such as race, gender, and religion (Greene et al., 2004). ====Social And Environmental Influences==== Social influence can be argued as the largest influence on future orientation due to the proximity of environmental situations throughout childhood and adolescence (Laursen & Faur, 2022). Research has shown socioeconomic determinants positively correlated to the future orientation of adolescents (Nurmi, 1991), particularly higher parental and personal education (Borowsky et al., 2009; Nguyen et al., 2012), whereas neighbourhood poverty and racial minorities demonstrate weaker future-oriented thinking (Kerpelman et al., 2004; Guthrie et al., 2009). Johnson et al. (2016) explored how parenting style, including acceptance and involvement, attachment security, and autonomy granting, has consistently demonstrated positive effects on adolescent future-oriented thinking. Interestingly, Kai and Tienda (1998) found that adolescent future orientation as represented through college aspirations did not rely on race as it did for other influencing factors. Chen and Vazsonyi (2013) conducted a multilevel study on the effects of schooling on future-oriented thought processes amongst nine thousand 9th to 12th graders. Findings showed adolescent problem behaviours negatively impacted future-oriented thinking, especially those in larger cohorts and schools with higher positivity towards future orientation (Chen & Vazsonyi, 2013). Interestingly, students with a low sense of future orientation experienced opposite effects than those with problem behaviours, displaying a significant effect from positive future orientation attitudes at school (Chen & Vazsonyi, 2013). Rydell and Brocki (2024) found similar effects, with problem behaviours and inattention around mid-adolescence negatively affecting future orientation. However, high levels of hyperactivity and positive social interactions and relationships had compensatory properties, improving future orientation. Continuing research on adolescence, Skinner et al. (2022) investigated the effects of parental authority on future influence throughout COVID-19. Parental control is known to have an influence on adolescent wellbeing and future orientation (Wang et al., 2024; Zhu et al., 2026; Roman et al., 2015). For instance, psychological control is characterised by ‘intrusive and manipulative behaviors [sic] aimed at children or adolescents’ thoughts and feelings, through which adults exert power by controlling their children’s psychological world’ (Pérez et al., 2021), and can hinder adolescent development and wellbeing (Mandara & Pikes, 2008; Xu and Ren, 2025), and is connected to the development of depressive symptoms (Skinner et al., 2022). Data collected from adolescents and their parents across three years from nine countries, accounting for confounding factors, demonstrated the importance of positive normative standards for future orientation through parent-child interactions, and an educated, targeted approach to overcome adverse effects of childhood (Merrick et al., 2017; Kerpelman et al. 2008), such as an ecological systems approach (Rosanbalm et al. 2020). Wang et al. (2024) reiterated these points, finding that parental control significantly influences future orientation and positively affects perceived significance of life, future orientation, and personal growth initiative. Interestingly, psychological control had the only negative effect on future orientation and no effect on perceived significance of life. Given these findings, the social influences on future orientation are of the utmost importance and should be diligently recognised for positive effects to occur (Rydell et al., 2024; Crespo et al., 2013), particularly in adolescents and the objective to minimise bad behaviour (Chen & Vazsonyi, 2011; Voisin et al., 2020) and decrease hopelessness (Giollabhui et al., 2018). It should be noted that social influences on adolescent health and wellbeing are connected to psychological wellbeing and perceived significance of life, and therefore can be argued as the component of the BPSM that takes priority, but not as most important (Long et al., 2021; Webster et al., 2021; World Health Organization [WHO], 2025). ===All Three Components=== The BPSM aims to put the ‘nature vs nurture’ debate to rest by highlighting primary considerations as equal contributors to behaviour. Within the ‘Nature’ conversation, particularly neuroscience, researchers dictate that ‘the capacity of the brain to change its connections. …must be the consequence of having molecular mechanisms that mediate those changes’. (United States Institute of Medicine [USIM]. Beauchaine et al. (2009) provided ten reasons to consider biological processes in psychological prevention and intervention in research, including biomarkers and endophenotypes in identifying priority prevention methods, how biological vulnerabilities moderate environmental experiences and differences, and interactions between environmental and neural systems sensitivity across lifespans. Similarly, Marder and Miller (2024) discuss how biological psychology is the focal point of psychophysiology, investigating the effects of biological markers in behaviour, accompanied by the influences of social trends creates the foundation of individual disposition. Eriksson et al. (2025) corroborated the multilevel consequences on behaviour and wellbeing, stating ‘It is essential to move beyond focusing solely on individual psychological and cognitive factors to an understanding of the complex processes involved in health behaviour change.’ Findings illustrated variance between health behaviours for different health identities amongst middle-aged men and women, guiding future socio-ecological interventions promoting healthier behaviours (Eriksson et al., 2025). Regarding the BPSM within future orientation and criminal behaviour, each element attracts differing levels of authority from many sources. For example, the majority of research in future orientation revolves around environmental influences (Wittmann & Sircova, 2018), compared to psychological and biological components. Conversely, criminal behaviour prioritises biological and psychological influences (Miller, 2014; Focquaert, 2019). ===Why is this particularly important for adolescents?=== Kovács et al. (2026) detail that future orientation is “not a singular construct but a multifaceted phenomenon operationalized [sic] through cognitive, affective, and contextual lenses”. This is particularly important for adolescents, as biological, social, and psychological factors interact dynamically, with each influencing the others and shaping development (Lindstrom Johnson et al., 2014). The BPSM is particularly relevant as stronger future orientation is associated with adolescent wellbeing and healthier behaviours, including greater engagement in health-promoting activities and reduced risk-taking, although these relationships vary across contexts (Kovács et al., 2026). Future orientation may support these outcomes by strengthening adolescents’ capacity to establish meaningful goals, anticipate the consequences of their decisions, and regulate present behaviour in pursuit of longer-term outcomes (Bandura, 2001; Nurmi, 1991). However, the extent to which future-oriented thinking translates into behaviour depends on individual motivation and adolescents’ perceived opportunities, social relationships, and environmental circumstances (Lindstrom Johnson et al., 2014). Future orientation should not be considered as universally protective, as an intense focus on future outcomes may also be associated with anxiety or heightened pressure to achieve in certain contexts (Kovács et al., 2026). This indicates the importance of the BPSM in future orientation, implying that no singular factor is the sole cause of development, and interventions should aim at promoting adolescent wellbeing beyond encouraging long-term goal setting to include the development of self-regulatory skills and supportive environments that enable adolescents to pursue their aspirations (Nurmi, 1991; Lindstrom Johnson et al., 2014). === Future Orientation’s Relevance to Criminal Activity === Relevant to criminal behaviour, higher levels of future-oriented cognition indicate further consideration of long-term consequences of offending (Kwon et al., 2025); in contrast, limited future orientation correlates with increased criminal behaviour by encouraging immediate gratification and reducing consideration of delayed consequences (Kübel et al., 2023). Clinkinbeard (2014) found that high future orientation and self-control predict adolescent criminal behaviour; particularly, higher future-oriented cognition mitigates negative effects of low self-control and lessens impulsivity, risky behaviour engagement, criminal cognition, and adolescent crime rates (Khetarpal et al., 2021; Squillaro & Bixter, 2025; Simmons et al., 2019). Importantly, Walters (2019) identified a gradual decrease in criminal behaviour with maturation, associated with increased future orientation and decreased adolescent hopelessness. Similarly, Wood et al. (2017) recognised that cognitive immaturity relates to engagement in aversive behaviours, and thereby suffering future negative life effects; promoting future orientation and positive childhood effects can mitigate these negative life effects and promote sustainability (Kübel et al., 2023; Cauffman et al., 2023; Mueller et al., 2023). Given the findings on future orientation, it is now important to consider the role it plays in criminal behaviour, and therefore how research can facilitate future orientation to prevent, mitigate or rehabilitate criminal behaviour. ;Quiz Choose your answers and click "Submit": <quiz display=simple> {A variety of influences contribute to future orientation: |type="(+)"} + True - False {Future-connectedness is defined by behaviours and cognition involving the engagement towards a person’s future: |type="(+)"} + True - False {Disconnect from the future self has been shown to have no aversive risks: |type="(+)"} + True - False {Hopelessness is a persistent and serious outcome of adolescence: |type="(+)"} + True - False {Future orientation is ''the'' primary indicator for engagement with criminal behaviour: |type="(-)"} + True - False </quiz> ==Key Points== * Future orientation is the tendency for a person to engage in thoughts and actions in favour of future goals. *Future-connectedness is defined by behaviours and cognition involving engagement towards a person’s future in all forms. *The severity of hopelessness in adolescence, and its high relation to adverse behaviours, indicates the importance of minimising adolescent hopelessness. * Higher levels of future-oriented cognition indicate further consideration of long-term consequences of offending. * Limited future orientation correlates with increased criminal behaviour. =Criminal Behaviour= Dent and Ward (2022) conceptualised criminal behaviour as ‘an action or set of actions... understood by the actor to be adaptive in relation to their functional needs, but...make it more difficult to function adaptively...to other sociocultural needs.’ Criminal behaviour can include minor crimes, such as ‘theft or traffic violations’, and major crimes, such as ‘physical abuse, sexual offense [sic], or murder’ (Prent et al., 2023), and is considered a ‘major social problem with complex causes... [and involves] a myriad of environmental, social, and psychological factors are associated with increased risk of convictions for this type of criminality’ (Prent et al., 2023; Levitt, 2012; Swaab & Meynen, 2023). ==Influences on Criminal Behaviour== Reiterating findings from Prent et al. (2023), criminal behaviour is shaped by many complex, interconnected facets, including social, psychological, biological, and environmental factors, each with influential components such as geographical location and socio-economic status. Importantly, future orientation has shown positive effects in minimising negative behaviour across different contributing factors (Giollabhui et al., 2018). ==Psychological Influences== In media and history, psychology has often been treated as the primary explanation for criminal acts; however, crime is not limited to a single discipline (South Australia News, 2026). While this is well established, it is often a misconception that one factor is more influential than the other. A longstanding theory explaining human behaviour, cognition, and reasoning is Nature v Nurture. Nurture, as a psychological construct, argues that a person’s circumstances stem from upbringing and environment rather than other factors. Hadsell (2026) stated that the ‘connection between parents and children has two pathways: biological, covering both genes and hormones, and environmental.’ ==Biological Influences== When discussing the Nature concept, it is theorised that an individual's biological makeup is the sole predictor of behaviour (Alper, 1995). Baschetti (2008) argued that genetic evidence proved Nature over Nurture. Drawing on Charles Darwin’s theories of evolution and biology, current and overwhelming research indicates that criminality is heritable, yet persistent (Baschetti, 2008; Pizzi, 2004). It should be noted that Nature can refer to the genetic contribution within mental illness. Rao (2007) argued that all humans have a ‘dark side that loves crime and violence... [while] we may all deny it, but the contrary is true...regardless of our age, caste, social status, region, religion or education.’ [''APA style: Direct quotes should use double quotation marks and include page numbers''] Relevant to crime rates, it can be reasoned that modern-day society is the ‘most violent culture in history in the number of crimes and...the nature of brutality.’ Ling et al. (2019) explored how psychophysiology, brain mechanisms, and genetics stimulate criminal behaviour. Particularly, disrupted prefrontal cortex-amygdala connectivity and changes in autonomic functioning damaging the hypothalamic-pituitary-adrenal axis correlate with increased criminal and anti-social tendencies (Gillespie et al., 2018; Critchley, 2005; Brzozowski & Mitchell, 2018) [''APA style: Arrange multiple citations in alphabetical order'']. Baschetti (2008) signified the importance of systematic changes acknowledging the existence of a biological influence to encourage appropriate reforms and minimise negative outcomes. ==Relationship Between Criminal Behaviour and Future Orientation== {{expand}} ==Does Future Orientation Influence the Risk of Criminal Activity?== Future orientation is an incredibly [''Avoid overly emotive language''] important skill, particularly for adolescents or those experiencing hopelessness. A lack of future-oriented cognition can exacerbate impulsive and risky behaviour and may lead to criminal behaviour (Squillaro & Bixter, 2025; Mueller et al., 2023). Given adolescents' vulnerability to criminal behaviours (Bui & Deakin, 2021) and hopelessness (Bolland, 2003; Tonkus et al., 2022), encouraging future orientation is highly relevant. Reiterating that future orientation positively relates to curbing adolescent criminal activity and juvenile incarceration (Walters, 2019; Simmons et al., 2019) reinforces future orientation as an imperative protective factor across the lifespan (Yang et al., 2026; Clinkinbeard, 2014). ==Book Chapter Direction== * Environmental influences in criminal activity. *Social influences in criminal activity. *Understanding the relationship between future orientation and criminal activity. *Protective factors within future orientation *Risk factors within future orientation *Future research directions ==Conclusion== ''''' Include key sentence on the importance of understanding future orientation in criminal behaviour''.''' Currently, there is substantial research into the effects and relationships of future orientation and adolescent criminal activity. However, conclusive findings on the relevance of using future orientation to mitigate criminal behaviour remain limited. Given the strong positive correlations between future-oriented thinking and reduced disturbances in life quality, it would be unwise to ignore how future orientation can help prevent rising crime rates among adolescents ''and'' adults. ==See also== {{expand}} ==References== {{Hanging indent|1= Alper, J. S. (1995). Biological influences on criminal behaviour: How good is the evidence? *BMJ, 310*(6975), 272–273. https://doi.org/10.1136/bmj.310.6975.272 Arora, S. (2023). *“It’s a future-me problem”: How future-self continuity and message framing affect climate change risk perceptions* [Doctoral dissertation, University of Guelph]. Atrium. https://atrium.lib.uoguelph.ca/items/4d97286e-5635-45b6-a1c3-e9be464b1e7c Bar, M. (2009). The proactive brain: Memory for predictions. *Philosophical Transactions of the Royal Society B: Biological Sciences, 364*(1521), 1235–1243. https://doi.org/10.1098/rstb.2008.0310 Baschetti, R. (2008). Genetic evidence that Darwin was right about criminality: Nature, not nurture. *Medical Hypotheses, 70*(6), 1092–1102. https://doi.org/10.1016/j.mehy.2008.01.005 Becker-Weidman, E. G., Jacobs, R. H., Reinecke, M. A., Silva, S. G., & March, J. S. (2010). Social problem-solving among adolescents treated for depression. *Behaviour Research and Therapy, 48*(1), 11–18. https://doi.org/10.1016/j.brat.2009.08.006 Benoit, R. G., Gilbert, S. J., & Burgess, P. W. (2011). A neural mechanism mediating the impact of episodic prospection on farsighted decisions. *The Journal of Neuroscience, 31*(18), 6771–6779. https://doi.org/10.1523/JNEUROSCI.6559-10.2011 Bickel, W. K., Athamneh, L. N., Basso, J. C., Mellis, A. M., DeHart, W. B., Craft, W. H., & Pope, D. (2019). Excessive discounting of delayed reinforcers as a trans-disease process: Update on the state of the science. *Current Opinion in Psychology, 30*, 59–64. https://doi.org/10.1016/j.copsyc.2019.01.005 Bolland, J. M. (2003). Hopelessness and risk behaviour among adolescents living in high-poverty inner-city neighbourhoods. *Journal of Adolescence, 26*, 145–158. https://doi.org/10.1016/S0140-1971(02)00136-7 Bui, L., & Deakin, J. (2021). What we talk about when we talk about vulnerability and youth crime: A narrative review. *Aggression and Violent Behavior, 58*, 101605. https://doi.org/10.1016/j.avb.2021.101605 Cauffman, E., Gillespie, M. L., Beardslee, J., Davis, F., Hernandez, M., & Williams, T. (2023). Adolescent contact, lasting impact? Lessons learned from two longitudinal studies spanning 20 years of developmental science research with justice-system-involved youths. *Psychological Science in the Public Interest, 24*(3), 133–161. https://doi.org/10.1177/15291006231205173 Chadwick, R. (n.d.). Genetics and crime. In M. Levitt (Ed.), *Encyclopedia of applied ethics* (2nd ed., pp. 462–469). Academic Press. https://doi.org/10.1016/B978-0-12-373932-2.000375 Clinkinbeard, S. S. (2014). What lies ahead: An exploration of future orientation, self-control, and delinquency. *Criminal Justice Review, 39*(1), 19–36. https://doi.org/10.1177/0734016813501193 Critchley, H. D. (2005). Neural mechanisms of autonomic, affective, and cognitive integration. *The Journal of Comparative Neurology, 493*(1), 154–166. https://doi.org/10.1002/cne.20749 Dent, H., & Ward, T. (2022). Emotion and correctional practice: Theoretical foundations of an enactive practice framework. *Aggression and Violent Behavior, 63*, 101672. https://doi.org/10.1016/j.avb.2021.101672 Faura-Garcia, J., Calvete, E., & Orue, I. (2024). Longitudinal associations between nonsuicidal self-injury, depressive symptoms, hopelessness, and emotional dysregulation in adolescents. *Archives of Suicide Research, 28*(3), 800–814. https://doi.org/10.1080/13811118.2023.2237075 Gillespie, S. M., Brzozowski, A., & Mitchell, I. J. (2018). Self-regulation and aggressive antisocial behaviour: Insights from amygdala-prefrontal and heart-brain interactions. *Psychology, Crime & Law, 24*(3), 243–257. https://doi.org/10.1080/1068316X.2017.1414816 Gilbert, D. T., & Wilson, T. D. (2007). Prospection: Experiencing the future. *Science, 317*(5843), 1351–1354. https://doi.org/10.1126/science.1144161 Goger, P., Nam, R. J., Lowry, N., Bell, K. A., Parvez, N., Pollak, O. H., Robinaugh, D. J., Schacter, D. L., & Cha, C. B. (2024). Testing the modifiability of episodic future thinking and episodic memory among suicidal and nonsuicidal adolescents. *JCPP Advances, 4*(3), e12236. https://doi.org/10.1002/jcv2.12236 Grekin, E. R., Thomas, H. A., Souweidane, M. A., & Stidham, J. L. (2025). The effects of future self-continuity interventions on behavioral outcomes in adults: A systematic review of the literature. *Personality Science, 6*. https://doi.org/10.1177/27000710251391610 Hadsell, L. (2026). Parents: Nature and nurture. In *Social economies*. Palgrave Macmillan. https://doi.org/10.1007/978-3-032-21916-9_18 Hershfield, H. E. (2011). Future self-continuity: How conceptions of the future self transform intertemporal choice. *Annals of the New York Academy of Sciences, 1235*, 30–43. https://doi.org/10.1111/j.1749-6632.2011.06201.x James, S., Reddy, S. P., Ellahebokus, A., Sewpaul, R., & Naidoo, P. (2017). The association between adolescent risk behaviours and feelings of sadness or hopelessness: A cross-sectional survey of South African secondary school learners. *Psychology, Health & Medicine, 22*(7), 778–789. https://doi.org/10.1080/13548506.2017.1300669 Johnson, S. R., Blum, R. W., & Cheng, T. L. (2014). Future orientation: A construct with implications for adolescent health and wellbeing. *International Journal of Adolescent Medicine and Health, 26*(4), 459–468. https://doi.org/10.1515/ijamh-2013-0333 Khetarpal, S. K., Szoko, N., Ragavan, M. I., & Culyba, A. J. (2021). Future orientation as a cross-cutting protective factor against multiple forms of violence. *The Journal of Pediatrics, 235*, 288–291. https://doi.org/10.1016/j.jpeds.2021.05.019 Kübel, L. S., Deitzer, R. J., Frankenhuis, E. W., Ribeaud, D., Eisner, P. M., & van Gelder, J.-L. (2023). The shortsighted victim: Short-term mindsets mediate the link between victimization and later offending. *Journal of Criminal Justice, 86*, 102062. https://doi.org/10.1016/j.jcrimjus.2023.102062 Kwon, K. Y., Sutin, A. R., & Kim, J. (2025). Juvenile delinquency and cognitive function in adulthood: Differentiating violent and nonviolent behaviors and exploring multiple mechanisms. *Journal of Research on Adolescence, 35*(4), e70095. https://doi.org/10.1111/jora.70095 Lester, D. (2015). Hopelessness in adolescents. *Journal of Affective Disorders, 173*, 221–225. https://doi.org/10.1016/j.jad.2014.10.048 Ling, S., Umbach, R., & Raine, A. (2019). Biological explanations of criminal behavior. *Psychology, Crime & Law, 25*(6), 626–640. https://doi.org/10.1080/1068316X.2019.1572753 Mac Giollabhui, N., Hamilton, J. L., Nielsen, J., Connolly, S. L., Stange, J. P., Varga, S., Burdette, E., Olino, T. M., Abramson, L. Y., & Alloy, L. B. (2018). Negative cognitive style interacts with negative life events to predict first onset of a major depressive episode in adolescence via hopelessness. *Journal of Abnormal Psychology, 127*(1), 1–11. https://doi.org/10.1037/abn0000301 McCue, R., McCormack, T., McElnay, J., Alto, A., & Feeney, A. (2019). The future and me: Imagining the future and the future self in adolescent decision making. *Cognitive Development, 50*, 142–156. https://doi.org/10.1016/j.cogdev.2019.04.001 Mueller, K. C., Cavitt, J., & Carey, M. T. (2023). Future orientation in justice-involved youth: The effects of adverse and positive childhood experiences, and being dual-status. *Child Abuse & Neglect, 141*, 106230. https://doi.org/10.1016/j.chiabu.2023.106230 Nurmi, E. J. (1991). How do adolescents see their future? A review of the development of future orientation and planning. *Developmental Review, 11*(1), 1–59. https://doi.org/10.1016/0273-2297(91)90002-6 Peterson, M. F. (2004). [Review of the book *Culture, leadership and organizations: The GLOBE study of 62 societies*, by R. J. House, M. Javidan, P. W. Dorfman, V. Gupta, & P. J. Hanges]. *Administrative Science Quarterly, 49*(4), 641–647. https://www.jstor.org/stable/4131493 Pizzi, W. J. (2004). Nature via nurture: Genes, experience, and what makes us human. *Journal of Undergraduate Neuroscience Education, 2*(2), R10–R11. Prent, N., Jonker, A. F., Schouws, N. T. M. S., & Jonker, C. (2023). The risk of criminal behavior in the elderly and patients with neurodegenerative disease. In H. Swaab & G. Meynen (Eds.), *Handbook of clinical neurology* (Vol. 197, pp. 181–196). Elsevier. https://doi.org/10.1016/B978-0-12-821375-9.00004-9 Sathyanarayana Rao, T. S. (2007). Psychiatrist and the science of criminology: Sociological, psychological and psychiatric analysis of the dark side. *Indian Journal of Psychiatry, 49*(1), 3–5. https://doi.org/10.4103/0019-5545.31511 Simmons, C., Rowan, Z., Knowles, A., Steinberg, L., Frick, P. J., & Cauffman, E. (2019). A life history approach to understanding juvenile offending and aggression. *Aggression and Violent Behavior, 49*, 101317. https://doi.org/10.1016/j.avb.2019.07.012 South Australia News. (2026, April 25). *Psychology of a criminal: Why do people commit crimes?* https://sanews.in/psychology-of-a-criminal/ Squillaro, D., & Bixter, M. T. (2025). The relationship between four indicators of future time orientation, criminal thinking style, and illegal behaviors. *Journal of Forensic Sciences, 70*(1), 200–214. https://doi.org/10.1111/1556-4029.15640 Steinberg, L. (2004). Risk taking in adolescence: What changes, and why? *Annals of the New York Academy of Sciences, 1021*, 51–58. https://doi.org/10.1196/annals.1308.005 Swaab, H., & Meynen, G. (2023). Introduction: On brain and crime. In H. Swaab & G. Meynen (Eds.), *Handbook of clinical neurology* (Vol. 197, pp. 3–9). Elsevier. https://doi.org/10.1016/B978-0-12-821375-9.00018-9 Turton, H., Berry, K., Danquah, A., & Pratt, D. (2021). The relationship between emotion dysregulation and suicide ideation and behaviour: A systematic review. *Journal of Affective Disorders Reports, 5*, 100136. https://doi.org/10.1016/j.jadr.2021.100136 Walters, G. D. (2019). Maturing out of delinquency: Unpacking the effects of identity achievement and future orientation on crime desistance. *Self and Identity, 18*(3), 267–283. https://doi.org/10.1080/15298868.2018.1439848 Wager, T. D., Waugh, C. E., Lindquist, M., Noll, D. C., Fredrickson, B. L., & Taylor, S. F. (2009). Brain mediators of cardiovascular responses to social threat: Part I: Reciprocal dorsal and ventral sub-regions of the medial prefrontal cortex and heart-rate reactivity. *NeuroImage, 47*(3), 821–835. https://doi.org/10.1016/j.neuroimage.2009.05.043 Wood, D., Crapnell, T., Lau, L., et al. (2017). Emerging adulthood as a critical stage in the life course. In N. Halfon, C. B. Forrest, R. M. Lerner, et al. (Eds.), *Handbook of life course health development*. Springer. https://www.ncbi.nlm.nih.gov/books/NBK543712/ Yang, E. S., Kim, S., Oh, S.-K., Kang, H., & Chung, I.-J. (2026). Disrupting the link between exposure to violence and adolescent delinquency: The protective role of expectations about the future. *Journal of Child and Family Studies*. https://doi.org/10.1007/s10826-026-03311-3 }} ==External links== {{expand}} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Cognitive]] [[Category:Motivation and emotion/Book/Forensic]] n4oq3h64s5gl960c5yjfxqt4j5xk4u9 10 330232 2842462 2842341 2026-10-24T22:37:46Z Jtneill 10242 2842462 wikitext text/x-wiki <noinclude>Feedback [[wikiversity:FAQ/Template|template]] for the [[Motivation and emotion/Assessment/Chapter|book chapter]] exercise for the [[motivation and emotion]] unit. Designed to be [[Help:Transclusion|transcluded]] on a chapter [[Help:Talk page|talk page]]. __NOTOC__</noinclude><includeonly> ==[[Motivation and emotion/Assessment/Chapter|Book chapter]] review and feedback== {{RoundBoxTop|theme=8}} This chapter has been reviewed according to the [[Motivation and emotion/Assessment/Chapter/Marking criteria|marking criteria]]. Written feedback is provided below, plus there is a [[Motivation and emotion/Assessment/Chapter/Feedback|general feedback]] page. Please also check the chapter's [[Help:Page history|page history]] to check for editing changes made whilst reviewing through the chapter. Chapter marks will be available via {{Motivation and emotion/Canvas}} along with [[Motivation and emotion/Assessment/Social contribution|social contribution]] marks and feedback. Keep an eye on Announcements. {{RoundBoxBottom}} {{RoundBoxTop|theme=}}[[File:Wikiuutiset logo typewriter.png|right|85px]] ===Overall=== {{{1|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Overview (5%)|Overview]]=== {{{2|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Theory (20%)|Theory]]=== {{{3|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Research (25%)|Research]]=== {{{4|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Integration (10%)|Integration]]=== {{{5|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Conclusion (5%)|Conclusion]]=== {{{6|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Style (20%)|Style]]=== {{{7|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Learning features (5%)|Learning features]]=== {{{8|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Social contribution (10%)|Social contribution]]=== {{{9|No comment}}} {{RoundBoxBottom}}</includeonly><noinclude>{{collapse top|Simple example}} ==Simple example== See also [[#Detailed example|detailed example]] <pre> <!-- Official feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # |2= <!-- Overview comments... --> # |3= <!-- Theory comments... --> # |4= <!-- Research comments... --> # |5= <!-- Integration comments... --> # |6= <!-- Conclusion comments... --> # |7= <!-- Style comments... --> # |8= <!-- Learning features comments... --> # |9= <!-- Social contribution comments... --> # }} ~~~~ </pre> gives <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # |2= <!-- Overview comments... --> # |3= <!-- Theory comments... --> # |4= <!-- Research comments... --> # |5= <!-- Integration comments... --> # |6= <!-- Conclusion comments... --> # |7= <!-- Style comments... --> # |8= <!-- Learning features comments... --> # |9= <!-- Social contribution comments... --> # }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:47, 10 October 2026 (UTC) {{Collapse bottom}} ==Detailed example== Example use of the template, with some common feedback comments: <small><small><small><pre> <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem # This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem # This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem # This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem # This is a reasonably good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem # This is a basic chapter # This is an insufficient chapter # The main area(s) for potential improvement: #* use the best psychological theory about the topic #* more detailed review of the best psychological research about the topic #* quality of written expression #* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]] #* integrate more strongly with other chapters by using embedded links to related topics #* overuse of unacknowledged [[Motivation and emotion/Assessment/Using generative AI]] (watch out for [[w:AI slop|AI slop]])—express more in your own words #* I suspect that the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were not invested in preparing this chapter <!-- Overall - Edit summaries --> #* Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits #* [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits #* Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit #* Insufficient use [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit <!-- Overall - GenAI --> #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged #* In some places, there is overreliance on genAI #* There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]]. <!-- Overall – Citations --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims # Only cite sources that you consult # All citations need to be in the References <!-- Overall – Word count --> # Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand # Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes. <!-- Overall – Copyedits --> # For additional feedback, see the following comments and [ these copyedits] |2= <!-- Overview – Comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study # Reasonably engaging scenario or case study # Basic scenario or case study # Use a more practical, real-life scenario; move review of research into a subsequent section # Consider using an example that a wider audience can relate to (less than 5% of the world population are university students) # Add an engaging case study or scenario <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario # Figure 1 could be more relevant to the scenario # Instead of a diagram, use an image which illustrates the scenario taking a place; use the diagram in a subsequent section # Include a relevant image <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box # Put the scenario in a feature box <!-- Overview – Explains problem --> # Clearly explains the psychological problem or phenomenon # Explains the psychological problem or phenomenon reasonably well/in a basic way # Briefly explains the psychological problem or phenomenon; provide more detail # Description of problem is too long/overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections. # Clarity of written expression can be improved <!-- Overview – Focus questions --> # The focus questions are excellent (clear and relevant)/very good/good/reasonably good/basic/promising/insufficient # The focus questions could be improved by: ## being more specific to the topic (i.e., the sub-title) ## matching the top-level headings more closely ## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]] ## splitting double-barrelled questions into separate questions ## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## being presented in a feature box to help guide the reader (fixed) # Add focus questions in a feature box # See copyedits for examples of possible improvements |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Excellent—key theories are very well explained and applied # Very good—key theories are well explained and applied; minor areas for improvements # Reasonably good—relevant theories are selected, described, and explained, with some room for improvement # Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement # A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic # Insufficient use of relevant psychological theory about this topic # Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question). <!-- Theory – Builds on --> # Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds on one [[w:|Wikipedia]] article and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Build more strongly on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Doesn't build on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # Key citations are well used # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) # If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source] # If you didn't consult an original source, don't cite it <!-- Theory – Examples --> # Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts # Consider using more examples to illustrate theoretical concepts # Use more examples to illustrate theoretical concepts # Insufficient use of examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research # Excellent emphasis on systematic reviews and/or meta-analyses # Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal # Insufficient use of relevant systematic reviews and/or meta-analyses # More detail about key studies would be ideal # Claims are well referenced # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) <!-- Research – Critical thinking --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out specific critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient integration between the most relevant theory(ies) and the best research # The chapter places more emphasis on theory than on research; strive for an integrated balance # Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence # The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> # Excellent/Very good/Good/Reasonably good/Basic summary and conclusion # Insufficient as a cohesive summary of the best psychological theory and research about the topic # Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words # Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the [[Motivation and emotion/Wikiversity/Edit summary|edit summary]]. # Highlights the importance of the problem or phenomenon of interest # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are well summarised # Key points are summarised in a basic way # Summarise key points <!-- Conclusion – Focus questions --> # The focus questions are addressed # The take-away messages for each focus question could be spelt out more clearly # Address the focus questions <!-- Conclusion – Take-home messages --> # Clear take-home message(s) # Add practical, take-home message(s) <!-- Conclusion – Word count --> # Not counted for marking purposes due to the chapter being over the maximum word count |7= <!-- Written expression – Style comments... --> <!-- Written expression – Written expression --> # Written expression ## The quality of written expression is excellent/very good/good/reasonably good/basic ## The quality of written expression is OK but there are several aspects which are below professional standard ## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills. ### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources ## Use active (e.g., "this chapter explores" or "this chapter explored") rather than passive voice (e.g., "this chapter will explore" or "this chapter has explored") [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/] ## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia]. ## Reduce excessive use of bold font <!-- Written expression – Sentences --> ## Some/Many sentences could be explained more clearly (e.g., see the {{explain}} and {{rewrite}} tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression ## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+. ## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence. <!-- Written expression – Paragraphs --> ## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences. ## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences. ## Bullet points are overused. Develop more of the bullet point statements into full sentences and paragraphs. <!-- Written expression – Language --> ## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., "it") instead of 1st (e.g., "we") or 2nd person (e.g., "you") in the main text. 1st or 2nd person can work well for case studies or feature boxes. ## Avoid phrases such as “as previously mentioned” or “as noted above,” as these add little value and can disrupt flow. If referencing another part of the chapter is necessary, use [[w:Help#Section linking|section linking]]. ## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]] ## "Individuals" is overused. "People" is usually a clearer term than "individuals". Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally. ## Remove [[w:weasel words|weasel words]]—they add bulk without improving meaning ## Use permanent, rather than relative, time references. For example, instead of "20 years ago", refer to something like "at the beginning of the 21st century". In this way, the text will survive better into the future, without needing to be rewritten. ## Avoid overly emotive language (e.g,. *) in science-based communication <!-- Written expression – Layout --> # Layout ## The chapter is well structured, with major sections using sub-sections ## The headings could be more clearly aligned with the focus questions ## The structure is overly complicated; simplify and integrate ## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion ## The chapter structure is underdeveloped; expand by using subheadings ## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings ## Use the default heading style (e.g., remove additional numbers, italics, bold, and/or change in font size) ## See earlier comment about [[#Heading casing|heading casing]] ## Provide more descriptive headings ## Move links from headings into their first mention in text ## Remove abbreviations/citations from headings ## Include an introductory paragraph before branching into the sub-sections (see {{expand}} tags) ## Integrate learning features rather than having it as a stand-alone section ## The Overview and Conclusion should not have subheadings (fixed) ## Remove colons (:) from the ends of headings <!-- Written expression – Grammar --> # Grammar and spelling are excellent # Grammar ## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback ## Improve the clarity and flow of the writing by checking and correct use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas]. ## Check and make correct use of [https://www.merriam-webster.com/grammar/em-dash-en-dash-how-to-use em dashes (instead of hyphens)] to set off an amplifying or explanatory element ## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats') ## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who] ## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect] ## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)] ## Use past tense when describing research studies, although implications of findings could be in the present tense <!-- Written expression – Abbreviations --> ## Abbreviations ### Only use abbreviations such as e.g., i.e., et al., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out ### Check and correct formatting of abbreviations (such as e.g., i.e., etc.) ### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms. ### Spell out abbreviations on their first use, to explain them to the reader ### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards ### Only introduce abbreviations which are subsequently used <!-- Written expression – Spelling --> # Spelling ## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour) ## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages. <!-- Written expression – Proofreading --> # Proofreading ## More proofreading is needed (e.g., fix punctuation and typographical errors) to bring the quality of written expression closer to a professional standard ## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info] <!-- Written expression – APA style --> # APA style ## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] ## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.] ## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)] <!-- Written expression – APA style - Quotes --> ## Use double (not single) quotation marks "to introduce a word or phrase used ... as slang, or as an invented or coined expression" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## "Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## Direct quotes need page numbers—even better, communicate about concepts in your own words ## Direct quotes are overused—it is better to communicate about concepts in your own words <!-- Written expression – Figures --> ## Figures ### Very well/Well/Reasonably well captioned ### Brief captions; provide more detail to help connect the figure to the text ### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing. [[Motivation and emotion/Assessment/Chapter/Figures|See example]]. ### Add captions <!-- Written expression – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style ### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., "(see Figure 1)"; do not use bold, italics, check and correct capitalisation). ### Refer to each Figure at least once within the main text (e.g., "(see Figure 1)") <!-- Written expression – Figures - Other --> ### Some image uploads were removed because of a lack of sufficient/appropriate copyright information ### Numbering needs correcting ### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size) <!-- Written expression – Tables --> ## Tables ### Very well/Well/Reasonably well captioned ### Brief caption(s); provide more explanatory detail about the table ### Add an APA style caption to each table ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) <!-- Written expression – Tables - Citations --> ### Each Table is referred to at least once within the main text using APA style ### Each Table is referred to at least once within the main text ### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation) ### Refer to each Table at least once within the main text (e.g., see Table 1) <!-- Written expression – Citations --> # Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags) ## If there are three or more authors, cite the first author followed by et al., then year. For example, either: ### in-text, Smith et al. (2020), or ### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020) ## Do not include author first name or initials ## Use ampersand (&) inside [[w:Bracket#Parentheses|parentheses]] and "and" outside parentheses ## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra & Aardvark, 2020) ## A full stop is needed after "et al" (i.e., "et al.") because it is an abbreviation of [[wikt:et alii|et alii]] ## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]] ## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point) ## Check and correct placement of full-stops ## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois ## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources # For citations, use APA style or wiki style, but not both # Secondary citations are overused; strive to consult primary sources <!-- Written expression – References --> ## References use excellent/very good/good/reasonably good/basic/poor APA style: ### Check and correct use of italicisation ### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization] ### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-) ### Include dois where available ### Hyperlink dois for 1-click access ### Provide the full titles of journals ### Remove "Retrieved from " ### Use alphabetical order ### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section ### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section ### Remove bullet-points; add hanging indent <!-- Feedback responsiveness ---> ## No evidence of responsiveness to feedback on talk page |8= <!-- Learning features comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features <!-- Learning features – Wikipedia embedded links --> # Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles # Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. # Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. <!-- Learning features – Wikiversity embedded links --> # Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]] # Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. # Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s) # Excellent/Very good/Good/Reasonably good/Basic/No use of table(s) # Excellent/Very good/Good/Reasonably good/Basic use of feature box(es) # No use of feature boxes other than the opening scenario # Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples # No use of scenarios, case studies, or examples, other than opening scenario <!-- Learning features – Quizzes --> # Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s) # The quiz questions could be improved by being more focused on the key points and/or take-home messages # The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end # The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]]) # No use of quiz(zes) and/or reflection question(s) <!-- Learning features – See also --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Also include links to [[Motivation and emotion/Book|related book chapters]] ## Also include links to related Wikipedia articles ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite ## Move external links into the [[{{PAGENAME}}#References|External links]] section ## Add more links # [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count <!-- Learning features – External links --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section ## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite ## Target an international audience ## Add more links ## Link to the top 3-6 external resources about this topic # [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count |9= <!-- Social contribution comments... --> # ~ logged, useful, mostly minor/moderate/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] # ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence. # Thanks for the Wiki Commons uploads # Thanks very much for your extensive contributions # Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # No logged contributions }} ~~~~ </pre></small></small></small> gives <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem # This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem # This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem # This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem # This is a reasonably good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem # This is a basic chapter # This is an insufficient chapter # The main area(s) for potential improvement: #* use the best psychological theory about the topic #* more detailed review of the best psychological research about the topic #* quality of written expression #* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]] #* integrate more strongly with other chapters by using embedded links to related topics #* overuse of unacknowledged [[Motivation and emotion/Assessment/Using generative AI]] (watch out for [[w:AI slop|AI slop]])—express more in your own words #* I suspect that the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were not invested in preparing this chapter <!-- Overall - Edit summaries --> #* Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits #* [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits #* Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit #* Insufficient use [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit <!-- Overall - GenAI --> #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged #* In some places, there is overreliance on genAI #* There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]]. <!-- Overall – Citations --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims # Only cite sources that you consult # All citations need to be in the References <!-- Overall – Word count --> # Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand # Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes. <!-- Overall – Copyedits --> # For additional feedback, see the following comments and [ these copyedits] |2= <!-- Overview – Comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study # Reasonably engaging scenario or case study # Basic scenario or case study # Use a more practical, real-life scenario; move review of research into a subsequent section # Consider using an example that a wider audience can relate to (less than 5% of the world population are university students) # Add an engaging case study or scenario <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario # Figure 1 could be more relevant to the scenario # Instead of a diagram, use an image which illustrates the scenario taking a place; use the diagram in a subsequent section # Include a relevant image <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box # Put the scenario in a feature box <!-- Overview – Explains problem --> # Clearly explains the psychological problem or phenomenon # Explains the psychological problem or phenomenon reasonably well/in a basic way # Briefly explains the psychological problem or phenomenon; provide more detail # Description of problem is too long/overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections. # Clarity of written expression can be improved <!-- Overview – Focus questions --> # The focus questions are excellent (clear and relevant)/very good/good/reasonably good/basic/promising/insufficient # The focus questions could be improved by: ## being more specific to the topic (i.e., the sub-title) ## matching the top-level headings more closely ## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]] ## splitting double-barrelled questions into separate questions ## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## being presented in a feature box to help guide the reader (fixed) # Add focus questions in a feature box # See copyedits for examples of possible improvements |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Excellent—key theories are very well explained and applied # Very good—key theories are well explained and applied; minor areas for improvements # Reasonably good—relevant theories are selected, described, and explained, with some room for improvement # Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement # A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic # Insufficient use of relevant psychological theory about this topic # Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question). <!-- Theory – Builds on --> # Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds on one [[w:|Wikipedia]] article and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Build more strongly on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Doesn't build on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # Key citations are well used # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) # If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source] # If you didn't consult an original source, don't cite it <!-- Theory – Examples --> # Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts # Consider using more examples to illustrate theoretical concepts # Use more examples to illustrate theoretical concepts # Insufficient use of examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research # Excellent emphasis on systematic reviews and/or meta-analyses # Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal # Insufficient use of relevant systematic reviews and/or meta-analyses # More detail about key studies would be ideal # Claims are well referenced # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) <!-- Research – Critical thinking --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out specific critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient integration between the most relevant theory(ies) and the best research # The chapter places more emphasis on theory than on research; strive for an integrated balance # Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence # The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> # Excellent/Very good/Good/Reasonably good/Basic summary and conclusion # Insufficient as a cohesive summary of the best psychological theory and research about the topic # Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words # Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the [[Motivation and emotion/Wikiversity/Edit summary|edit summary]]. # Highlights the importance of the problem or phenomenon of interest # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are well summarised # Key points are summarised in a basic way # Summarise key points <!-- Conclusion – Focus questions --> # The focus questions are addressed # The take-away messages for each focus question could be spelt out more clearly # Address the focus questions <!-- Conclusion – Take-home messages --> # Clear take-home message(s) # Add practical, take-home message(s) <!-- Conclusion – Word count --> # Not counted for marking purposes due to the chapter being over the maximum word count |7= <!-- Written expression – Style comments... --> <!-- Written expression – Written expression --> # Written expression ## The quality of written expression is excellent/very good/good/reasonably good/basic ## The quality of written expression is OK but there are several aspects which are below professional standard ## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills. ### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources ## Use active (e.g., "this chapter explores" or "this chapter explored") rather than passive voice (e.g., "this chapter will explore" or "this chapter has explored") [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/] ## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia]. ## Reduce excessive use of bold font <!-- Written expression – Sentences --> ## Some/Many sentences could be explained more clearly (e.g., see the {{explain}} and {{rewrite}} tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression ## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+. ## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence. <!-- Written expression – Paragraphs --> ## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences. ## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences. ## Bullet points are overused. Develop more of the bullet point statements into full sentences and paragraphs. <!-- Written expression – Language --> ## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., "it") instead of 1st (e.g., "we") or 2nd person (e.g., "you") in the main text. 1st of 2nd person can work well for case studies or feature boxes. ## Avoid phrases such as “as previously mentioned” or “as noted above,” as these add little value and can disrupt flow. If referencing another part of the chapter is necessary, use [[w:Help#Section linking|section linking]].## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]] ## "Individuals" is overused. "People" is usually a clearer term than "individuals". Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally. ## Remove [[w:weasel words|weasel words]]—they add bulk without improving meaning ## Use permanent, rather than relative, time references. For example, instead of "20 years ago", refer to something like "at the beginning of the 21st century". In this way, the text will survive better into the future, without needing to be rewritten. ## Avoid overly emotive language (e.g,. *) in science-based communication <!-- Written expression – Layout --> # Layout ## The chapter is well structured, with major sections using sub-sections ## The headings could be more clearly aligned with the focus questions ## The structure is overly complicated; simplify and integrate ## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion ## The chapter structure is underdeveloped; expand by using subheadings ## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings ## Use the default heading style (e.g., remove additional numbers, italics, bo]ld, and/or change in font size) ## See earlier comment about [[#Heading casing|heading casing]] ## Provide more descriptive headings ## Move links from headings into their first mention in text ## Remove abbreviations/citations from headings ## Include an introductory paragraph before branching into the sub-sections (see {{expand}} tags) ## Integrate learning features rather than having it as a stand-alone section ## The Overview and Conclusion should not have subheadings (fixed) ## Remove colons (:) from the ends of headings <!-- Written expression – Grammar --> # Grammar and spelling are excellent # Grammar ## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback ## Improve the clarity and flow of the writing by checking and correct use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas]. ## Check and make correct use of [https://www.merriam-webster.com/grammar/em-dash-en-dash-how-to-use em dashes (instead of hyphens)] to set off an amplifying or explanatory element ## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats') ## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who] ## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect] ## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)] ## Use past tense when describing research studies, although implications of findings could be in the present tense <!-- Written expression – Abbreviations --> ## Abbreviations ### Only use abbreviations such as e.g., i.e., et al., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out ### Check and correct formatting of abbreviations (such as e.g., i.e., etc.) ### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms. ### Spell out abbreviations on their first use, to explain them to the reader ### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards ### Only introduce abbreviations which are subsequently used <!-- Written expression – Spelling --> # Spelling ## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour) ## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages. <!-- Written expression – Proofreading --> # Proofreading ## More proofreading is needed (e.g., fix punctuation and typographical errors) to bring the quality of written expression closer to a professional standard ## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info] <!-- Written expression – APA style --> # APA style ## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] ## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.] ## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)] <!-- Written expression – APA style - Quotes --> ## Use double (not single) quotation marks "to introduce a word or phrase used ... as slang, or as an invented or coined expression" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## "Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## Direct quotes need page numbers—even better, communicate about concepts in your own words ## Direct quotes are overused—it is better to communicate about concepts in your own words <!-- Written expression – Figures --> ## Figures ### Very well/Well/Reasonably well captioned ### Brief captions; provide more detail to help connect the figure to the text ### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing. [[Motivation and emotion/Assessment/Chapter/Figures|See example]]. ### Add captions <!-- Written expression – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style ### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., "(see Figure 1)"; do not use bold, italics, check and correct capitalisation). ### Refer to each Figure at least once within the main text (e.g., "(see Figure 1)") <!-- Written expression – Figures - Other --> ### Some image uploads were removed because of a lack of sufficient/appropriate copyright information ### Numbering needs correcting ### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size) <!-- Written expression – Tables --> ## Tables ### Very well/Well/Reasonably well captioned ### Brief caption(s); provide more explanatory detail about the table ### Add an APA style caption to each table ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) <!-- Written expression – Tables - Citations --> ### Each Table is referred to at least once within the main text using APA style ### Each Table is referred to at least once within the main text ### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation) ### Refer to each Table at least once within the main text (e.g., see Table 1) <!-- Written expression – Citations --> # Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags) ## If there are three or more authors, cite the first author followed by et al., then year. For example, either: ### in-text, Smith et al. (2020), or ### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020) ## Do not include author first name or initials ## Use ampersand (&) inside [[w:Bracket#Parentheses|parentheses]] and "and" outside parentheses ## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra & Aardvark, 2020) ## A full stop is needed after "et al" (i.e., "et al.") because it is an abbreviation of [[wikt:et alii|et alii]] ## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]] ## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point) ## Check and correct placement of full-stops ## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois ## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources # For citations, use APA style or wiki style, but not both # Secondary citations are overused; strive to consult primary sources <!-- Written expression – References --> ## References use excellent/very good/good/reasonably good/basic/poor APA style: ### Check and correct use of italicisation ### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization] ### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-) ### Include dois where available ### Hyperlink dois for 1-click access ### Provide the full titles of journals ### Remove "Retrieved from " ### Use alphabetical order ### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section ### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section ### Remove bullet-points; add hanging indent <!-- Feedback responsiveness ---> ## No evidence of responsiveness to feedback on talk page |8= <!-- Learning features comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features <!-- Learning features – Wikipedia embedded links --> # Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles # Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. # Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. <!-- Learning features – Wikiversity embedded links --> # Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]] # Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. # Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s) # Excellent/Very good/Good/Reasonably good/Basic/No use of table(s) # Excellent/Very good/Good/Reasonably good/Basic use of feature box(es) # No use of feature boxes other than the opening scenario # Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples # No use of scenarios, case studies, or examples, other than opening scenario <!-- Learning features – Quizzes --> # Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s) # The quiz questions could be improved by being more focused on the key points and/or take-home messages # The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end # The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]]) # No use of quiz(zes) and/or reflection question(s) <!-- Learning features – See also --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Also include links to [[Motivation and emotion/Book|related book chapters]] ## Also include links to related Wikipedia articles ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite ## Move external links into the [[{{PAGENAME}}#References|External links]] section ## Add more links # [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count <!-- Learning features – External links --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section ## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite ## Target an international audience ## Add more links ## Link to the top 3-6 external resources about this topic # [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count |9= <!-- Social contribution comments... --> # ~ logged, useful, mostly minor/moderate/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] # ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence. # Thanks for the Wiki Commons uploads # Thanks very much for your extensive contributions # Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # No logged contributions }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:47, 10 October 2026 (UTC) ==See also== * [[Motivation and emotion/Assessment/Chapter|Book chapter guidelines]] * [[Template:METF]] * [[Template:MEMF]] [[Category:Motivation and emotion/Admin/2026]] [[Category:Motivation and emotion/Assessment/Chapter]] </noinclude> a52b8i3kltq47g2t1r9xbsrcwh0bd6h 2842551 2842462 2026-10-25T07:39:16Z Jtneill 10242 2842551 wikitext text/x-wiki <noinclude>Feedback [[wikiversity:FAQ/Template|template]] for the [[Motivation and emotion/Assessment/Chapter|book chapter]] exercise for the [[motivation and emotion]] unit. Designed to be [[Help:Transclusion|transcluded]] on a chapter [[Help:Talk page|talk page]]. __NOTOC__</noinclude><includeonly> ==[[Motivation and emotion/Assessment/Chapter|Book chapter]] review and feedback== {{RoundBoxTop|theme=8}} This chapter has been reviewed according to the [[Motivation and emotion/Assessment/Chapter/Marking criteria|marking criteria]]. Written feedback is provided below, plus there is a [[Motivation and emotion/Assessment/Chapter/Feedback|general feedback]] page. Please also check the chapter's [[Help:Page history|page history]] to check for editing changes made whilst reviewing through the chapter. Chapter marks will be available via {{Motivation and emotion/Canvas}} along with [[Motivation and emotion/Assessment/Social contribution|social contribution]] marks and feedback. Keep an eye on Announcements. {{RoundBoxBottom}} {{RoundBoxTop|theme=}}[[File:Wikiuutiset logo typewriter.png|right|85px]] ===Overall=== {{{1|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Overview (5%)|Overview]]=== {{{2|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Theory (20%)|Theory]]=== {{{3|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Research (25%)|Research]]=== {{{4|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Integration (10%)|Integration]]=== {{{5|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Conclusion (5%)|Conclusion]]=== {{{6|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Style (20%)|Style]]=== {{{7|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Learning features (5%)|Learning features]]=== {{{8|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Social contribution (10%)|Social contribution]]=== {{{9|No comment}}} {{RoundBoxBottom}}</includeonly><noinclude>{{collapse top|Simple example}} ==Simple example== See also [[#Detailed example|detailed example]] <pre> <!-- Official feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # |2= <!-- Overview comments... --> # |3= <!-- Theory comments... --> # |4= <!-- Research comments... --> # |5= <!-- Integration comments... --> # |6= <!-- Conclusion comments... --> # |7= <!-- Style comments... --> # |8= <!-- Learning features comments... --> # |9= <!-- Social contribution comments... --> # }} ~~~~ </pre> gives <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # |2= <!-- Overview comments... --> # |3= <!-- Theory comments... --> # |4= <!-- Research comments... --> # |5= <!-- Integration comments... --> # |6= <!-- Conclusion comments... --> # |7= <!-- Style comments... --> # |8= <!-- Learning features comments... --> # |9= <!-- Social contribution comments... --> # }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:47, 10 October 2026 (UTC) {{Collapse bottom}} ==Detailed example== Example use of the template, with some common feedback comments: <small><small><small><pre> <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem # This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem # This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem # This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem # This is a reasonably good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem # This is a basic chapter # This is an insufficient chapter # The main area(s) for potential improvement: #* use the best psychological theory about the topic #* more detailed review of the best psychological research about the topic #* quality of written expression #* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]] #* integrate more strongly with other chapters by using embedded links to related topics #* overuse of unacknowledged [[Motivation and emotion/Assessment/Using generative AI]] (watch out for [[w:AI slop|AI slop]])—express more in your own words #* I suspect that the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were not invested in preparing this chapter <!-- Overall - Edit summaries --> #* Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits #* [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits #* Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit #* Insufficient use [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit <!-- Overall - GenAI --> #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged #* In some places, there is overreliance on genAI #* There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]]. <!-- Overall – Citations --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims # Only cite sources that you consult # All citations need to be in the References <!-- Overall – Word count --> # Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand # Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes. <!-- Overall – Copyedits --> # For additional feedback, see the following comments and [ these copyedits] |2= <!-- Overview – Comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study # Reasonably engaging scenario or case study # Basic scenario or case study # Use a more practical, real-life scenario; move review of research into a subsequent section # Consider using an example that a wider audience can relate to (less than 5% of the world population are university students) # Add an engaging case study or scenario <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario # Figure 1 could be more relevant to the scenario # Instead of a diagram, use an image which illustrates the scenario taking a place; use the diagram in a subsequent section # Include a relevant image <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box # Put the scenario in a feature box <!-- Overview – Explains problem --> # Clearly explains the psychological problem or phenomenon # Explains the psychological problem or phenomenon reasonably well/in a basic way # Briefly explains the psychological problem or phenomenon; provide more detail # Description of problem is too long/overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections. # Clarity of written expression can be improved <!-- Overview – Focus questions --> # The focus questions are excellent (clear and relevant)/very good/good/reasonably good/basic/promising/insufficient # The focus questions could be improved by: ## being more specific to the topic (i.e., the sub-title) ## matching the top-level headings more closely ## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]] ## splitting double-barrelled questions into separate questions ## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## being presented in a feature box to help guide the reader (fixed) # Add focus questions in a feature box # See copyedits for examples of possible improvements |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Excellent—key theories are very well explained and applied # Very good—key theories are well explained and applied; minor areas for improvements # Reasonably good—relevant theories are selected, described, and explained, with some room for improvement # Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement # A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic # Insufficient use of relevant psychological theory about this topic # Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question). <!-- Theory – Builds on --> # Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds on one [[w:|Wikipedia]] article and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Build more strongly on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Doesn't build on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # Key citations are well used # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) # If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source] # If you didn't consult an original source, don't cite it <!-- Theory – Examples --> # Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts # Consider using more examples to illustrate theoretical concepts # Use more examples to illustrate theoretical concepts # Insufficient use of examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research # Excellent emphasis on systematic reviews and/or meta-analyses # Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal # Insufficient use of relevant systematic reviews and/or meta-analyses # More detail about key studies would be ideal # Claims are well referenced # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) <!-- Research – Critical thinking --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out specific critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient integration between the most relevant theory(ies) and the best research # The chapter places more emphasis on theory than on research; strive for an integrated balance # Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence # The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> # Excellent/Very good/Good/Reasonably good/Basic summary and conclusion # Insufficient as a cohesive summary of the best psychological theory and research about the topic # Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words # Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the [[Motivation and emotion/Wikiversity/Edit summary|edit summary]]. # Highlights the importance of the problem or phenomenon of interest # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are well summarised # Key points are summarised in a basic way # Summarise key points <!-- Conclusion – Focus questions --> # The focus questions are addressed # The take-away messages for each focus question could be spelt out more clearly # Address the focus questions <!-- Conclusion – Take-home messages --> # Clear take-home message(s) # Add practical, take-home message(s) <!-- Conclusion – Word count --> # Not counted for marking purposes due to the chapter being over the maximum word count |7= <!-- Written expression – Style comments... --> <!-- Written expression – Written expression --> # Written expression ## The quality of written expression is excellent/very good/good/reasonably good/basic ## The quality of written expression is OK but there are several aspects which are below professional standard ## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills. ### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources ## Use active (e.g., "this chapter explores" or "this chapter explored") rather than passive voice (e.g., "this chapter will explore" or "this chapter has explored") [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/] ## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia]. ## Reduce excessive use of bold font <!-- Written expression – Sentences --> ## Some/Many sentences could be explained more clearly (e.g., see the {{explain}} and {{rewrite}} tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression ## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+. ## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence. <!-- Written expression – Paragraphs --> ## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences. ## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences. ## Bullet points are overused. Develop more of the bullet point statements into full sentences and paragraphs. <!-- Written expression – Language --> ## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., "it") instead of 1st (e.g., "we") or 2nd person (e.g., "you") in the main text. 1st or 2nd person can work well for case studies or feature boxes. ## Avoid phrases such as “as previously mentioned” or “as noted above,” as these add little value and can disrupt flow. If referencing another part of the chapter is necessary, use [[w:Help#Section linking|section linking]]. ## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]] ## "Individuals" is overused. "People" is usually a clearer term than "individuals". Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally. ## Remove [[w:weasel words|weasel words]]—they add bulk without improving meaning ## Use permanent, rather than relative, time references. For example, instead of "20 years ago", refer to something like "at the beginning of the 21st century". In this way, the text will survive better into the future, without needing to be rewritten. ## Avoid overly emotive language (e.g,. *) in science-based communication <!-- Written expression – Layout --> # Layout ## The chapter is well structured, with major sections using sub-sections ## The headings could be more clearly aligned with the focus questions ## The structure is overly complicated; simplify and integrate ## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion ## The chapter structure is underdeveloped; expand by using subheadings ## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings ## Use the default heading style (e.g., remove additional numbers, italics, bold, and/or change in font size) ## See earlier comment about [[#Heading casing|heading casing]] ## Provide more descriptive headings ## Move links from headings into their first mention in text ## Remove abbreviations/citations from headings ## Include an introductory paragraph before branching into the sub-sections (see {{expand}} tags) ## Integrate learning features rather than having it as a stand-alone section ## The Overview and Conclusion should not have subheadings (fixed) ## Remove colons (:) from the ends of headings <!-- Written expression – Grammar --> # Grammar and spelling are excellent # Grammar ## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback ## Improve the clarity and flow of the writing by checking and correct use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas]. ## Check and make correct use of [https://www.merriam-webster.com/grammar/em-dash-en-dash-how-to-use em dashes (instead of hyphens)] to set off an amplifying or explanatory element ## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats') ## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who] ## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect] ## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)] ## Use past tense when describing research studies, although implications of findings could be in the present tense <!-- Written expression – Abbreviations --> ## Abbreviations ### Only use abbreviations such as e.g., i.e., et al., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out ### Check and correct formatting of abbreviations (such as e.g., i.e., etc.) ### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms. ### Spell out abbreviations on their first use, to explain them to the reader ### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards ### Only introduce abbreviations which are subsequently used <!-- Written expression – Spelling --> # Spelling ## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour) ## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages. <!-- Written expression – Copy editing --> # Copy editing ## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors). ## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info] <!-- Written expression – APA style --> # APA style ## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] ## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.] ## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)] <!-- Written expression – APA style - Quotes --> ## Use double (not single) quotation marks "to introduce a word or phrase used ... as slang, or as an invented or coined expression" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## "Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## Direct quotes need page numbers—even better, communicate about concepts in your own words ## Direct quotes are overused—it is better to communicate about concepts in your own words <!-- Written expression – Figures --> ## Figures ### Very well/Well/Reasonably well captioned ### Brief captions; provide more detail to help connect the figure to the text ### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing. [[Motivation and emotion/Assessment/Chapter/Figures|See example]]. ### Add captions <!-- Written expression – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style ### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., "(see Figure 1)"; do not use bold, italics, check and correct capitalisation). ### Refer to each Figure at least once within the main text (e.g., "(see Figure 1)") <!-- Written expression – Figures - Other --> ### Some image uploads were removed because of a lack of sufficient/appropriate copyright information ### Numbering needs correcting ### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size) <!-- Written expression – Tables --> ## Tables ### Very well/Well/Reasonably well captioned ### Brief caption(s); provide more explanatory detail about the table ### Add an APA style caption to each table ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) <!-- Written expression – Tables - Citations --> ### Each Table is referred to at least once within the main text using APA style ### Each Table is referred to at least once within the main text ### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation) ### Refer to each Table at least once within the main text (e.g., see Table 1) <!-- Written expression – Citations --> # Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags) ## If there are three or more authors, cite the first author followed by et al., then year. For example, either: ### in-text, Smith et al. (2020), or ### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020) ## Do not include author first name or initials ## Use ampersand (&) inside [[w:Bracket#Parentheses|parentheses]] and "and" outside parentheses ## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra & Aardvark, 2020) ## A full stop is needed after "et al" (i.e., "et al.") because it is an abbreviation of [[wikt:et alii|et alii]] ## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]] ## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point) ## Check and correct placement of full-stops ## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois ## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources # For citations, use APA style or wiki style, but not both # Secondary citations are overused; strive to consult primary sources <!-- Written expression – References --> ## References use excellent/very good/good/reasonably good/basic/poor APA style: ### Check and correct use of italicisation ### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization] ### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-) ### Include dois where available ### Hyperlink dois for 1-click access ### Provide the full titles of journals ### Remove "Retrieved from " ### Use alphabetical order ### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section ### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section ### Remove bullet-points; add hanging indent <!-- Feedback responsiveness ---> ## No evidence of responsiveness to feedback on talk page |8= <!-- Learning features comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features <!-- Learning features – Wikipedia embedded links --> # Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles # Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. # Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. <!-- Learning features – Wikiversity embedded links --> # Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]] # Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. # Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s) # Excellent/Very good/Good/Reasonably good/Basic/No use of table(s) # Excellent/Very good/Good/Reasonably good/Basic use of feature box(es) # No use of feature boxes other than the opening scenario # Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples # No use of scenarios, case studies, or examples, other than opening scenario <!-- Learning features – Quizzes --> # Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s) # The quiz questions could be improved by being more focused on the key points and/or take-home messages # The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end # The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]]) # No use of quiz(zes) and/or reflection question(s) <!-- Learning features – See also --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Also include links to [[Motivation and emotion/Book|related book chapters]] ## Also include links to related Wikipedia articles ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite ## Move external links into the [[{{PAGENAME}}#References|External links]] section ## Add more links # [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count <!-- Learning features – External links --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section ## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite ## Target an international audience ## Add more links ## Link to the top 3-6 external resources about this topic # [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count |9= <!-- Social contribution comments... --> # ~ logged, useful, mostly minor/moderate/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] # ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence. # Thanks for the Wiki Commons uploads # Thanks very much for your extensive contributions # Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # No logged contributions }} ~~~~ </pre></small></small></small> gives <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem # This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem # This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem # This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem # This is a reasonably good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem # This is a basic chapter # This is an insufficient chapter # The main area(s) for potential improvement: #* use the best psychological theory about the topic #* more detailed review of the best psychological research about the topic #* quality of written expression #* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]] #* integrate more strongly with other chapters by using embedded links to related topics #* overuse of unacknowledged [[Motivation and emotion/Assessment/Using generative AI]] (watch out for [[w:AI slop|AI slop]])—express more in your own words #* I suspect that the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were not invested in preparing this chapter <!-- Overall - Edit summaries --> #* Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits #* [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits #* Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit #* Insufficient use [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit <!-- Overall - GenAI --> #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged #* In some places, there is overreliance on genAI #* There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]]. <!-- Overall – Citations --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims # Only cite sources that you consult # All citations need to be in the References <!-- Overall – Word count --> # Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand # Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes. <!-- Overall – Copyedits --> # For additional feedback, see the following comments and [ these copyedits] |2= <!-- Overview – Comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study # Reasonably engaging scenario or case study # Basic scenario or case study # Use a more practical, real-life scenario; move review of research into a subsequent section # Consider using an example that a wider audience can relate to (less than 5% of the world population are university students) # Add an engaging case study or scenario <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario # Figure 1 could be more relevant to the scenario # Instead of a diagram, use an image which illustrates the scenario taking a place; use the diagram in a subsequent section # Include a relevant image <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box # Put the scenario in a feature box <!-- Overview – Explains problem --> # Clearly explains the psychological problem or phenomenon # Explains the psychological problem or phenomenon reasonably well/in a basic way # Briefly explains the psychological problem or phenomenon; provide more detail # Description of problem is too long/overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections. # Clarity of written expression can be improved <!-- Overview – Focus questions --> # The focus questions are excellent (clear and relevant)/very good/good/reasonably good/basic/promising/insufficient # The focus questions could be improved by: ## being more specific to the topic (i.e., the sub-title) ## matching the top-level headings more closely ## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]] ## splitting double-barrelled questions into separate questions ## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## being presented in a feature box to help guide the reader (fixed) # Add focus questions in a feature box # See copyedits for examples of possible improvements |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Excellent—key theories are very well explained and applied # Very good—key theories are well explained and applied; minor areas for improvements # Reasonably good—relevant theories are selected, described, and explained, with some room for improvement # Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement # A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic # Insufficient use of relevant psychological theory about this topic # Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question). <!-- Theory – Builds on --> # Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds on one [[w:|Wikipedia]] article and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Build more strongly on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Doesn't build on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # Key citations are well used # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) # If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source] # If you didn't consult an original source, don't cite it <!-- Theory – Examples --> # Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts # Consider using more examples to illustrate theoretical concepts # Use more examples to illustrate theoretical concepts # Insufficient use of examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research # Excellent emphasis on systematic reviews and/or meta-analyses # Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal # Insufficient use of relevant systematic reviews and/or meta-analyses # More detail about key studies would be ideal # Claims are well referenced # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) <!-- Research – Critical thinking --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out specific critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient integration between the most relevant theory(ies) and the best research # The chapter places more emphasis on theory than on research; strive for an integrated balance # Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence # The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> # Excellent/Very good/Good/Reasonably good/Basic summary and conclusion # Insufficient as a cohesive summary of the best psychological theory and research about the topic # Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words # Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the [[Motivation and emotion/Wikiversity/Edit summary|edit summary]]. # Highlights the importance of the problem or phenomenon of interest # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are well summarised # Key points are summarised in a basic way # Summarise key points <!-- Conclusion – Focus questions --> # The focus questions are addressed # The take-away messages for each focus question could be spelt out more clearly # Address the focus questions <!-- Conclusion – Take-home messages --> # Clear take-home message(s) # Add practical, take-home message(s) <!-- Conclusion – Word count --> # Not counted for marking purposes due to the chapter being over the maximum word count |7= <!-- Written expression – Style comments... --> <!-- Written expression – Written expression --> # Written expression ## The quality of written expression is excellent/very good/good/reasonably good/basic ## The quality of written expression is OK but there are several aspects which are below professional standard ## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills. ### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources ## Use active (e.g., "this chapter explores" or "this chapter explored") rather than passive voice (e.g., "this chapter will explore" or "this chapter has explored") [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/] ## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia]. ## Reduce excessive use of bold font <!-- Written expression – Sentences --> ## Some/Many sentences could be explained more clearly (e.g., see the {{explain}} and {{rewrite}} tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression ## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+. ## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence. <!-- Written expression – Paragraphs --> ## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences. ## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences. ## Bullet points are overused. Develop more of the bullet point statements into full sentences and paragraphs. <!-- Written expression – Language --> ## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., "it") instead of 1st (e.g., "we") or 2nd person (e.g., "you") in the main text. 1st of 2nd person can work well for case studies or feature boxes. ## Avoid phrases such as “as previously mentioned” or “as noted above,” as these add little value and can disrupt flow. If referencing another part of the chapter is necessary, use [[w:Help#Section linking|section linking]].## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]] ## "Individuals" is overused. "People" is usually a clearer term than "individuals". Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally. ## Remove [[w:weasel words|weasel words]]—they add bulk without improving meaning ## Use permanent, rather than relative, time references. For example, instead of "20 years ago", refer to something like "at the beginning of the 21st century". In this way, the text will survive better into the future, without needing to be rewritten. ## Avoid overly emotive language (e.g,. *) in science-based communication <!-- Written expression – Layout --> # Layout ## The chapter is well structured, with major sections using sub-sections ## The headings could be more clearly aligned with the focus questions ## The structure is overly complicated; simplify and integrate ## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion ## The chapter structure is underdeveloped; expand by using subheadings ## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings ## Use the default heading style (e.g., remove additional numbers, italics, bo]ld, and/or change in font size) ## See earlier comment about [[#Heading casing|heading casing]] ## Provide more descriptive headings ## Move links from headings into their first mention in text ## Remove abbreviations/citations from headings ## Include an introductory paragraph before branching into the sub-sections (see {{expand}} tags) ## Integrate learning features rather than having it as a stand-alone section ## The Overview and Conclusion should not have subheadings (fixed) ## Remove colons (:) from the ends of headings <!-- Written expression – Grammar --> # Grammar and spelling are excellent # Grammar ## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback ## Improve the clarity and flow of the writing by checking and correct use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas]. ## Check and make correct use of [https://www.merriam-webster.com/grammar/em-dash-en-dash-how-to-use em dashes (instead of hyphens)] to set off an amplifying or explanatory element ## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats') ## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who] ## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect] ## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)] ## Use past tense when describing research studies, although implications of findings could be in the present tense <!-- Written expression – Abbreviations --> ## Abbreviations ### Only use abbreviations such as e.g., i.e., et al., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out ### Check and correct formatting of abbreviations (such as e.g., i.e., etc.) ### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms. ### Spell out abbreviations on their first use, to explain them to the reader ### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards ### Only introduce abbreviations which are subsequently used <!-- Written expression – Spelling --> # Spelling ## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour) ## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages. <!-- Written expression – Copy editing --> # Copy editing ## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors).## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info] <!-- Written expression – APA style --> # APA style ## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] ## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.] ## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)] <!-- Written expression – APA style - Quotes --> ## Use double (not single) quotation marks "to introduce a word or phrase used ... as slang, or as an invented or coined expression" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## "Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## Direct quotes need page numbers—even better, communicate about concepts in your own words ## Direct quotes are overused—it is better to communicate about concepts in your own words <!-- Written expression – Figures --> ## Figures ### Very well/Well/Reasonably well captioned ### Brief captions; provide more detail to help connect the figure to the text ### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing. [[Motivation and emotion/Assessment/Chapter/Figures|See example]]. ### Add captions <!-- Written expression – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style ### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., "(see Figure 1)"; do not use bold, italics, check and correct capitalisation). ### Refer to each Figure at least once within the main text (e.g., "(see Figure 1)") <!-- Written expression – Figures - Other --> ### Some image uploads were removed because of a lack of sufficient/appropriate copyright information ### Numbering needs correcting ### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size) <!-- Written expression – Tables --> ## Tables ### Very well/Well/Reasonably well captioned ### Brief caption(s); provide more explanatory detail about the table ### Add an APA style caption to each table ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) <!-- Written expression – Tables - Citations --> ### Each Table is referred to at least once within the main text using APA style ### Each Table is referred to at least once within the main text ### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation) ### Refer to each Table at least once within the main text (e.g., see Table 1) <!-- Written expression – Citations --> # Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags) ## If there are three or more authors, cite the first author followed by et al., then year. For example, either: ### in-text, Smith et al. (2020), or ### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020) ## Do not include author first name or initials ## Use ampersand (&) inside [[w:Bracket#Parentheses|parentheses]] and "and" outside parentheses ## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra & Aardvark, 2020) ## A full stop is needed after "et al" (i.e., "et al.") because it is an abbreviation of [[wikt:et alii|et alii]] ## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]] ## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point) ## Check and correct placement of full-stops ## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois ## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources # For citations, use APA style or wiki style, but not both # Secondary citations are overused; strive to consult primary sources <!-- Written expression – References --> ## References use excellent/very good/good/reasonably good/basic/poor APA style: ### Check and correct use of italicisation ### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization] ### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-) ### Include dois where available ### Hyperlink dois for 1-click access ### Provide the full titles of journals ### Remove "Retrieved from " ### Use alphabetical order ### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section ### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section ### Remove bullet-points; add hanging indent <!-- Feedback responsiveness ---> ## No evidence of responsiveness to feedback on talk page |8= <!-- Learning features comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features <!-- Learning features – Wikipedia embedded links --> # Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles # Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. # Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. <!-- Learning features – Wikiversity embedded links --> # Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]] # Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. # Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s) # Excellent/Very good/Good/Reasonably good/Basic/No use of table(s) # Excellent/Very good/Good/Reasonably good/Basic use of feature box(es) # No use of feature boxes other than the opening scenario # Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples # No use of scenarios, case studies, or examples, other than opening scenario <!-- Learning features – Quizzes --> # Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s) # The quiz questions could be improved by being more focused on the key points and/or take-home messages # The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end # The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]]) # No use of quiz(zes) and/or reflection question(s) <!-- Learning features – See also --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Also include links to [[Motivation and emotion/Book|related book chapters]] ## Also include links to related Wikipedia articles ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite ## Move external links into the [[{{PAGENAME}}#References|External links]] section ## Add more links # [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count <!-- Learning features – External links --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section ## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite ## Target an international audience ## Add more links ## Link to the top 3-6 external resources about this topic # [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count |9= <!-- Social contribution comments... --> # ~ logged, useful, mostly minor/moderate/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] # ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence. # Thanks for the Wiki Commons uploads # Thanks very much for your extensive contributions # Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # No logged contributions }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:47, 10 October 2026 (UTC) ==See also== * [[Motivation and emotion/Assessment/Chapter|Book chapter guidelines]] * [[Template:METF]] * [[Template:MEMF]] [[Category:Motivation and emotion/Admin/2026]] [[Category:Motivation and emotion/Assessment/Chapter]] </noinclude> mbo15bprf4lj00wxi30vq8cjzy4dusn 2842552 2842551 2026-10-25T08:19:38Z Jtneill 10242 # Somewhat clear take-home message(s) 2842552 wikitext text/x-wiki <noinclude>Feedback [[wikiversity:FAQ/Template|template]] for the [[Motivation and emotion/Assessment/Chapter|book chapter]] exercise for the [[motivation and emotion]] unit. Designed to be [[Help:Transclusion|transcluded]] on a chapter [[Help:Talk page|talk page]]. __NOTOC__</noinclude><includeonly> ==[[Motivation and emotion/Assessment/Chapter|Book chapter]] review and feedback== {{RoundBoxTop|theme=8}} This chapter has been reviewed according to the [[Motivation and emotion/Assessment/Chapter/Marking criteria|marking criteria]]. Written feedback is provided below, plus there is a [[Motivation and emotion/Assessment/Chapter/Feedback|general feedback]] page. Please also check the chapter's [[Help:Page history|page history]] to check for editing changes made whilst reviewing through the chapter. Chapter marks will be available via {{Motivation and emotion/Canvas}} along with [[Motivation and emotion/Assessment/Social contribution|social contribution]] marks and feedback. Keep an eye on Announcements. {{RoundBoxBottom}} {{RoundBoxTop|theme=}}[[File:Wikiuutiset logo typewriter.png|right|85px]] ===Overall=== {{{1|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Overview (5%)|Overview]]=== {{{2|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Theory (20%)|Theory]]=== {{{3|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Research (25%)|Research]]=== {{{4|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Integration (10%)|Integration]]=== {{{5|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Conclusion (5%)|Conclusion]]=== {{{6|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Style (20%)|Style]]=== {{{7|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Learning features (5%)|Learning features]]=== {{{8|No comment}}} ===[[Motivation and emotion/Assessment/Chapter/Marking criteria#Social contribution (10%)|Social contribution]]=== {{{9|No comment}}} {{RoundBoxBottom}}</includeonly><noinclude>{{collapse top|Simple example}} ==Simple example== See also [[#Detailed example|detailed example]] <pre> <!-- Official feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # |2= <!-- Overview comments... --> # |3= <!-- Theory comments... --> # |4= <!-- Research comments... --> # |5= <!-- Integration comments... --> # |6= <!-- Conclusion comments... --> # |7= <!-- Style comments... --> # |8= <!-- Learning features comments... --> # |9= <!-- Social contribution comments... --> # }} ~~~~ </pre> gives <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # |2= <!-- Overview comments... --> # |3= <!-- Theory comments... --> # |4= <!-- Research comments... --> # |5= <!-- Integration comments... --> # |6= <!-- Conclusion comments... --> # |7= <!-- Style comments... --> # |8= <!-- Learning features comments... --> # |9= <!-- Social contribution comments... --> # }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:47, 10 October 2026 (UTC) {{Collapse bottom}} ==Detailed example== Example use of the template, with some common feedback comments: <small><small><small><pre> <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem # This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem # This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem # This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem # This is a reasonably good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem # This is a basic chapter # This is an insufficient chapter # The main area(s) for potential improvement: #* use the best psychological theory about the topic #* more detailed review of the best psychological research about the topic #* quality of written expression #* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]] #* integrate more strongly with other chapters by using embedded links to related topics #* overuse of unacknowledged [[Motivation and emotion/Assessment/Using generative AI]] (watch out for [[w:AI slop|AI slop]])—express more in your own words #* I suspect that the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were not invested in preparing this chapter <!-- Overall - Edit summaries --> #* Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits #* [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits #* Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit #* Insufficient use [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit <!-- Overall - GenAI --> #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged #* In some places, there is overreliance on genAI #* There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]]. <!-- Overall – Citations --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims # Only cite sources that you consult # All citations need to be in the References <!-- Overall – Word count --> # Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand # Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes. <!-- Overall – Copyedits --> # For additional feedback, see the following comments and [ these copyedits] |2= <!-- Overview – Comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study # Reasonably engaging scenario or case study # Basic scenario or case study # Use a more practical, real-life scenario; move review of research into a subsequent section # Consider using an example that a wider audience can relate to (less than 5% of the world population are university students) # Add an engaging case study or scenario <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario # Figure 1 could be more relevant to the scenario # Instead of a diagram, use an image which illustrates the scenario taking a place; use the diagram in a subsequent section # Include a relevant image <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box # Put the scenario in a feature box <!-- Overview – Explains problem --> # Clearly explains the psychological problem or phenomenon # Explains the psychological problem or phenomenon reasonably well/in a basic way # Briefly explains the psychological problem or phenomenon; provide more detail # Description of problem is too long/overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections. # Clarity of written expression can be improved <!-- Overview – Focus questions --> # The focus questions are excellent (clear and relevant)/very good/good/reasonably good/basic/promising/insufficient # The focus questions could be improved by: ## being more specific to the topic (i.e., the sub-title) ## matching the top-level headings more closely ## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]] ## splitting double-barrelled questions into separate questions ## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## being presented in a feature box to help guide the reader (fixed) # Add focus questions in a feature box # See copyedits for examples of possible improvements |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Excellent—key theories are very well explained and applied # Very good—key theories are well explained and applied; minor areas for improvements # Reasonably good—relevant theories are selected, described, and explained, with some room for improvement # Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement # A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic # Insufficient use of relevant psychological theory about this topic # Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question). <!-- Theory – Builds on --> # Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds on one [[w:|Wikipedia]] article and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Build more strongly on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Doesn't build on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # Key citations are well used # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) # If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source] # If you didn't consult an original source, don't cite it <!-- Theory – Examples --> # Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts # Consider using more examples to illustrate theoretical concepts # Use more examples to illustrate theoretical concepts # Insufficient use of examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research # Excellent emphasis on systematic reviews and/or meta-analyses # Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal # Insufficient use of relevant systematic reviews and/or meta-analyses # More detail about key studies would be ideal # Claims are well referenced # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) <!-- Research – Critical thinking --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out specific critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient integration between the most relevant theory(ies) and the best research # The chapter places more emphasis on theory than on research; strive for an integrated balance # Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence # The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> # Excellent/Very good/Good/Reasonably good/Basic summary and conclusion # Insufficient as a cohesive summary of the best psychological theory and research about the topic # Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words # Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the [[Motivation and emotion/Wikiversity/Edit summary|edit summary]]. # Highlights the importance of the problem or phenomenon of interest # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are well summarised # Key points are summarised in a basic way # Summarise key points <!-- Conclusion – Focus questions --> # The focus questions are addressed # The take-away messages for each focus question could be spelt out more clearly # Address the focus questions <!-- Conclusion – Take-home messages --> # Clear take-home message(s) # Somewhat clear take-home message(s) # Add practical, take-home message(s) <!-- Conclusion – Word count --> # Not counted for marking purposes due to the chapter being over the maximum word count |7= <!-- Written expression – Style comments... --> <!-- Written expression – Written expression --> # Written expression ## The quality of written expression is excellent/very good/good/reasonably good/basic ## The quality of written expression is OK but there are several aspects which are below professional standard ## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills. ### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources ## Use active (e.g., "this chapter explores" or "this chapter explored") rather than passive voice (e.g., "this chapter will explore" or "this chapter has explored") [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/] ## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia]. ## Reduce excessive use of bold font <!-- Written expression – Sentences --> ## Some/Many sentences could be explained more clearly (e.g., see the {{explain}} and {{rewrite}} tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression ## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+. ## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence. <!-- Written expression – Paragraphs --> ## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences. ## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences. ## Bullet points are overused. Develop more of the bullet point statements into full sentences and paragraphs. <!-- Written expression – Language --> ## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., "it") instead of 1st (e.g., "we") or 2nd person (e.g., "you") in the main text. 1st or 2nd person can work well for case studies or feature boxes. ## Avoid phrases such as “as previously mentioned” or “as noted above,” as these add little value and can disrupt flow. If referencing another part of the chapter is necessary, use [[w:Help#Section linking|section linking]]. ## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]] ## "Individuals" is overused. "People" is usually a clearer term than "individuals". Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally. ## Remove [[w:weasel words|weasel words]]—they add bulk without improving meaning ## Use permanent, rather than relative, time references. For example, instead of "20 years ago", refer to something like "at the beginning of the 21st century". In this way, the text will survive better into the future, without needing to be rewritten. ## Avoid overly emotive language (e.g,. *) in science-based communication <!-- Written expression – Layout --> # Layout ## The chapter is well structured, with major sections using sub-sections ## The headings could be more clearly aligned with the focus questions ## The structure is overly complicated; simplify and integrate ## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion ## The chapter structure is underdeveloped; expand by using subheadings ## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings ## Use the default heading style (e.g., remove additional numbers, italics, bold, and/or change in font size) ## See earlier comment about [[#Heading casing|heading casing]] ## Provide more descriptive headings ## Move links from headings into their first mention in text ## Remove abbreviations/citations from headings ## Include an introductory paragraph before branching into the sub-sections (see {{expand}} tags) ## Integrate learning features rather than having it as a stand-alone section ## The Overview and Conclusion should not have subheadings (fixed) ## Remove colons (:) from the ends of headings <!-- Written expression – Grammar --> # Grammar and spelling are excellent # Grammar ## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback ## Improve the clarity and flow of the writing by checking and correct use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas]. ## Check and make correct use of [https://www.merriam-webster.com/grammar/em-dash-en-dash-how-to-use em dashes (instead of hyphens)] to set off an amplifying or explanatory element ## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats') ## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who] ## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect] ## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)] ## Use past tense when describing research studies, although implications of findings could be in the present tense <!-- Written expression – Abbreviations --> ## Abbreviations ### Only use abbreviations such as e.g., i.e., et al., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out ### Check and correct formatting of abbreviations (such as e.g., i.e., etc.) ### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms. ### Spell out abbreviations on their first use, to explain them to the reader ### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards ### Only introduce abbreviations which are subsequently used <!-- Written expression – Spelling --> # Spelling ## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour) ## Some words are misspelt (e.g., see the {{sp}} tags). Spell-checking tools are available in most internet browsers and word processing software packages. <!-- Written expression – Copy editing --> # Copy editing ## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors). ## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info] <!-- Written expression – APA style --> # APA style ## Use [[w:Serial comma|serial comma]]s[https://www.buzzfeed.com/adamdavis/the-oxford-comma-is-extremely-important-and-everyone-should][https://www.youtube.com/watch?v=gBx8ooDupXY] ## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.] ## [https://apastyle.apa.org/style-grammar-guidelines/numbers Express numbers under 10 using words (e.g., two) and 10 and over using numerals (e.g., 99)] <!-- Written expression – APA style - Quotes --> ## Use double (not single) quotation marks "to introduce a word or phrase used ... as slang, or as an invented or coined expression" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## "Use quotation marks only for the first occurrence of the word or phrase, not for subsequent occurrences" ([https://apastyle.apa.org/products/publication-manual-7th-edition APA Style 7th ed.], 2020, p. 159) ## Direct quotes need page numbers—even better, communicate about concepts in your own words ## Direct quotes are overused—it is better to communicate about concepts in your own words <!-- Written expression – Figures --> ## Figures ### Very well/Well/Reasonably well captioned ### Brief captions; provide more detail to help connect the figure to the text ### Use this format for captions: '''Figure X'''. Descriptive caption goes here in sentence casing. [[Motivation and emotion/Assessment/Chapter/Figures|See example]]. ### Add captions <!-- Written expression – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style ### Each Figure is referred to at least once within the main text. Refer to each Figure using APA style (e.g., "(see Figure 1)"; do not use bold, italics, check and correct capitalisation). ### Refer to each Figure at least once within the main text (e.g., "(see Figure 1)") <!-- Written expression – Figures - Other --> ### Some image uploads were removed because of a lack of sufficient/appropriate copyright information ### Numbering needs correcting ### Adjust some image sizes to make them easier to read (increase size) and/or less dominant (decrease size) <!-- Written expression – Tables --> ## Tables ### Very well/Well/Reasonably well captioned ### Brief caption(s); provide more explanatory detail about the table ### Add an APA style caption to each table ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) <!-- Written expression – Tables - Citations --> ### Each Table is referred to at least once within the main text using APA style ### Each Table is referred to at least once within the main text ### Refer to each Table using APA style (e.g., do not use bold, italics, check and correct capitalisation) ### Refer to each Table at least once within the main text (e.g., see Table 1) <!-- Written expression – Citations --> # Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # Citations use very good/good/reasonably good/basic/poor [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] # In some/many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags) ## If there are three or more authors, cite the first author followed by et al., then year. For example, either: ### in-text, Smith et al. (2020), or ### in [[w:Bracket#Parentheses|parentheses]] (Smith et al., 2020) ## Do not include author first name or initials ## Use ampersand (&) inside [[w:Bracket#Parentheses|parentheses]] and "and" outside parentheses ## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra & Aardvark, 2020) ## A full stop is needed after "et al" (i.e., "et al.") because it is an abbreviation of [[wikt:et alii|et alii]] ## Use a comma between the author(s) and year for citations in [[w:Bracket#Parentheses|parentheses]] ## Select up to a maximum of three citations per point (i.e., avoid citing four or more citations to support a single point) ## Check and correct placement of full-stops ## Move embedded links to academic peer-reviewed sources into the [[{{PAGENAME}}#References|References]] as APA style citations with hyperlinked dois ## Move embedded links to non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section; only cite peer-reviewed sources # For citations, use APA style or wiki style, but not both # Secondary citations are overused; strive to consult primary sources <!-- Written expression – References --> ## References use excellent/very good/good/reasonably good/basic/poor APA style: ### Check and correct use of italicisation ### Check and correct use of capitalisation[https://apastyle.apa.org/style-grammar-guidelines/capitalization] ### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-) ### Include dois where available ### Hyperlink dois for 1-click access ### Provide the full titles of journals ### Remove "Retrieved from " ### Use alphabetical order ### Move Wikipedia links into the [[{{PAGENAME}}#See also|See also]] section ### Move non-peer reviewed sources into the [[{{PAGENAME}}#External links|External links]] section ### Remove bullet-points; add hanging indent <!-- Feedback responsiveness ---> ## No evidence of responsiveness to feedback on talk page |8= <!-- Learning features comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features <!-- Learning features – Wikipedia embedded links --> # Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles # Very good/Good/Reasonably good/Basic/One use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. # Use [[m:Help:Interwiki linking|interwiki links]] (rather than external links) to Wikipedia articles, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. <!-- Learning features – Wikiversity embedded links --> # Excellent use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]] # Very good/Good/Reasonably good/Basic/One use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. # Use in-text [[m:Help:Interwiki linking|interwiki links]], rather than external links to Wikiversity chapters, per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # Move embedded links to non-peer-reviewed sources to the [[{{PAGENAME}}#External links|External links]] section # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of figure(s) # Excellent/Very good/Good/Reasonably good/Basic/No use of table(s) # Excellent/Very good/Good/Reasonably good/Basic use of feature box(es) # No use of feature boxes other than the opening scenario # Excellent/Very good/Good/Reasonably good/Basic use of scenarios, case studies, or examples # No use of scenarios, case studies, or examples, other than opening scenario <!-- Learning features – Quizzes --> # Excellent/Very good/Good/Reasonably good/Basic use of quiz(zes) and/or reflection question(s) # The quiz questions could be improved by being more focused on the key points and/or take-home messages # The quiz questions could be more effective as learning prompts by being embedded as single questions within each corresponding section rather than as a set of questions at the end # The reflection questions could be made more interactive by using a drop-down box that the reader can expand to see an answer/prompt to figure out the answers to the questions (e.g., see [[Motivation and emotion/Book/2026/Immersive therapy for PTSD treatment|this chapter]]) # No use of quiz(zes) and/or reflection question(s) <!-- Learning features – See also --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#See also|See also]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use internal linking style per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Also include links to [[Motivation and emotion/Book|related book chapters]] ## Also include links to related Wikipedia articles ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move peer-reviewed articles into the [[{{PAGENAME}}#References|References]] section and cite ## Move external links into the [[{{PAGENAME}}#References|External links]] section ## Add more links # [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count <!-- Learning features – External links --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Rename links per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] ## Use alphabetical order ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link ## Move Wikipedia/Wikiversity links to the [[{{PAGENAME}}#See also|See also]] section ## Move peer-reviewed articles to the [[{{PAGENAME}}#References|References]] section and cite ## Target an international audience ## Add more links ## Link to the top 3-6 external resources about this topic # [[{{PAGENAME}}#External links|External links]] section not counted for marking purposes due to being over the maximum word count |9= <!-- Social contribution comments... --> # ~ logged, useful, mostly minor/moderate/major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] # ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence. # Thanks for the Wiki Commons uploads # Thanks very much for your extensive contributions # Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # No logged contributions }} ~~~~ </pre></small></small></small> gives <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is an outstanding chapter that successfully integrates psychological theory and research in a highly readable way to address a practical, real-world phenomenon or problem # This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem # This is a very good chapter that makes very good use of psychological theory and research to address a real-world phenomenon or problem # This is a good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem # This is a reasonably good chapter that makes good use of psychological theory and research to address a real-world phenomenon or problem # This is a basic chapter # This is an insufficient chapter # The main area(s) for potential improvement: #* use the best psychological theory about the topic #* more detailed review of the best psychological research about the topic #* quality of written expression #* tackle the target topic more directly; this chapter [[wikt:beat around the bush|beats around the bush]] #* integrate more strongly with other chapters by using embedded links to related topics #* overuse of unacknowledged [[Motivation and emotion/Assessment/Using generative AI]] (watch out for [[w:AI slop|AI slop]])—express more in your own words #* I suspect that the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were not invested in preparing this chapter <!-- Overall - Edit summaries --> #* Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits #* [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits #* Use more specific [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in each edit #* Insufficient use [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit <!-- Overall - GenAI --> #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] is appropriately acknowledged #* In some places, there is overreliance on genAI #* There is insufficient acknowledgement in [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] when adding [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; this violates academic integrity principles #* [[Motivation and emotion/Assessment/Using generative AI|GenAI use]] has not been appropriately acknowledged in [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] with links to the conversation sources despite previous feedback about this issue. Unacknowledged use of GenAI-generated content raises concerns about compliance with academic integrity principles and assessment requirements and [[Wikiversity:Artificial intelligence|Wikiversity's artificial intelligence policy]]. <!-- Overall – Citations --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of academic, peer-reviewed citations to support claims # Only cite sources that you consult # All citations need to be in the References <!-- Overall – Word count --> # Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand # Over the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]]. Content beyond 4,000 words has been ignored for marking purposes. <!-- Overall – Copyedits --> # For additional feedback, see the following comments and [ these copyedits] |2= <!-- Overview – Comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study # Reasonably engaging scenario or case study # Basic scenario or case study # Use a more practical, real-life scenario; move review of research into a subsequent section # Consider using an example that a wider audience can relate to (less than 5% of the world population are university students) # Add an engaging case study or scenario <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario # Figure 1 could be more relevant to the scenario # Instead of a diagram, use an image which illustrates the scenario taking a place; use the diagram in a subsequent section # Include a relevant image <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box # Put the scenario in a feature box <!-- Overview – Explains problem --> # Clearly explains the psychological problem or phenomenon # Explains the psychological problem or phenomenon reasonably well/in a basic way # Briefly explains the psychological problem or phenomenon; provide more detail # Description of problem is too long/overly complicated—explain the psychological problem or phenomenon in a simpler way. Move detail into subsequent sections. # Clarity of written expression can be improved <!-- Overview – Focus questions --> # The focus questions are excellent (clear and relevant)/very good/good/reasonably good/basic/promising/insufficient # The focus questions could be improved by: ## being more specific to the topic (i.e., the sub-title) ## matching the top-level headings more closely ## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]] ## splitting double-barrelled questions into separate questions ## using bullet points as taught in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] ## being presented in a feature box to help guide the reader (fixed) # Add focus questions in a feature box # See copyedits for examples of possible improvements |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Excellent—key theories are very well explained and applied # Very good—key theories are well explained and applied; minor areas for improvements # Reasonably good—relevant theories are selected, described, and explained, with some room for improvement # Basic—a basic range of relevant theories are selected, described, and explained; there is considerable room for improvement # A promising range of ideas are presented but it is far from clear how this material is derived from a first person reading of the best peer-reviewed psychological theory and research about this topic # Insufficient use of relevant psychological theory about this topic # Reduce general theoretical background (e.g., definitions). Instead, summarise and link to related resources (i.e., other book chapters and/or Wikipedia articles). Increase emphasis on [[wikt:substantive|substantive]] aspects of theory that relate directly to the specific topic (i.e., the sub-title question). <!-- Theory – Builds on --> # Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds effectively on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds reasonably well on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds somewhat on other [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds in a basic way on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Builds on one [[w:|Wikipedia]] article and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Build more strongly on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # Doesn't build on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Insightful/Very good/Good/Reasonably good/Basic/Insufficient depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient/No use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # Key citations are well used # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) # If you didn't consult an original source (e.g., ?), cite it as a [https://apastyle.apa.org/style-grammar-guidelines/citations/secondary-sources secondary source] # If you didn't consult an original source, don't cite it <!-- Theory – Examples --> # Excellent/Very good/Good/Reasonably good/Basic use of examples to illustrate theoretical concepts # Consider using more examples to illustrate theoretical concepts # Use more examples to illustrate theoretical concepts # Insufficient use of examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient review of relevant research # Excellent emphasis on systematic reviews and/or meta-analyses # Some emphasis on systematic reviews and/or meta-analyses; more emphasis would be ideal # Insufficient use of relevant systematic reviews and/or meta-analyses # More detail about key studies would be ideal # Claims are well referenced # In some/many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic. # Insufficient use of academic, peer-reviewed citations (e.g., see the {{f}} tags) <!-- Research – Critical thinking --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out specific critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient integration between the most relevant theory(ies) and the best research # The chapter places more emphasis on theory than on research; strive for an integrated balance # Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence # The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> # Excellent/Very good/Good/Reasonably good/Basic summary and conclusion # Insufficient as a cohesive summary of the best psychological theory and research about the topic # Reads like generic [[Motivation and emotion/Assessment/Using generative AI|genAI output]]; write more compellingly in your own words # Is this section based on [[Motivation and emotion/Assessment/Using generative AI|genAI output]]? If so, this was not acknowledged in the [[Motivation and emotion/Wikiversity/Edit summary|edit summary]]. # Highlights the importance of the problem or phenomenon of interest # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are well summarised # Key points are summarised in a basic way # Summarise key points <!-- Conclusion – Focus questions --> # The focus questions are addressed # The take-away messages for each focus question could be spelt out more clearly # Address the focus questions <!-- Conclusion – Take-home messages --> # Clear take-home message(s) # Somewhat clear take-home message(s) # Add practical, take-home message(s) <!-- Conclusion – Word count --> # Not counted for marking purposes due to the chapter being over the maximum word count |7= <!-- Written expression – Style comments... --> <!-- Written expression – Written expression --> # Written expression ## The quality of written expression is excellent/very good/good/reasonably good/basic ## The quality of written expression is OK but there are several aspects which are below professional standard ## The quality of written expression is below professional standard. [https://www.canberra.edu.au/current-students/study-skills UC Study Skills] assistance is recommended to help improve writing skills. ### GenAI is over-used; write more in your own words, based on your own knowledge based on reading the primary sources ## Use active (e.g., "this chapter explores" or "this chapter explored") rather than passive voice (e.g., "this chapter will explore" or "this chapter has explored") [https://apastyle.apa.org/style-grammar-guidelines/grammar/active-passive-voice][https://www.grammarly.com/blog/active-vs-passive-voice/] ## Target an ''international'' rather than domestic national audience. [http://www.worldometers.info/world-population/australia-population/ Only 0.3% of the world human population lives in Australia]. ## Reduce excessive use of bold font <!-- Written expression – Sentences --> ## Some/Many sentences could be explained more clearly (e.g., see the {{explain}} and {{rewrite}} tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression ## Some/Many sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+. ## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence. <!-- Written expression – Paragraphs --> ## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences. ## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences. ## Bullet points are overused. Develop more of the bullet point statements into full sentences and paragraphs. <!-- Written expression – Language --> ## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., "it") instead of 1st (e.g., "we") or 2nd person (e.g., "you") in the main text. 1st of 2nd person can work well for case studies or feature boxes. ## Avoid phrases such as “as previously mentioned” or “as noted above,” as these add little value and can disrupt flow. If referencing another part of the chapter is necessary, use [[w:Help#Section linking|section linking]].## Embed direct quotes within sentences and paragraphs, rather than presenting them [[wikt:holus-bolus|holus-bolus]] ## "Individuals" is overused. "People" is usually a clearer term than "individuals". Use ''individuals'' to highlight each person separately (e.g., “individual test scores”) and ''people'' when referring to humans more generally. ## Remove [[w:weasel words|weasel words]]—they add bulk without improving meaning ## Use permanent, rather than relative, time references. For example, instead of "20 years ago", refer to something like "at the beginning of the 21st century". In this way, the text will survive better into the future, without needing to be rewritten. ## Avoid overly emotive language (e.g,. *) in science-based communication <!-- Written expression – Layout --> # Layout ## The chapter is well structured, with major sections using sub-sections ## The headings could be more clearly aligned with the focus questions ## The structure is overly complicated; simplify and integrate ## The structure is overly complicated; aim for 3 to 6 top-level headings between the Introduction and Conclusion ## The chapter structure is underdeveloped; expand by using subheadings ## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings ## Use the default heading style (e.g., remove additional numbers, italics, bo]ld, and/or change in font size) ## See earlier comment about [[#Heading casing|heading casing]] ## Provide more descriptive headings ## Move links from headings into their first mention in text ## Remove abbreviations/citations from headings ## Include an introductory paragraph before branching into the sub-sections (see {{expand}} tags) ## Integrate learning features rather than having it as a stand-alone section ## The Overview and Conclusion should not have subheadings (fixed) ## Remove colons (:) from the ends of headings <!-- Written expression – Grammar --> # Grammar and spelling are excellent # Grammar ## The grammar for some/many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback ## Improve the clarity and flow of the writing by checking and correct use of commas. See [https://www.grammarly.com/blog/comma/ Grammarly's guide to using commas]. ## Check and make correct use of [https://www.merriam-webster.com/grammar/em-dash-en-dash-how-to-use em dashes (instead of hyphens)] to set off an amplifying or explanatory element ## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats') ## Check and correct use of [https://www.google.com.au/search?q=grammar+that+vs+who that vs. who] ## Check and correct use of [https://www.google.com.au/search?q=affect+vs.+effect+grammar affect vs. effect] ## Check and correct use of [http://www.colonsemicolon.com/ semicolons (;) and colons (:)] ## Use past tense when describing research studies, although implications of findings could be in the present tense <!-- Written expression – Abbreviations --> ## Abbreviations ### Only use abbreviations such as e.g., i.e., et al., etc. inside [[w:Bracket#Parentheses|parentheses]], otherwise spell them out ### Check and correct formatting of abbreviations (such as e.g., i.e., etc.) ### Use abbreviations sparingly. Do not use abbreviations for minor/infrequently used terms. ### Spell out abbreviations on their first use, to explain them to the reader ### Once an abbreviation has been established (e.g., PTSD), use it consistently afterwards ### Only introduce abbreviations which are subsequently used <!-- Written expression – Spelling --> # Spelling ## Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., hypothesize vs. hypothesise; behavior vs. behaviour) ## Some words are misspelt (e.g., see the {{sp}} tags). 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See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence. # Thanks for the Wiki Commons uploads # Thanks very much for your extensive contributions # Use a numbered list per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] # No logged contributions }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:47, 10 October 2026 (UTC) ==See also== * [[Motivation and emotion/Assessment/Chapter|Book chapter guidelines]] * [[Template:METF]] * [[Template:MEMF]] [[Category:Motivation and emotion/Admin/2026]] [[Category:Motivation and emotion/Assessment/Chapter]] </noinclude> gg1hj9tdf06enq9kfa9r29cyuwe7zll 0 331021 2842523 2841760 2026-10-25T04:06:37Z StretchBeyond 3105744 I have gone back through the chapter to address minor grammar and factual queries raised during marking review. I have also included a link to the 2026 book chapter addressing fear extinction. 2842523 wikitext text/x-wiki {{title|Immersive therapy for PTSD treatment:<br>How does it work and what are the effects?}} __TOC__ == Overview == {{RoundBoxTop|theme=2}} [[File:Andriy 3MDR enhanced.gif|right|250x250px|'''Figure 1.''' Andriy moves through his 3MDR treatment.]] '''Scenario: Immersive PTSD therapies''' Andriy, a soldier, harnessed to a treadmill, walks towards an image he spent months avoiding, his clinical therapist beside him. This is multi-modular motion assisted memory desensitisation and reconsolidation (3MDR) therapy, one of a new generation of immersive therapies for post-traumatic stress disorder. Learn about immersive therapy and Andriy’s* experience below (*Andriy: a fictional name). {{RoundBoxBottom}} [[w:Post-traumatic_stress_disorder|Post-traumatic stress disorder]] (PTSD) develops after severe or life-threatening trauma and carries a substantial personal and societal cost (Figure 2), with military personnel disproportionately represented (Boska et al., 2025). PTSD's emotional impact is shaped by [[w:Emotional_dysregulation|emotional dysregulation]], a difficulty in managing intense feelings such as guilt, fear or shame (Westphal et al., 2017). This commonly triggers [[Cognitive psychology|cognitive]] and behavioural avoidance that offers short-term relief but prevents traumatic memory from being adaptively processed, trapping individuals in cycles of avoidance and chronic hyper-arousal (de Haart et al., 2026; van Gelderen et al., 2018). Immersive interventions, including [[w:Virtual_reality_therapy|virtual reality exposure therapy]] (VRET) and 3MDR, aim to break this avoidance cycle by creating controlled environments in which trauma cues can be safely approached rather than avoided (van Gelderen et al., 2018; Wiederhold & Wiederhold, 2025). This chapter explains the psychological theory behind these approaches, reviews the research evidence for their effects and considers their limitations. {{RoundBoxTop|theme=2}} '''Focus questions'''[[File:Crystal Clear app ktip.svg|left|20px|]] * Why is emotional processing important in PTSD? * How can immersive therapies influence the emotional processes underlying PTSD? * What does the research evidence show? * What are the costs and limitations of immersive therapies? {{RoundBoxBottom}} == Why is emotional processing important in PTSD? == [[File:Post-traumatic_stress_disorder_world_map_-_DALY_-_WHO2004.svg|alt=|thumb|271x271px|'''Figure 2.''' The World Health Organisation (2024) estimated that [https://www.who.int/news-room/fact-sheets/detail/post-traumatic-stress-disorder 3.9% of the world's population had PTSD] at some stage.]] [[File:PTSD.png|thumb|'''Figure 3.''' PTSD can deeply affect our emotions.]] PTSD is formally diagnosed according to [[w:DSM-5|DSM-5]] criteria and is characterised by disrupted executive and emotional processing systems, heightened threat perception, hyper-vigilance and persistent negative emotional states as depicted in Figure 3 (Kukharuk et al., 2025; López-Ojeda & Hurley, 2022; Osman et al., 2016). Emotional processing theory, together with emotion regulation frameworks and contemporary learning models, provides the theoretical foundation for understanding these disruptions and their clinical treatment. === Understanding PTSD and emotions === Emotional processing theory suggests that recovery requires the trauma memory network to be reactivated and integrated with with corrective information. In PTSD, this process is blocked and to manage distress, many individuals adopt cognitive and behavioural avoidance mechanisms (López-Ojeda & Hurley, 2022). Avoidance offers short-term relief, but prevents the traumatic memory from being reactivated, so it cannot be updated. Trauma reminders such as flashbacks continue to trigger extreme distress, trapping the individual in a maladaptive, self-reinforcing avoidance cycle (Figure 4) (Vermetten, Burback, et al., 2025b). At a neural level, Westphal et al. (2017) linked this to [[wikipedia:Transdiagnostic_process|transdiagnostic]] [[w:Emotional_dysregulation|emotion dysregulation]], in which the traumatic memory network remains isolated from the brain's [[w:Salience_network|salience]] and central executive networks. Effective treatment requires reactivating this network so the memory can be integrated (Vermetten, Burback, et al., 2025b; Westphal et al., 2017). This is empirically supported: A 2025 study of Danish military veterans (''n''= 142) found that emotional regulation difficulties explained an additional 28% of the variance in PTSD symptoms; combined with [[w:Comorbidity|comorbid]] symptoms, these factors accounted for 52% of the variance in severity (Elklit & Dahl, 2025). === Why treatment can be difficult === [[File:Avoidance and Processing Cycles.png|450x450px|right|'''Figure 4'''. Avoidance cycles and the anticipated positive responses associated with immersive therapies. Based on Lopez-Ojeda et al. (2022), Felemban et al. (2026), and Vermetten et. al. (2025b).]] Trauma-focused psychotherapies such as [[w:Prolonged_exposure_therapy|prolonged exposure]] (PE) and [[w:Cognitive_processing_therapy|cognitive processing therapy]] (CPT) ask patients to actively engage with distressing memories to generate [[Motivation and emotion/Book/2026/Fear extinction|fear extinction]], precisely what avoidance prevents (van Toorenburg et al., 2020). Many patients cannot tolerate the emotional exposure these therapies require (Lopes et al., 2025; van Gelderen et al., 2018; Vermetten, Burback, et al., 2025b). Because the trigger is never confronted, the brain cannot experience a prediction error needed to learn the threat has passed, so symptoms persist (de Haart et al., 2026; López-Ojeda & Hurley, 2022). This is reflected in outcomes where an estimated 39% of patients fail to respond to standard trauma-focused therapy, and dropout rates range from 16-48% (de Haart et al., 2026; Vermetten, Burback, et al., 2025b). Among military veterans, younger patients show heightened severity when trauma is central to their identity, while others turn to poor diet or substance abuse, further eroding emotional regulation (Efremov, 2025; Niles et al., 2023). [[w:Emotional_dysregulation|Emotional dysregulation]] was historically viewed as a fixed barrier requiring lengthy stabilisation before treatment could begin (van Toorenburg et al., 2020). More recent evidence suggests otherwise; although emotional regulation contributes to PTSD symptoms, it is also being increasingly viewed as a modifiable process with dynamic capacity, one able to improve as a natural consequence of successful memory processing (van Toorenburg et al., 2020). This reframes the clinical challenge as helping an individual safely approach and process the traumatic memory, not correcting a fixed deficit. [[File:UC PhD VR study.png|right|thumb|'''Figure 5.''' PhD work by Selvakumaran (2025) examining cultural adaptation using the US-''Bravemind'' system.]] Cultural context adds another layer of difficulty. Doctoral research at the [[University of Canberra]] is integrating exposure therapy with gamified, posture-adaptive cognitive rehabilitation to support physical and emotional recovery (Figure 5). Selvakumaran (2025) is examining how existing protocols such as used in the US-centric ''[https://medvr.ict.usc.edu/projects/bravemind.html Bravemind]'' system need cultural adaptation for Australian veterans and first responders, because their operational context and backgrounds differ from their US counterparts (Selvakumaran, 2025). These treatment gaps carry a substantial economic and personal cost, part of the motivation for developing alternatives such as immersive therapies. {{RoundBoxTop|theme=3}}[[Image:Crystal Clear app help index.svg|left|50px]] ;Predict the outcome A soldier with PTSD encounters a trauma-related image expecting: '''TRAUMA CUE → DANGER → DISTRESS → AVOID''' However, during immersive treatment, the expected danger does not occur. What is the most likely consequence?<quiz display=simple> { |type="()"} - Fear increases permanently. - Memory cannot change. + Prediction error creates opportunities for new learning. - Emotional processing stops.  } </quiz> <div style="text-align:right; color:red; font-weight:bold;"> Click "show" below to understand more⤵ </div> {{Hidden begin|title=Please consider the answer before opening this section}}The answer is '''C'''. <div style="text-align:left; color:black; font-weight:regular;"> Mismatches between expected and actual danger create a '''prediction error'''. This may contribute to fear extinction, emotion regulation, and memory reconsolidation. '''Think about it:''' If approaching rather than avoiding trauma can create new learning, what role might emotion regulation, fear extinction, prediction error and memory reconsolidation play? </div> {{Hidden end}} <div style="text-align:left; color:black; font-weight:regular;"> </div> {{RoundBoxBottom}} == How can immersive therapies influence emotional processes? == Immersive PTSD treatments mark a shift from passive, sedentary therapy towards active, embodied, interactive approaches (van Gelderen et al., 2018). Immersive therapy uses three psychological mechanisms and their interaction: multisensory presence, embodied cognition, and divergent thinking (López-Ojeda & Hurley, 2022; van Gelderen et al., 2018). Embodied cognition is how physical states of the body can directly modify states of the mind (van Gelderen et al., 2018). Immersion therapy goes beyond simple visual replication of a trauma memory; instead, it captures the participant’s visceral and cognitive focus by limiting distractions (Macey et al., 2026). === What is immersive therapy? === [[w:Immersion_therapy|Immersive therapies]] use [[w:Extended_reality|extended reality]] (XR) platforms, encompassing [[virtual reality]] (VR), [[Augmented Reality|augmented reality]] (AR), and [[w:Mixed_reality_game|mixed reality]] (MR), to create customisable, controlled, and standardised therapeutic environments (López-Ojeda & Hurley, 2022; Wiederhold & Wiederhold, 2025). This ecosystem is depicted in Figure 6. This chapter focuses on two applications: * VRET where therapists reconstruct traumatic scenarios in safe, graded environments, and * 3MDR, which extends this by having the patient move on a treadmill beside their therapist, rather than in a stationary, face-to-face session (de Haart et al., 2026; Felemban et al., 2026). [[File:XR and Human Senses.png|thumb|'''Figure 6.''' The Extended Reality environment. |center|750x750px]] === Presence and embodied cognition === Immersive therapy’s distinguishing feature is its ability to generate presence, or the psychological illusion of being ‘''there’''. The illusion is amplified by integrating synchronised audio, visual, [[w:Olfactory|olfactory]], [[wikipedia:Haptic_perception|haptic]], and movement stimuli (Lopes et al., 2025; López-Ojeda & Hurley, 2022). This increases engagement with the trauma memory and supports emotional processing (van Gelderen et al., 2018). In 3MDR, presence combines with cognition. The working principle is that physical states of the body can directly shape states of the mind (van Gelderen et al., 2018). Walking towards a virtual trauma reminder functions as a fear antagonistic action and rather than retreating in avoidance, the patient approaches, converting passive helplessness into active, empowered participation and disrupting the rigid and repetitive trauma narratives common in PTSD (de Haart et al., 2026; Osman et al., 2016; van Gelderen et al., 2018). Notably, extinction learning normally requires moderate-to-high-intensity activity to stimulate [[w:Brain-derived_neurotrophic_factor|brain-derived neurotrophic factor]] (BDNF), but 3MDR's walking pace (< 4 km/h) is too slow to generate meaningful BDNF secretion, making a major exercise-induced BDNF mechanism unlikely, although physiological factors cannot be ruled out. The approach action and the cognitive restructuring it enables may be more important psychological drivers (de Haart et al., 2026). === Prediction error and inhibitory learning === Immersive therapy integrates both inhibitory learning and memory reconsolidation models of exposure. While traditional exposure generates a competing safety association to suppress fear responses, physically approaching a trauma cue creates a prediction error between the anticipated, life-threatening event and actual reality, encountering safety instead of the expected catastrophe. This mismatch can destabilise the memory trace, allowing the memory network itself to be updated and reconsolidated in a less distressing form (van Gelderen et al., 2018; Vermetten, Burback, et al., 2025b; Wiederhold & Wiederhold, 2025). === Memory reconsolidation === According to [[w:Memory_consolidation|memory reconsolidation]] theory, traumatic memories retrieved in a safe immersive context can become [[w:Malleability_of_intelligence|malleable]], allowing new, safe information to reconsolidate the memory in a non-threatening form (van Gelderen et al., 2018; Vermetten, Burback, et al., 2025b). To prevent the patient from becoming overwhelmed, 3MDR uses dual-attention tasks such as tracking an oscillating ball (Figure 1), which tax working memory resources and reduce the vividness and emotional intensity of the memory (Vermetten, Burback, et al., 2025b). Emerging linguistic research suggests this processing is reflected in patients' language. Across successive 3MDR sessions, [[w:Affect_labeling|affective labelling]] of feelings such as guilt shifted from past-tense to present-tense narration, consistent with a renewed ability to integrate traumatic memories into present-moment awareness (Vermetten, Barcaro, et al., 2025a). == What does the research evidence show? == Addressing these barriers is important for any PTSD population, but especially critical for military populations with high treatment failure and dropout rates (van Gelderen et al., 2018). Immersive therapies, as outlined in Table 1, can tailor patient-selected trauma cues to improve access to traumatic memory networks (Vermetten, Burback, et al., 2025b). === Virtual reality exposure therapy === [[w:Virtual_reality_therapy|VRET]] reconstructs traumatic events in a structured context (Felemban et al., 2026; López-Ojeda & Hurley, 2022). This may help bypass imagination challenges such as emotional numbing or amnesia that can prevent patients from engaging with traditional therapy (Macey et al., 2026). A meta-analysis of nine VRET randomised control studies (involving 444 military veterans) for PTSD studies found substantial symptom reductions, averaging a 33.73-point decrease in the 0-80 point [[w:Clinician_Administered_PTSD_Scale|Clinician-Administered PTSD Scale]] and a 20.96-point decrease in the 0-80 point [https://www.ptsd.va.gov/professional/assessment/adult-sr/ptsd-checklist.asp PTSD Checklist PCL-5) scale] (Felemban et al., 2026). Because changes of 10-20 points on these scales are considered clinically significant, this could mean the difference between severe functional impairment and mild or subclinical symptoms (Boska et al., 2025; de Haart et al., 2026; Felemban et al., 2026). Comparative effects against other active PTSD treatments remain modest, but VRET appears to be a more engaging alternative to conventional treatment (Felemban et al., 2026). === 3MDR === 3MDR takes the same multisensory effect in VRET and adds an activating context. Rather than a sedentary, face-to-face session, the patient and the therapist face the virtual display together (van Gelderen et al., 2018; Vermetten, Burback, et al., 2025b). It follows a three-phase protocol: pre-platform preparation, platform treadmill exposure, and post-platform re-consolidation (Vermetten, Burback, et al., 2025b). In a trial involving treatment-resistant PTSD; 3MDR showed large effect sizes from pre-treatment to six-months (''n'' = 134, ''d'' = 1.0) and high-acceptability, with 7-20% dropout rates (de Haart et al., 2026; van Gelderen et al., 2018; Vermetten, Burback, et al., 2025b). This improved treatment acceptability result is substantially lower than the 16-48% typical trauma-focused therapy in military populations (Lewis et al., 2020). Improvements are not limited to PTSD symptoms. A trial of 62 adults with severe PTSD, including childhood sexual trauma, found intensive trauma-focused treatment improved emotion-regulation abilities regardless of PTSD outcome (van Toorenburg et al., 2020). Some researchers link this improvement to positive psychology [[w:Broaden-and-build|broaden-and-build theory]], theorising that as patients regain a sense of safety and control, this may support a positive spiral of emotional flexibility that reinforces recovery (Fredrickson, 2001; Niles et al., 2023; Westphal et al., 2017). Critically, the exact causation remains unknown. Given the recent innovations in immersive therapies and small study samples, it remains unclear whether they work through exposure-based extinction, enhanced emotional engagement, physiological factors, embodiment, or positive cognitive restructuring. These questions are relevant to both basic and applied research and have not yet been adequately addressed by current trial designs. '''Table 1''' ''Treatment Effects and Theorised Psychological Mechanisms Involved'' {| class="wikitable mw-collapsible" | valign="top" |'''Clinical Dimension''' | valign="top" |'''Traditional''' | valign="top" |'''Immersive''' | valign="top" |'''Psychological Mechanisms''' |- | valign="top" |'''Context''' | valign="top" |'''''Sedentary'''''. Face-to-face, verbally describes trauma. | valign="top" |'''''Activating.''''' Dynamic, multi-sensory environment. | valign="top" |'''''Fear Antagonistic Action.''''' '''''Approach behaviours''''' '''''Prediction Errors''''' |- | valign="top" |'''Trauma cue delivery''' | valign="top" |'''''Imaginary Retrieval''''' Patient capacity | valign="top" |'''''Multisensory Immersion''''' Highly tailored | valign="top" |'''''External Scaffolding.''''' Bypasses internal barriers to activate memory networks. |- | valign="top" |'''Processing''' | valign="top" |'''''Convergent processing.''''' Repeated narration and fear habituation. | valign="top" |'''''Active Narrative Processing.''''' Interactive, real-time affective labelling and dual-attention tasks. | valign="top" |'''''Working Memory.''''' Memory taxation reduces vividness and emotional intensity. |- | valign="top" |'''Engagement''' | valign="top" |'''''Attrition.''''' Dropout rates 16-48%. | valign="top" |'''''Acceptability.''''' Attractive. Dropout rates 7-20%. | valign="top" |'''''Sustained Motivation.''''' Presence and safety in immersive environment. |} === Applied example: 3MDR treatment in Ukraine === {{RoundBoxTop|theme=2}} '''Scenario: Ukraine War: an applied 3MDR example''' Andriy*, a Ukrainian Armed Forces volunteer, is struggling with hyper-vigilance, anxiety, and depression (Kukharuk et al., 2025). Traditional ‘talk’ therapy feels impossible; his mind remains in a constant state of combat readiness, even in quiet rooms. In the pre-platform phase, Andriy identifies ‘hotspot’ memories in photographs. On the treadmill, harnessed and walking, he faces a panoramic screen as personalised warm-up music plays, selected to keep him connected to his traumatic memory network (Vermetten et al., 2025b). As his hotspot image fills the screen, his therapist asks three questions (Vermetten et al., 2025b): 1. What do you '''SEE''' ? 2. What does it '''TELL''' you? 3. What do you '''FEEL''' in your body NOW? When Andriy identifies a surge of shame, the word '''GUILT''' is displayed as an affective label {{ic|why not show '''SHAME'''?}}, and a dual-attention task begins. He tracks an oscillating, numbered ball while he stays with the emotion, taxing his working memory and reducing the intensity of the recalled trauma. [[File:Andriy 3MDR enhanced.gif|center|600x600px|'''Figure 7.''' Andriy's 3MDR treatment.]] By the end of the session, Andriy had walked toward what he used to avoid, creating a mismatch between his expectation of threat and his safety and turning a rigid, stuck memory into a manageable narrative. *''Andriy, a fictional name given to one of 69 Ukrainian veterans, participating in a 2023 randomised controlled trial. Many demobilised after one year because of mental or neurological injuries'' (Kukharuk et al., 2025). {{RoundBoxBottom}} == What are the costs and limitations of immersive therapies? == PTSD carries a substantial economic burden. PTSD was associated with an estimated US$232 billion in excess costs in the United States in 2018 and over £40 billion in the United Kingdom in 2020-21, 92% of which were indirect rather than direct clinical costs (Davis et al., 2022; Montgomery-Marks et al., 2025). In Australia, the annual cost of PTSD per military veteran was estimated at $112,172 in 2025 (Magnusson & Dey, 2025). In addition, individual healthcare costs surge by 142% in the year following a PTSD diagnosis, with comorbidities tripling this effect (Bothe et al., 2020). These figures span several countries and years and should be read as an indication of scale rather than as directly comparable totals. Immersive therapy’s ability to move from small trials into mainstream PTSD treatment remains limited by methodological [[w:Homogeneity_and_heterogeneity|heterogeneity]], small sample sizes, and a lack of long-term data (Felemban et al., 2026). Many trials rely on military-heavy, treatment-resistant samples with few active comparison conditions, which limits the generalisation of findings to civilian or initial treatment populations. Practical issues such as [[w:Virtual_reality_sickness|cyber sickness]] or motion sickness can disrupt participation (Kukharuk et al., 2025). Structural barriers include workforce training and equipment costs. Basic VR systems cost around US$3,500 per provider headset annually, with advanced simulation environments costing up to US$200,000 (Garrett et al., 2018). Despite this, adoption of immersive therapy is scaling. The [[w:United_States_Department_of_Veterans_Affairs|United States Veterans Affairs]] have expanded VR use from five medical centres in 2017 to over 154 centres and 2,300 staff, with applications now including over 40 documented clinical interventions such as chronic pain and suicide intervention (Bailey et al., 2024). Reflecting this potential growth trend, some market analysts estimate the global PTSD-focused VR therapy market was worth US$1.59 billion in 2025, forecasted to reach US$5.94 billion by 2032, driven largely by growing mental health awareness and the absence of standard clinical protocols (Stratistics MRC, 2025, an industry market report). These limitations do not undermine the case for immersive therapy, but they show its evidence base and infrastructure are still maturing. Demonstrating rigorously how these therapies work, rather than assuming their novelty accounts for their effects is essential to avoid misallocating resources to unproven interventions and may support an emerging shift towards greater emphasis on post-traumatic growth and renewed optimism (Trejo et al., 2015; Vermetten, Burback, et al., 2025b; Wiederhold & Wiederhold, 2025). {{Robelbox|theme=2|title=Quiz}} <quiz display=simple> {Which of the following is an expected outcome of immersive therapy? |type="()"} - Patients are at risk because of an uncontrolled environment. - The environments represent traditional face-to-face treatments. - Failure and dropout rates are higher than traditional PTSD treatment. + Patients often broaden and build an upward spiral of emotion and optimism. </quiz> {{Robelbox/close}} == Conclusion == PTSD traps people in a [[#Understanding PTSD and emotions|self-reinforcing cycle of avoidance]] that blocks the adaptive processing necessary for traumatic memories to be resolved. Standard trauma-focused therapies require patients to confront exactly what this cycle causes them to avoid, which contributes to [[#Why treatment can be difficult|high non-response and dropout rates]]. Immersive therapies such as VRET and 3MDR address this by using presence, embodied cognition, prediction error and memory reconsolidation to help patients safely approach trauma cues rather than avoid them]]. The technology itself is only a delivery mechanism that enables the processes; the positive benefits appear to arise from the psychological mechanisms that immersive therapies facilitate, rather than the novelty of technology alone. The [[#What does the research evidence show?|evidence to date]] shows meaningful reductions in PTSD symptoms and comparatively low drop-out rates, alongside broader gains in emotional regulation. However, small samples, methodological variation, and a lack of long-term data mean the evidence base is still developing, and [[#What are the costs and limitations of immersive therapies?|cost and infrastructure barriers remain significant]]. {{RoundBoxTop|theme=11}} [[File:Nuvola apps kuser.svg|Nuvola apps kuser|left|20px]] ''' Take-home message:''' Immersive therapies do not heal PTSD trauma through technological novelty or digital feedback. Instead, they create a safe, dynamic space that lets individuals confront trauma and reprocess memories into something they can live with. In doing so, people’s lives may once again shift toward value, optimism, and wellness. {{RoundBoxBottom}} == See also == '''Sub-pages''' * [[/Non-invasive brain stimulation techniques/|Non-invasive brain stimulation techniques and PTSD]] '''Wikiversity''' * [[Motivation and emotion/Textbook/Emotion/Anxiety|Anxiety]] (Book chapter, 2010) * [[Evidence-based assessment/Posttraumatic stress disorder (disorder portfolio)|Evidence-based PTSD assessment]] * [[Motivation and emotion/Book/2026/Fear extinction|Fear extinction]] (Book chapter, 2026) * [[Motivation and emotion/Book/2024/Sense hacking|Sense hacking]] (Book chapter, 2024) '''Wikipedia''' * [[w:Affect_labeling|Affective labelling]] * [[Augmented Reality|Augmented reality]] * [[w:Brain-derived_neurotrophic_factor|Brain-derived neurotrophic factor]] * [[w:Broaden-and-build|Broaden-and-build theory]] * [[w:Cognitive_processing_therapy|Cognitive processing therapy]] * [[w:Virtual_reality_sickness|Cyber sickness]] * [[w:DSM-5|DSM-5]] * [[w:Emotional_dysregulation|Emotional dysregulation]] * [[w:Extended_reality|Extended reality]] * [[w:Malleability_of_intelligence|Malleability of intelligence]] * [[w:Memory_consolidation|Memory reconsolidation]] * [[w:Mixed_reality_game|Mixed reality]] * [[w:Post-traumatic_stress_disorder|Post-traumatic stress disorder]] * [[w:Prolonged_exposure_therapy|Prolonged exposure therapy]] * [[wikipedia:Virtual_reality_therapy|Virtual reality exposure therapy]] == References == {{Hanging indent|1= Bailey, A. L., Kirsh, S., Rawlins, C., Persky, S., & Clancy, C. (2024). Early scaling of immersive technology within the Veterans Health Administration. ''NEJM Catalyst Innovations in Care Delivery'', 5(4). https://doi.org/10.1056/cat.23.0356 Boska, R. L., Bishop, T. M., Capron, D. W., Paxton Willing, M. M., & Ashrafioun, L. (2025). Difficulties with emotion regulation within PTSD clusters and moral injury subtypes. ''Military Psychology'', 37(2), 159-167. https://doi.org/10.1080/08995605.2024.2322904 de Haart, R., Daniels, J. K., Timmerman, M. E., Cath, D. C., & Lommen, M. J. J. (2026). Augmenting virtual reality exposure for PTSD with physical activity: Study protocol of a randomised controlled trial. ''European'' ''Journal of Psychotraumatology'', 17(1), 2605803. https://doi.org/10.1080/20008066.2025.2605803 Efremov, A. (2025). Age-specific mental health profiles of combat veterans: Post-traumatic stress disorder and related disorders. ''Journal of Rational-Emotive & Cognitive-Behavior Therapy'', 44(4), 1-15. https://doi.org/10.1007/s10942-025-00637-7 Elklit, A., & Dahl, N. H. (2025). Emotion regulation difficulties, aggression, and PTSD symptoms in Danish treatment-seeking veterans. ''Scandinavian Journal of Military Studies'', 8(1), 308-326. https://doi.org/10.31374/sjms.264 Felemban, R. G., Alzahrani, R. R., Alrefaei, N. F., Alharbi, N. M., Alghamdi, A. S., & Alqadi, S. (2026). Efficacy of virtual reality-based exposure therapy for post-traumatic stress disorder in military veterans: A systematic review and meta-analysis. ''Frontiers in Psychiatry'', 17, 1857109. https://doi.org/10.3389/fpsyt.2026.1857109 Fredrickson, B. L. (2001). The role of positive emotions in positive psychology: The broaden-and-build theory of positive emotions. ''American Psychologist'', 56(3), 218-226. https://doi.org/10.1037/0003-066X.56.3.218 Garrett, B., Taverner, T., Gromala, D., Tao, G., Cordingley, E., & Sun, C. (2018). Virtual reality clinical research: Promises and challenges. ''JMIR Serious Games'', 6(4), e10839. https://doi.org/10.2196/10839 Kukharuk, O., Tkalich, K., Kamash, N., & Georgiou, O. (2025). Effectiveness of immersive VR therapy in reducing stress-associated symptoms in Ukraine. ''European Journal of Psychotraumatology'', 16(1), 2488097. https://doi.org/10.1080/20008066.2025.2488097 Lewis, C., Roberts, N. P., Andrew, M., Starling, E., & Bisson, J. I. (2020). Psychological therapies for post-traumatic stress disorder in adults: Systematic review and meta-analysis. ''European Journal of Psychotraumatology'', 11(1), 1729633. https://doi.org/10.1080/20008198.2020.1729633 Lopes, M. K. S., Perreault, L., de Jesus, B. Jr., Roberge, M. C., & Falk, T. H. (2025). Subjective and objective evaluation of the benefits of multisensory virtual nature immersion for patients with post-traumatic stress disorder. In ''Proceedings of the 17th International Conference on Quality of Multimedia Experience'' (QoMEX) (pp.1-5). IEEE. https://doi.org/10.1109/QoMEX65720.2025.11219945 López-Ojeda, W., & Hurley, R. A. (2022). Extended reality technologies: Expanding therapeutic approaches for PTSD. ''The Journal of Neuropsychiatry and Clinical Neurosciences'', 34(1), 1-5. https://doi.org/10.1176/appi.neuropsych.21100244 Macey, A.-L., Macey, J., & Hamari, J. (2026). Emotion regulation in immersive virtual reality environments: A scoping review. ''Interacting with Computers'', 29, 1-20. https://doi.org/10.1093/iwc/iwag029 Niles, B., Lang, A., & Olff, M. (2023). Complementary and integrative interventions for PTSD. ''European Journal of Psychotraumatology'', 14(2), 2247888. https://doi.org/10.1080/20008066.2023.2247888 Osman, A., Paczynski, M., & Jha, A. P. (2016). Affective expectations influence neural responses to stressful images in soldiers. ''Military Psychology'', 29(1), 41-57. https://doi.org/10.1037/mil0000128 Selvakumaran, R. V. (2025). Developing virtual reality (VR) simulations with embedded user analytics for cognitive rehabilitation in PTSD veterans. In ''Proceedings of the 27th International Conference on Multimodal Interaction'' (pp. 740-744). ACM. https://doi.org/10.1145/3716553.3750826 Trejo, B. C., Richard, E. M., van Driel, M., & McDonald, D. P. (2015). Cross-cultural competence: The role of emotion regulation ability and optimism. ''Military Psychology'', 27(5), 276-286. https://doi.org/10.1037/mil0000081 van Gelderen, M. J., Nijdam, M. J., & Vermetten, E. (2018). An innovative framework for delivering psychotherapy to patients with treatment-resistant posttraumatic stress disorder: Rationale for interactive motion-assisted therapy. ''Frontiers in Psychiatry'', 9, 176. https://doi.org/10.3389/fpsyt.2018.00176 van Toorenburg, M. M., Sanches, S. A., Linders, B., Rozendaal, L., Voorendonk, E. M., Van Minnen, A., & De Jongh, A. (2020). Do emotion regulation difficulties affect outcome of intensive trauma-focused treatment of patients with severe PTSD? ''European Journal of Psychotraumatology'', 11(1), 1724417. https://doi.org/10.1080/20008198.2020.1724417 Vermetten, E., Barcaro, S., Espejo, E., Bellini, P., Roy, M. J., & Bremault-Phillips, S. (2025a). Linguistic analysis of patients’ labels during 3MDR psychotherapy. ''Psychiatry and Clinical Psychopharmacology'', 35(Suppl. 1), S29. https://doi.org/10.5152/pcp.2025.241024 Vermetten, E., Burback, L., Sevigny, P. R., Nijdam, M. J., Winkler, O., Espejo, E., Sessoms, P., Bright, K., Roy, M. J., & Brémault-Phillips, S. (2025b). Brief manual for multi-modal motion-assisted memory desensitization and reconsolidation therapy for the treatment of post-traumatic stress disorder. ''Psychiatry and Clinical Psychopharmacology'', 35(Suppl. 1), S122. https://doi.org/10.5152/pcp.2025.241028 Westphal, M., Aldao, A., & Jackson, C. (2017). Emotion dysregulation in comorbid posttraumatic stress disorder and substance use disorders: A narrative review. ''Military Psychology'', 29(3), 216-233. https://doi.org/10.1037/mil0000157 Wiederhold, B. K., & Wiederhold, M. D. (2025). Virtual reality therapy combined with physiological monitoring provides effective treatment, with objective metrics, for post-traumatic stress disorder. ''Expert Review of Medical Devices'', 22(2), 117-119. https://doi.org/10.1080/17434440.2025.2454930 }} == External links == * '''Web:''' [https://www.healthdirect.gov.au/post-traumatic-stress-disorder-ptsd Post traumatic stress disorder] (Australian Government) * '''Web:''' [https://www.ptsd.va.gov/index.asp National Center for PTSD information home page] (US Government) * '''Web:''' [https://www.innovation.va.gov/hil/views/immersive/immersive-programs.html US Veterans Affairs Immersive Programs Innov][https://www.innovation.va.gov/hil/views/immersive/immersive-programs.html ation] (US Government) * '''Web:''' [https://defenceveteransuicide.royalcommission.gov.au/publications/final-report Australian Royal Commission into Defence and Veteran Suicide - Final Report] [https://defenceveteransuicide.royalcommission.gov.au/publications/final-report](Australian Government) * '''Web:''' [https://www.strategymrc.com/report/virtual-reality-therapy-for-ptsd-market Virtual reality therapy for PTSD market forecasts to 2032:] (Stratistics MRC, a 2025 industry market report). * '''Podcast:''' [https://shows.acast.com/5c3353e484e2e79370e1d135/5c3353ef84e2e79370e1d13c PTSD Podcast] (Peace of Mind: Mental Health and Psychiatry, ACAST, 43 min) * '''Video:''' [https://www.youtube.com/watch?v=bD43R_oa6qo 3MDR: Virtual reality treatment for veterans] (National Centre for Mental Health,Youtube, 2:46 min) * '''Video:''' [https://www.youtube.com/watch?v=jL2bKmniMTc VR exposure for combat PTSD] (PsyTech VR, Youtube, 2:20 min) * '''Final Report:''' [https://www.ncmh.info/wp-content/uploads/2020/05/Cardiff-3MDR-Study-Final-Report-with-cover-21.5.20.pdf 3MDR randomised control trial - Final-Report] (Cardiff University) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Psychotherapy]] [[Category:Motivation and emotion/Book/Trauma]] gz94icr1xmq9rbq19ry8o9s515cqjpg 0 331095 2842560 2840999 2026-10-25T11:14:40Z Isa Murphy 3105869 Grammatical correction to multiple in-text references. 2842560 wikitext text/x-wiki {{title|Alcohol use for emotion regulation:<br>Why and how do people use alcohol to regulate their emotions?}} __TOC__ == Overview == {{RoundBoxTop|theme=4}} [[File:Alcohol-bar-party-cocktail (24218389642).jpg|right|thumb|200px|'''Figure 1'''. An alcoholic beverage on a bar bench.]] ;Consider this scenario It is Friday night; you've just completed a long week balancing university assessments, work and social engagements. You're feeling overwhelmed, stressed and mentally exhausted. You go out with some friends to celebrate the weekend and decide to have a few drinks. You start to notice your nervous system relaxing; you feel less tired, stop thinking about the stressful week you've had, and find your mind and body relaxing. What started as a few social drinks with friends turns into every night. You now find yourself dependent on alcohol to relax, drinking whenever you are overwhelmed or stressed. Now think: Are you actually reliant on the alcohol or the relief of overwhelming emotions that alcohol provides? {{RoundBoxBottom}} Alcohol consumption is a widespread phenomenon that is often associated with a variety of contexts, including events, celebrations and recreational activity. However, people may also use alcohol to manage emotional experiences. Alcohol use for emotion regulation refers to drinking that is motivated, at least in part, by a desire to change, reduce or escape an emotional state (Cooper, 1994). Understanding this pattern of alcohol consumption requires consideration not only of the emotional experiences that precede drinking but also the perceived effects that follow it. Emotion regulation is the process an individual uses to influence which emotion they experience, when those emotions occur, and how and why they are experienced, expressed and responded to (Gross, 1998). Although individuals can use strategies such as problem solving, cognitive reappraisal and seeking social support, some may rely on alcohol when they perceive their emotions as difficult to manage. Alcohol can produce temporary changes in mood, reduce inhibition and create feelings of relaxation, potentially reinforcing its use as a coping strategy. Understanding the relationship between alcohol and emotion regulation is important because it can provide greater insight into why alcohol use can become difficult to control and why effective approaches to problematic drinking may need to address both alcohol consumption and the underlying difficulties with emotion regulation. This book chapter explores the psychological concept of emotion regulation, including why and how people use alcohol to regulate their emotions. It also examines what happens in the brain and body after alcohol consumption, including its effects on neurotransmitter systems involved in inhibition, excitation, reward and emotional processing. Finally, it compares short-term emotional relief with long-term emotional regulation and examines healthier alternative strategies for managing difficult emotions. {{RoundBoxTop|theme=3}} '''Focus questions''' * What is emotional regulation and how is it related to alcohol use? * Why do people use alcohol to regulate emotions? * What are the outcomes of excessive alcohol use? * How does alcohol influence the nervous system? {{RoundBoxBottom}} =='''Defining emotion regulation and alcohol use'''== To understand the relationship between emotional regulation and alcohol use, the core mechanisms underlying each concept are considered independently. === What is emotion regulation? === Emotion regulation is a fundamental psychological process that enables individuals to manage and manipulate their emotional responses. In line with Gross (1998), emotion regulation is the process through which people influence the emotions they have, when they have them, and how they experience them. To understand emotion regulation, it is essential to understand emotions. Emotions are adaptive responses that help individuals respond to events relevant to their goals, needs and values. They influence attention, cognition, physiological activity and behaviour. However, emotional responses may sometimes be more intense, persistent or difficult to manage than a situation requires. As a result, people develop strategies to regulate and adapt to their emotions (Barrett et al., 2007). Emotion regulation works best when strategies fit the situation and support long-term goals. Distraction can offer temporary relief from overwhelming situations, while problem-solving suits distress when a situation or problem can be solved. Cognitive reappraisal is linked to positive emotional outcomes and greater well-being (Gross & John, 2003). Thus, emotion regulation is a flexible process, not a dependence on a single strategy. Difficulties with emotion regulation can arise when individuals struggle to understand their emotions, tolerate distress, control emotional responses, or select effective strategies. When negative emotions feel particularly intense or difficult to manage, individuals may seek strategies that provide rapid relief. This is relevant to alcohol use, as rather than addressing the source of an emotion, drinking may offer temporary distraction or relief from distress. Consequently, alcohol can function as an emotion-regulation strategy, particularly when an individual has learned to associate drinking with feeling calmer, less anxious, or less emotionally distressed. Understanding emotion regulation is therefore important for explaining why some people may turn to alcohol when experiencing negative emotions and why this pattern of behaviour can become reinforced over time. ==== Process model of emotion regulation ==== Gross's (1998) foundational model of emotion regulation proposed 5 main emotion regulation strategies: selecting the situation, modifying the situation, deploying attention, changing cognitions, and modulating responses. Situation selection involves approaching or avoiding situations based on their expected emotional consequences, while situation modification involves altering aspects of a situation to influence its emotional impact. Attentional deployment involves directing attention towards or away from particular aspects of an experience. Cognitive change, commonly referred to as cognitive reappraisal, involves changing how a situation is interpreted to change its emotional meaning. Finally, response modulation involves attempting to influence an emotional response after it has occurred, such as reducing outward expression or altering physiological arousal (Gross, 1998). Individuals may use several of these strategies in everyday life. For example, a person experiencing stress about an examination might manage it by creating a study schedule, reframing the examination as an opportunity to demonstrate knowledge, temporarily distracting themselves, or seeking support from another person. These emotion-regulation strategies vary in intensity, timing, and effectiveness across individuals and contexts (Gross, 1998). == '''The effects of alcohol use on emotion regulation''' == People may use alcohol to regulate their emotions because they expect it to provide temporary relief from negative emotional experiences. To understand the motivations for alcohol use, it is essential to examine the psychological processes that reinforce it. === Why do people use alcohol to regulate their emotions? === Emotion regulation requires changing the intensity, duration, meaning or expression of an emotional response (Gross, 1998). People use many different strategies to regulate emotions, including seeking social support, problem solving, cognitive reappraisal, exercise and avoidance. Alcohol can become another strategy when individuals believe that drinking will provide rapid relief from distress. The reasons people consume alcohol are referred to as drinking motives, including social, enhancement, conformity, and coping motives (Cooper, 1994). Coping motives are particularly relevant to emotion regulation because they involve drinking to manage unpleasant emotional experiences (Messman-Moore et al., 2014). People who drink to cope may use alcohol to reduce negative emotions, escape unpleasant thoughts, or make difficult emotional experiences feel more manageable. Importantly, drinking to cope does not mean that alcohol reliably improves emotional well-being. Repeated reliance on alcohol can contribute to interpersonal, psychological and behavioural difficulties, which may create further negative emotions and reinforce continued drinking (Petit et al., 2015). The relationship between alcohol and emotion regulation is therefore complex. Alcohol can produce short-term changes in emotional experience, including reduced tension, increased relaxation and decreased inhibition. These effects can make drinking appear effective as an emotion-regulation strategy (Messman-Moore et al., 2014). However, alcohol also affects cognitive functioning, behavioural control and physiological systems involved in emotion regulation. Furthermore, alcohol's effects on mood can vary according to the amount consumed, the individual's expectations, personality, drinking history and the emotional context in which drinking occurs (Petit et al., 2015). === '''Consequences of alcohol use for emotion regulation''' === Although alcohol may provide temporary emotional relief, this does not necessarily mean that it is an effective or healthy long-term emotion-regulation strategy. The immediate reduction in distress can reinforce drinking behaviour, increasing the likelihood that an individual will use alcohol again when experiencing similar emotions. Over time, repeated alcohol use can contribute to tolerance, dependence and changes in the brain's systems involved in stress, reward and emotional processing. Consequently, a cycle may develop in which negative emotions lead to alcohol consumption, alcohol provides temporary relief, and the alcohol-related effects contribute to further emotional difficulties and continued drinking. {{f}} The distinction between drinking to enhance positive emotions and drinking to cope with negative emotions is particularly important. Someone may drink to make a social experience more exciting, whereas another individual may drink after a difficult day to stop feeling stressed. Although both individuals consume alcohol, the psychological function of the behaviour is different (Kuntsche et al., 2005). === '''What happens when alcohol enters the body''' === When alcohol first enters the body through oral consumption, it travels into the stomach and intestine and then passes into the bloodstream. Once in the bloodstream, alcohol distributes throughout the body's water systems until it reaches organs, specifically the brain. Once in the brain, alcohol interferes with the communication between neurotransmitter systems, which results in alterations in mood, behaviour, coordination, impulse control and memory (Cargiulo, 2007; Goldman et al., 2025) {{ic|Use APA style for multiple citations}}. The effects of alcohol can initially be experienced as relaxing or calming, which may make it appealing to individuals experiencing stress, anxiety, sadness or other forms of emotional distres {{f}}. Research has demonstrated an association between difficulties in emotion regulation and substance use, suggesting that problems managing emotions may contribute to the use of substances as a method of coping (Cargiulo, 2007). Repeated alcohol consumption can affect sleep, physical health, relationships, work or study and psychological well-being (Cargiulo, 2007). These consequences can create additional sources of stress. Consequently, alcohol may become part of a cycle in which it is used to manage problems that are partly maintained by alcohol use itself. == '''The long-term and short-term effects of alcohol as emotion regulation''' == Alcohol interferes with the development of alternative coping strategies {{f}}. Repeatedly drinking in response to distress limits opportunities to practise other emotion-regulation methods, potentially making alcohol the most familiar response {{f}}. Thus, the concern is not only alcohol’s immediate emotional effects but also how reliance on it may weaken emotion regulation over time. Excessive, prolonged drinking can also severely affect the nervous system, especially the brain, altering emotions, perception, and personality (Oscar-Berman et al., 2026). Individuals who experience difficulties with emotion regulation are most susceptible to using alcohol as a method of self-regulation and, in turn, will experience the greatest susceptibility to developing an alcohol use disorder (Rydzewska et al., 2023; Brown & Melas, 2024). For example, consider an individual who experiences significant anxiety after a stressful workday. They consume alcohol and temporarily feel calmer and less preoccupied with their worries. The immediate reduction in anxiety provides reinforcement for drinking. The next time the person experiences similar anxiety, they may remember the previous relief and be more likely to drink again. Over time, alcohol can become associated with emotional distress, creating a learned pattern in which negative emotions trigger alcohol consumption. <quiz display="simple"> {Alcohol can be used to make someone feel relaxed for a short period of time, but it doesn’t solve the problems that is causing the stress: |type="()"} + True - False {Why might an individual decide to use alcohol for emotional regulation? |type="()"} - To increase their ability to problem solve. + To temporarily reduce or escape unpleasant emotions. - To improve their long-term emotional regulation - To permanently eliminate negative emotions </quiz> === Drinking to cope with negative emotions === Drinking to cope with negative emotions is among the most extensively examined psychological motives for alcohol consumption. Individuals may drink in response to a wide range of unpleasant experiences. The underlying assumption is that alcohol will alter the emotional state or make the experience easier to tolerate. Cooper et al. (1995) demonstrated {{how}} that coping motives are associated with alcohol-related problems, suggesting that the psychological function of drinking is relevant to understanding excessive consumption. One explanation is that coping-motivated drinking can create a cycle involving negative emotion → alcohol consumption → short-term relief → longer-term consequences → further negative emotion. A stressful or upsetting event produces emotional distress. The person drinks to reduce that distress. Alcohol produces temporary changes in emotional experience, reinforcing the behaviour. However, intoxication may result in interpersonal problems, impaired functioning, sleep disruption, regret or other consequences. These outcomes can subsequently contribute to further distress, increasing the motivation to drink again. {{f}} This cycle is particularly important because the immediate and delayed effects of alcohol may differ. The person may focus on the immediate emotional benefit, which is most salient when deciding to drink. This does not mean that every person who drinks when upset will develop problematic alcohol use (Khantzian, 1997). Emotionally motivated drinking exists on a continuum, and many people consume alcohol without developing alcohol-related disorders. Risk depends on factors such as drinking frequency, quantity consumed, individual vulnerability, available coping strategies, and the broader social environment. === The effects of alcohol on emotion === Alcohol can influence emotional experiences in several ways. At lower levels of intoxication, individuals may report relaxation, increased sociability and reduced anxiety. As consumption increases, however, cognitive and behavioural impairment becomes more pronounced (Rydzewska et al., 2023). Emotional responses may also become less controlled. The effects of alcohol are not consistent across individuals or situations. Alcohol can sometimes reduce negative affect, but it can also increase emotional instability or negative mood. Experimental and review research has highlighted the importance of context, expectations and individual differences when examining alcohol's effects on emotion (Curtin et al., 2001). {{ABCDE|Question=Test your knowledge: <quiz display="simple"> {What is emotion regulation?} - Eliminating all emotions - Increasing alcohol consumption + Managing or changing emotional responses {What is coping-motivated drinking?} + Drinking to manage or escape difficult emotions - Drinking only at social events - Drinking to improve physical health {What is cognitive reappraisal?} + Changing how a situation is interpreted to influence an emotional response - Avoiding every stressful situation - Drinking to forget an experience </quiz>}} === Positive and negative reinforcement === The reinforcement model {{f}} provides an important framework for explaining why alcohol use can become habitual. Positive reinforcement occurs when a behaviour increases because it produces a desirable outcome. For example, a person may drink because alcohol makes a celebration feel more enjoyable. Negative reinforcement occurs when drinking reduces an undesirable state, such as anxiety or emotional distress. Both forms of reinforcement can contribute to repeated drinking, but coping-motivated drinking is especially relevant to negative reinforcement. The individual learns that alcohol is associated with relief from unpleasant emotional states. This process is consistent with learning theories of addictive behaviour. Repeated pairing of alcohol with emotional relief can strengthen associations between distress and alcohol consumption. Over time, drinking may become increasingly automatic rather than the result of deliberate decision-making. {{f}} However, alcohol use is not determined solely by conditioning. Individuals remain influenced by their beliefs, goals, social environments and available alternatives. If a person has developed effective non-alcohol coping strategies, they may be less likely to rely on alcohol following stressful experiences. Conversely, limited access to social support or effective coping resources may increase reliance on alcohol (Russell & Mehrabian, 1975). === Individual differences in alcohol use for emotion regulation === Not everyone responds to negative emotions by drinking. Individual differences are therefore essential for understanding emotion-regulation drinking. One important factor is personality {{f}}. Traits associated with impulsivity, negative emotionality and difficulties regulating emotions have been linked with greater vulnerability to problematic alcohol use {{f}}. Individuals who experience intense negative emotions may be particularly motivated to find rapid methods of reducing distress (Russell & Mehrabian, 1975). Stress exposure is another important factor. Chronic stress can increase the frequency and intensity of negative emotional states, providing more opportunities for coping-motivated drinking. People experiencing repeated adversity may therefore be more likely to use short-term strategies that provide immediate relief. Social learning also plays an important role. People observe how family members, peers and broader cultural environments use alcohol. If drinking is consistently presented as a normal way of dealing with stress, celebrating, relaxing or socialising, individuals may learn that alcohol is an appropriate emotional-regulation tool (Curtin et al., 2001). Importantly, the social meaning of alcohol varies across cultures and communities. In some contexts, drinking is strongly associated with social connection and celebration, whereas other communities may place less emphasis on alcohol. Consequently, emotion-regulation drinking should be understood within the cultural and social environment in which it occurs. ==== Alcohol expectancies ==== Alcohol expectancies refer to beliefs about the effects of drinking. They can involve expectations that alcohol will increase sociability, reduce anxiety, improve mood or help people cope with stress. Expectancies are important because drinking behaviour is partly shaped by anticipated outcomes (Khantzian, 1997). A person who expects alcohol to make them feel relaxed may be more motivated to drink when experiencing stress. If drinking then appears to confirm the expectation, the belief can become stronger (Goldman et al., 1999). This creates a feedback loop: Expectation of emotional relief → drinking → perceived emotional change → strengthened expectation → future drinking (Goldman et al., 1999). The loop can persist even when the actual pharmacological effect of alcohol is more complex than the person perceives. Expectancies can shape attention to emotional changes and how people interpret them. Expectancies also help explain why individuals respond differently to alcohol. Two people may consume the same amount but report different emotional experiences because they enter the situation with different beliefs. Changing maladaptive alcohol expectancies can therefore be relevant to prevention and intervention. If an individual believes that alcohol is their only reliable way to manage anxiety, developing awareness of alternative strategies may weaken the perceived necessity of drinking (Goldman et al., 1999). == '''Alternative strategies to manage emotion regulation''' == Understanding alcohol use as emotion regulation informs prevention and intervention. Drinking for social enjoyment may require different approaches from drinking to cope with anxiety or distress. Developing alternative emotion-regulation strategies can address the distress driving alcohol use and reduce reliance on alcohol (Kozubal et al., 2023). For people experiencing alcohol dependence or substance use disorder, {{ic|what sort of?}} therapy can help them examine and reflect on why they drink and what they expect from it, including whether they drink to relax, forget problems, ease anxiety, or escape unwanted thoughts (Aurora & Klanecky, 2016). [[File:Model of Alcohol use as an emotion management strategy.jpg|thumb|300px|Visual model to help explain emotion menagement {{sp}} strategy.]] Cognitive-behavioural therapy approaches help people connect situations, emotions and thoughts with drinking, identify triggers, and develop alternative responses {{f}}. Motivational approaches explore how alcohol’s temporary relief can contribute to problems people want to avoid. Prevention can challenge beliefs that alcohol reliably manages stress or anxiety, but education works best alongside practical coping strategies (Magill et al., 2019; Aurora & Klanecky, 2016). Treatment should reduce alcohol-related harm by addressing social and environmental factors, such as availability, norms, advertising, and cultural practices. It should also teach practical problem-solving and encourage social support (Aurora & Klanecky, 2016). Mindfulness helps people tolerate emotions without escaping distress, while exercise, sleep, structured activities, and social connection support broader well-being {{f}}. The aim is to build flexible coping strategies, not simply replace alcohol with another avoidance behaviour. ==Conclusion== Alcohol use for emotion regulation occurs when individuals drink because they expect it to alter an unwanted emotional experience. People may drink to cope with stress, anxiety, anger, despair or other forms of psychological distress. The immediate effects of alcohol can make this strategy seem effective, particularly when drinking produces temporary relaxation, reduced inhibition or distraction from unpleasant thoughts. The concept of negative reinforcement helps explain why coping-motivated drinking can become habitual. When alcohol temporarily reduces distress, that reduction itself can reinforce future drinking. Over time, emotional states may become cues for alcohol consumption, creating learned associations between negative emotions and drinking. Short-term emotional relief does not necessarily mean effective long-term regulation. Repeated alcohol use can cause health, social and psychological problems, which may fuel further distress and create a cycle of drinking to manage emotions worsened by alcohol’s consequences. Drinking motives, expectancies, personality, stress, social learning and emotion-regulation skills help explain why some people turn to alcohol when distressed. Alcohol use varies; people drink for different reasons, and the same behaviour can serve different purposes. Understanding why people drink is as important as knowing how much they drink. Ask what they expect alcohol to do, which emotions trigger drinking, and what follows. This perspective distinguishes recreational use from alcohol serving as a primary emotion-regulation strategy. Effective regulation requires flexibility: negative emotions cannot always be prevented, and trying to eliminate them immediately may cause further problems. Cognitive reappraisal, problem solving, social support, mindfulness, and behavioural coping offer alternatives to alcohol. Understanding these processes helps reduce alcohol-related harm and promote healthier ways to manage emotions. ==See also== * [[Motivation and emotion/Book/2013/Alcohol and emotion|Alcohol and emotion]] (Book chapter, 2013) * [[Motivation and emotion/Book/2015/Alcohol addiction and emotion|Alcohol addiction and emotion]] (Book chapter, 2015) * [[Motivation and emotion/Book/2020/Cocaine and emotion regulation|Cocaine and emotion regulation]] (Book chapter, 2020) * [[Motivation and emotion/Book/2026/Emotion dysregulation|Emotion dysregulation]] (Book chapter, 2026) * [[Motivation and emotion/Book/2026/Positive emotion dysregulation|Positive emotion dysregulation]] (Book chapter, 2026) ==References== {{Hanging indent|1= Aurora, P., Klanecky, A. K., & Klanecky, A. K. (2016, February 23). Drinking motives mediate emotion regulation difficulties and problem drinking in college students. The American Journal of Drug and Alcohol Abuse; Taylor & Francis. https://www.tandfonline.com/doi/full/10.3109/00952990.2015.1133633?utm_ Barrett, L. F., Mesquita, B., Ochsner, K. N., & Gross, J. J. (2007). The experience of emotion. Annual Review of Psychology, 58, 373–403. https://doi.org/10.1146/annurev.psych.58.110405.085709 Brown, W. C., & Melas, M. T. (2024). A Narrative Review of Emotion Regulation Abilities, Strategy Use, and Their Relationship to Alcohol. Current Addiction Reports, 11(2), 199–209. https://doi.org/10.1007/s40429-024-00553-3 Cargiulo, T. (2007). Understanding the health impact of alcohol dependence. American Journal of Health-System Pharmacy, 64(5_Supplement_3), S5–S11. https://doi.org/10.2146/ajhp060647 Cooper, M. L. (1994). Motivations for alcohol use among adolescents: Development and validation of a four-factor model. Psychological Assessment, 6(2), 117–128. https://doi.org/10.1037/1040-3590.6.2.117 Cooper, M. L., Frone, M. R., Russell, M., & Mudar, P. (1995). Drinking to regulate positive and negative emotions: A motivational model of alcohol use. Journal of Personality and Social Psychology, 69(5), 990–1005. https://doi.org/10.1037/0022-3514.69.5.990 Curtin, J. J., Patrick, C. J., Lang, A. R., Cacioppo, J. T., & Birbaumer, N. (2001). Alcohol Affects Emotion Through Cognition. Psychological Science, 12(6), 527–531. https://doi.org/10.1111/1467-9280.00397 Goldman, M. S., Del Boca, F. K., & Darkes, J. (1999). Alcohol expectancy theory: The application of cognitive neuroscience. In K. E. Leonard & H. T. Blane (Eds.), Psychological theories of drinking and alcoholism (2nd ed., pp. 203–246). The Guilford Press. Goldman, M. R., Molina-Castro, M., Etkins, J. C., Koide, T. L., Ramchandani, V. A., Plawecki, M. H., Mennella, J. A., & Pepino, M. Y. (2025). Recent advances in alcohol metabolism: from the gut to the brain. Physiological Reviews, 105(4), 2501–2535. https://doi.org/10.1152/physrev.00053.2024 Gross, J. J. (1998). The emerging field of emotion regulation: An integrative review. Review of General Psychology, 2(3), 271–299. https://doi.org/10.1037/1089-2680.2.3.271 Gross, J. J., & John, O. P. (2003). Individual differences in two emotion regulation processes: Implications for affect, relationships, and well-being. Journal of Personality and Social Psychology, 85(2), 348–362. https://doi.org/10.1037/0022-3514.85.2.348 Khantzian, E. J. (1997). The self-medication hypothesis of substance use disorders: A reconsideration and recent applications. Harvard Review of Psychiatry, 4(5), 231–244. https://doi.org/10.3109/10673229709030550 Kozubal, M., Szuster, A., & Wielgopolan, A. (2023). Emotional regulation strategies in daily life: the intensity of emotions and regulation choice. Frontiers in Psychology, 14. https://doi.org/10.3389/fpsyg.2023.1218694 Kuntsche, E., Knibbe, R., Gmel, G., & Engels, R. (2005). Why do young people drink? A review of drinking motives. Clinical Psychology Review, 25(7), 841–861. https://doi.org/10.1016/j.cpr.2005.06.002 Magill, M., Ray, L., Kiluk, B., Hoadley, A., Bernstein, M., Tonigan, J. S., & Carroll, K. (2019). A meta-analysis of cognitive-behavioral therapy for alcohol or other drug use disorders: Treatment efficacy by contrast condition. Journal of Consulting and Clinical Psychology, 87(12), 1093–1105. https://doi.org/10.1037/ccp0000447 Messman-Moore, T., Ward, R. M., Zerubavel, N., Chandley, R. B., & Barton, S. N. (2014). Emotion Dysregulation and Drinking to Cope as Predictors and Consequences of Alcohol-Involved Sexual Assault. Journal of Interpersonal Violence, 30(4), 601–621. https://doi.org/10.1177/0886260514535259 Oscar-Berman, M., Shagrin, B., Evert, D. L., & Epstein, C. (2026). Impairments of Brain and Behavior: The Neurological Effects of Alcohol. Alcohol Health and Research World, 21(1), 65. https://pmc.ncbi.nlm.nih.gov/articles/PMC6826797/ Petit, G., Luminet, O., Maurage, F., Tecco, J., Lechantre, S., Ferauge, M., Gross, J. J., & de Timary, P. (2015). Emotion Regulation in Alcohol Dependence. Alcoholism: Clinical and Experimental Research, 39(12), 2471–2479. https://doi.org/10.1111/acer.12914 Russell, J. A., & Mehrabian, A. (1975). The mediating role of emotions in alcohol use. Journal of Studies on Alcohol, 36(11), 1508–1536. https://doi.org/10.15288/jsa.1975.36.1508 Rydzewska, M., Zaorska, J., & Jakubczyk, A. (2023). The regulation of emotions and problematic alcohol use: a review of literature. Alcoholism and Drug Addiction, 36(2), 113–140. https://doi.org/10.5114/ain.2023.132441 }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Drugs/Alcohol]] [[Category:Motivation and emotion/Book/Emotion regulation]] sq4ao88zbkj14xzyb9k91b44pmbbgzn 0 331107 2842463 2841180 2026-10-24T22:45:00Z Jtneill 10242 Copyediting 2842463 wikitext text/x-wiki {{title|Charismatic leadership and follower motivation:<br>How does charismatic leadership inspire follower motivation?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}} [[File:German football team 2016 Olympics women.jpg|right|thumb|200px|'''Figure 1'''. The soccer team you have just been named captain of {{g}}.]] ; Imagine this... You have just been named captain of your soccer team (see Figure 1), but this is the first time you have been put in a position like this {{g}}. You have always been told that you are rather charismatic, and you have heard of this thing called charismatic leadership. Is there a way you can use this apparent charisma you have and turn it into charismatic leadership to help rally your team so you can win? {{RoundBoxBottom}} It is important for a good leader to know how to inspire motivation in their followers, and there are various methods of doings this. Charisma is often spoken about as one such method, and is an effective tool, used by many successful leaders, fictional and real, both current and historical. It is therefore important to understand the role that charisma plays in leadership, how it can be utilised to inspire follower motivation, and how all of this ties together in two concepts known as charismatic leadership and follower motivation. {{ic|Introduce what psychological science (theory and research) is relevant to addressing this topic.}} {{RoundBoxTop|theme=3}} '''Focus questions''' * What is charismatic leadership? * What is follower motivation? * How do traits of the leader influence the effects of charismatic leadership? * How do follower traits influence the effects of charismatic leadership? * How do leader and follower traits interact to influence the effects of charismatic leadership? * How do all of these things come together? {{RoundBoxBottom}} == What is Charismatic Leadership? == Charismatic leadership is a form of [[w:Leadership|leadership]] that is also sometimes referred to as [[w:Charismatic_authority|charismatic authority]] {{f}}. It is a form of leadership that is based in the individual charisma of the leader in question. It has been used by figures, both fictional and real who seek a devoted support base. These people often take on the roles of political or [[w:Cult|cult]] leaders. Examples of people who have used their charisma to gain mass political power for greatly destructive purposes include, on the fictional side, [[w:Palpatine|Emperor Sheev Palpatine]] (''[[w:Star_Wars|Star Wars]]'') and [[w:Coriolanus_Snow|President Coriolanus Snow]] (''[[w:The_Hunger_Games_(franchise)|The Hunger Games]]'') or, in the real world, [[w:Adolf_Hitler|Adolf Hitler]] and [[w:Donald_Trump|Donald Trump]]. {{f}} From a more research and theory based perspective, charismatic leadership has a tendency to be ill-defined, despite frequently being stated to improve follower motivation across various peer-reviewed sources (Antonakis et al., 2021; Sy et al., 2018). Sy et al. (2018) chose to take this lack of specific definition and understanding to develop a theoretical framework in an attempt to better understand charismatic leadership and its underlying mechanisms. Through this work they developed a theory of ''fellowship-relevant emotions'' (FREs), framed around research on moral emotions that describes how leaders utilise emotions to influence how their followers act (Sy et al., 2018). From there, they discussed the Elicit-Channel (EC) model of charismatic leadership, positing a five-step cyclical process that formed the basis of, what they called, the charismatic cycle (Sy et al., 2018) {{expand}}. From the purely research-based side of things {{awkward}}, there is the work of Antonakis et al. (2021) and Paul et al. (2001), who looked at the effects of charismatic leadership when applied to speeches and more generally, respectively {{missing}}. Antonakis et al. (2021) looked at the effects on followers who were given a "charismatic speech" versus a "standard speech" before being set to work. They found that the workers given a "charismatic speech" had a 17% greater work output than those who were given a "standard speech", thus determining that charismatic leadership, when applied to speeches given to workers, had a significant impact when being compared to a more "normal" approach (Antonakis et al., 2021) {{ic|repetitive}}. Paul et al. (2001), however, looked at the effects of both charismatic leadership and, what they referred to as, "integrative" leadership styles on follower self-concept, supporting a self-concept-based theory of charismatic leadership. This "integrative" leadership style they utilised combined both charismatic and individualised considerations when dealing with followers (Paul et al., 2001). They found that messages based in both charismatic and integrated leadership styles improved collective follower self-concept when compared with routinised messages, determining that these findings underpinned the importance of charismatic messages, even when said {{awkward}} messages have become routinised (Paul et al., 2001). <quiz display="simple"> {How many steps are in the cyclical process of the Elicit-Channel (EC) model of charismatic leadership? |type="()"} - 3 - 4 + 5 - 6 </quiz> == What is Follower Motivation? == Follower motivation, although an overall simpler concept than charismatic leadership, is still a key aspect needing to be understood, in order to be able to establish how it is influenced by charismatic leadership. Follower motivation is, in its simplest terms, what motivates an individual, or group of individuals to follow a specific leader or leadership structure, depending on the circumstances of the specific situation. {{f}} The role leaders play in follower motivation has been at the forefront of research in the field for {{which}} decades, and yet remains as one of the most complex and difficult to truly understand organisational behaviours (Berson et al., 2015). A key example of follower motivation and, thus, undergoing of certain behaviours can be traced back to the [[w:Nuremburg_trials|Nuremburg trials]], wherein assorted Nazis used a plea that is now, arguably, most commonly known as the [[w:Superior_orders|Nuremburg defense]], due to its infamy in this particular example. This plea essentially argues on the basis of followers being unable to resist the compulsion to follow a given leader or leaders, who are simply too motivating to deny. {{f}} Follower motivation does not always have to be manipulated in such dark scenarios, {{g}} it can also be used to better productivity and prosocial behaviour. As Christensen-Salem et al. (2024) noted, prosocial follower motivation and understanding its roots from a scientific level, is key to being able to manipulate it. This deeper understanding would also be of use in counteracting some of the darker uses people might have for manipulating follower motivation. == Effect of Leader and Follower Traits on Charismatic Leadership == The [[w:Trait_theory|traits]] of both leaders and followers in a given situation heavily influence the outcome of that given situation and the effectiveness of the charismatic leadership being employed. {{f}} === Leader Traits and Charismatic Leadership === The traits of the leader in question in a given scenario are key to the effectiveness of said {{awkward}} leader's attempts to employ charismatic leadership. Christensen-Salem et al. (2024) investigated the influence of abusive supervision {{ic|explain what is meant by abusive supervision}} on charismatic leadership, including taking into account other factors, such as, prosocial follower motivation. They found followers' prosocial motivation to be positively influenced by charismatic leadership when abusive supervision is low, however, the same was not found to be true when abusive supervision was high (Christensen-Salem et al., 2024). Expanding on these findings, they found prosocial motivation to be a mediating factor between organisational citizenship behaviour and charismatic leadership (Christensen-Salem et al., 2024). However, they, again, found this phenomenon only occurred when abusive supervision was low, and not when it was high (Christensen-Salem et al., 2024). '''Reflection question: Charisma versus trust''' Think about a leader who inspires people through confidence and persuasive communication. Does being charismatic necessarily mean they are trustworthy or ethical? Consider how a leader's actions, values, and treatment of others might influence whether followers feel genuinely motivated or pressured to follow them. <quiz display="simple"> {Followers' prosocial motivation is positively influenced by charismatic leadership when abusive supervision is _____: |type="()"} - high + low </quiz> === Follower Traits and Charismatic Leadership === It is not only leader traits that influence the effectiveness of charismatic leadership, but also those of the follower or followers in question. Yan et al. (2025) is an example of a study that chose to focus specifically focus on the traits of {{who}} employees and charismatic leadership. In this study, they found that charismatic leadership positively influences both employees' perceived support and employees' goal-directed energy (Yan et al., 2025). They also found that when employees perceive themselves as being supported, that perceived support then acts as a mediating factor {{ic|explain direction of mediation}} between charismatic leadership and employees' goal-directed energy (Yan et al., 2025). Although, counterintuitively to the positive influence noted, they also found duration under current tenure to negatively moderate both charismatic leadership and employees' perceived support (Yan et al., 2025). <quiz display="simple"> {Charismatic leadership has been found to ______ influence both employees' perceived support and employees' goal-directed energy: |type="()"} + Positively - Negatively </quiz> === Interaction Between Leader and Follower Traits on Charismatic Leadership === A study that investigated the influence of leader traits on the effectiveness of charismatic leadership, in tandem with those of followers was Brown & Treviño (2009). They focussed on the importance of value congruence between followers and a socialised charismatic leader {{ic|explain what is meant by a "socialised charismatic leader"}} (Brown & Treviño, 2009). They found that congruent values held between followers and a socialised charismatic leader positively influenced the effectiveness of said {{awkward}} leader's charismatic leadership (Brown & Treviño, 2009). They also found that these values held by both the follower and leader can be occupation-linked, therefore, those who choose to work in a given field, should, theoretically, share, at least some, congruent values (Brown & Treviño, 2009). == Interaction Between Charismatic Leadership, Follower Motivation, and Traits of Both Leaders and Followers == Tying together the Yan et al. (2025) study and the one by Brown & Treviño (2009) and real world examples, historical charismatic leaders, such as Hitler and Trump, have utilised a portrayal of values held by their followers, that, at the very least, these followers see as congruent with those that their chosen leader portray as their own, combined with a follower displeasure of the seemingly "long-tenured" system in place at the time they each sought to gain power, in order to gain said power {{ic|overly long sentence}}. This {{what}} example also touches on the importance of follower motivation, as the {{which}} followers in question were all displeased with the previous system, and therefore motivated to seek out a new regime. In these {{which}} examples, the followers were motivated by assorted factors, such as, {{g}} what they perceived to be abusive supervision from the powers they lived under (Christensen-Salem et al., 2024). In the case of Hitler and the Nazis, this was the result of treaty of Versailles, the power the allies held, and the restrictions placed on Germany {{f}}. In the case of Trump and his "MAGA" supporters, this perceived abusive supervision lay with the democrats under Obama and then Biden (Christensen-Salem et al., 2024). Both of these regimes were able to come to power because of the traits of all parties concerned, but also, particularly because the respective leaders were able to identify the disdain their followers held for the system, and utilise their own charisma to manipulate the motivation of their eventual followers into becoming said followers. ==Conclusion== Charismatic leadership is a form of leadership that is also sometimes known as charismatic authority {{ic|yes, but what is charismatic leadership? This hasn't been sufficiently defined}} and, despite being often talked about in scientific literature {{vague}} {f}}, is typically not so well-defined {{ic|but how is is described by psychological science?}}, although more specific models and studies are being done in more recent years {{vague}} (Antonakis et al., 2021; Paul et al., 2001; Sy et al., 2018). It is heavily reliant on follower motivation {{ic|this hasn't been defined}}, a concept which is, overall simpler than that of charismatic leadership, however, understanding it and its roots are both key in being able to effectively employ charismatic leadership {{vague}} (Berson et al., 2015; Christensen-Salem et al., 2024). Traits relating to and held by both followers and charismatic leaders have been found to influence the effectiveness of charismatic leadership, resulting in assorted outcomes {{vague}} (Brown & Treviño, 2009; Christensen-Salem et al., 2024; Yan et al., 2025). Overall, charismatic leadership is a style of leadership that is reliant, not only, on the charisma {{ic|What is charisma; this hasn't been defined}} of a given leader, but also on various other factors {{vague}}, both of the leader and follower(s) in question, including follower motivation. ==See also== * [[w:Charismatic_authority|Charismatic authority]] (Wikipedia) * [[Motivation and emotion/Book/2015/Leadership motivation|Leadership motivation]] (Book chapter, 2015) ==References== {{Hanging indent|1= Antonakis, J., d’Adda, G., Weber, R. A., & Zehnder, C. (2021). “Just words? Just speeches?” On the economic value of charismatic leadership. ''Management Science'', ''68''(9). https://doi.org/10.1287/mnsc.2021.4219 Berson, Y., Halevy, N., Shamir, B., & Erez, M. (2015). Leading from different psychological distances: A construal-level perspective on vision communication, goal setting, and follower motivation. "The Leadership Quarterly", "26"(2), 143–155. https://doi.org/10.1016/j.leaqua.2014.07.011 Brown, M. E., & Treviño, L. K. (2009). Leader–follower values congruence: Are socialized charismatic leaders better able to achieve it? ''Journal of Applied Psychology'', ''94''(2), 478–490. https://doi.org/10.1037/a0014069 Christensen-Salem, A., Hartnell, C. A., Walumbwa, F. O., Chiang, F., & Birtch, T. (2024). When charismatic leadership loses its luster: Examining the interactive effect of charismatic leadership and abusive supervision on follower prosocial motivation and citizenship behavior. ''Journal of Leadership & Organizational Studies'', ''31''(3), 304–318. https://doi.org/10.1177/15480518241263913 Paul, J., Costley, D. L., Howell, J. P., Dorfman, P. W., & Trafimow, D. (2001). The effects of charismatic leadership on followers’ self-concept accessibility. ''Journal of Applied Social Psychology'', ''31''(9), 1821–1842. https://doi.org/10.1111/j.1559-1816.2001.tb00206.x Sy, T., Horton, C., & Riggio, R. (2018). Charismatic leadership: Eliciting and channeling follower emotions. ''The Leadership Quarterly'', ''29''(1), 58–69. https://doi.org/10.1016/j.leaqua.2017.12.008 Yan, K., Guo, W., Latukha, M., & Fefelov, D. (2025). How does charismatic leadership foster employees’ goal-directed energy and perceived support? The moderating role of charismatic leadership tenure. ''Journal of East-West Business'', ''31''(4), 1–29. https://doi.org/10.1080/10669868.2025.2504988 }} ==External links== * [https://www.youtube.com/watch?v=_3hSmaOhGWU Charismatic leadership style (explained in 3 minutes)] (Social Science Explainer, via YouTube) * [https://www.coachhub.com/blog/the-impact-of-charismatic-leadership-on-organizational-transformation The impact of charismatic leadership on organisational transformation] (CoachHub) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Leadership]] [[Category:Motivation and emotion/Book/Personality]] q80rrktj929506owy0acbxb18u9rpoc 0 331239 2842481 2839328 2026-10-25T00:17:16Z Jtneill 10242 Copyediting 2842481 wikitext text/x-wiki {{title|Developing a growth mindset:<br> How can a growth mindset be cultivated and sustained?}} __TOC__ ==Overview== {{RoundBoxTop|theme=2}} [[File:Openminded.png|right|thumb|150px|'''Figure 1'''. Illustration of being open-minded]] ; Imagine this... Two childhood best friends find themselves in two different places when they finish high school. Though living in the same neighbourhood, attending the same schools, enjoying the same hobbies and receiving equal opportunities, one is earning a scholarship to study their dream degree, and the other is falling behind. How could two people with the same opportunities have vastly separate life outcomes? Well, one evaded challenge for fear of failing, while the other appraised these failures as opportunities to learn. The margin between one and the other was not created based on ability, but from mindset (see Figure 1). {{RoundBoxBottom}} Developing a growth mindset is not a single process; adopting the belief that your abilities and intelligence can change does not necessarily guarantee it will hold over time. In the face of failure, setbacks, negative feedback or difficult environmental challenges, growth-oriented beliefs may waver. Adopting maladaptive approaches to adversity can lead individuals to believe in their own incompetence and avoid challenges necessary for lifelong learning and success. The scenario raises an important question that remains: "how can a growth mindset be sustained beyond its initial development?" Addressing this question requires seeing a growth mindset not as something linear once adopted, but as something that requires ongoing cultivation and reinforcement. To sustain a growth mindset beyond its initial development, we must explore the internal and external factors that cultivate and reinforce this framework over time. {{RoundBoxTop|theme=4}} '''Focus questions''' * How does a growth mindset differ from a fixed mindset? * How do mastery-oriented goal types sustain growth mindset attributes? * How does person-focused feedback inhibit the development of a growth mindset? * How can feedback be used to reinforce a growth mindset over time? * How do environmental factors, such as parental and educational interventions, inform the trajectory of growth mindset attributes over time? {{RoundBoxBottom}} == Mindsets == A [[wikipedia:Mindset|mindset]] is a mental framework or set of cognitive patterns that manages an individual's thinking, feeling and behaving, and secondary consequences of information processing and decision making (Reeve, 2024). Adopting a specific mindset creates a downstream influence on one's perception and beliefs of personal success, flaws, qualities, and other’s [[wikipedia:Belief|beliefs]] of your own [[wikipedia:Ability|abilities]]. Implementing a mindset gives way to creating altering motivational systems that manage differing ways of coping, and habits implemented on the basis of how external experiences are appraised. Dweck and Yeager (2019) suggest that mindsets are the core of meaning systems. Meaning, {{g}} mindsets allow us to create meaning systems that serve to organise [[wikipedia:Goal|goals]], beliefs and behaviours with ‘effort beliefs’ being a core concept that dictates how an individual approaches effort as a positive belief to grow or a negative fixed belief. === '''Growth versus fixed mindset's''' {{g}} === American psychologist, [[wikipedia:Carol_Dweck|Carol Dweck]], characterised that an individual can adopt two categories of mindsets, a growth or fixed mindset (Mueller & Dweck, 1988). A growth or fixed mindset’s {{g}} concern the framework in which an individual views their abilities, personal qualities and intelligence on a continuum between something that is fixed or transformative. An individual with a fixed mindset may perceive their abilities as inherent characteristics that define what they can achieve. However, an individual with a growth mindset is more likely to perceive their abilities as malleable and capable of being developed through [[wikipedia:Learning|learning]] and effort. {{g}} A person that {{g}} adopts a fixed mindset believes that their [[wikipedia:Skill|skills]], and [[wikipedia:Intelligence|intelligence]] is something that is concrete and unlikely to change (Mueller & Dweck, 1988). The [[wikipedia:Implicit_theories_of_intelligence|implicit theory of intelligence]] suggests a fixed mindset as an ‘entity theory’ leading an individual to understand their intelligence as stagnant and a product of genetic predisposition, as they perceive that these qualities are a physical body that is attached to an individual (Dweck & Leggett, 1988; Reeve, 2024). From this perspective, an individual may interpret their abilities as an intrinsic level of competence. But when faced with challenges or failure, individuals with this mindset are more likely to discern these experiences are evidence of incompetence, potentially limiting their enthusiasm to sustain or engage in challenging goal. Conversely, if a person adopts a growth mindset, they believe that with persistence, their skills and intelligence can strengthen and increase. Contradictory to an entity theorist, the implicit theory of intelligence suggests that a growth mindset is referred to as an ‘incremental theorist’, proposing that intelligence and personal capabilities are malleable traits with the possibility to improve with motivation and dedication (Dweck & Leggett, 1988). Rather than perceiving their current abilities as a fixed depiction of their greatest potential, incremental theorists perceive skills and intelligence as characteristics that can be enhanced whereby learning and effort is persistent. For an incremental theorist, though their realistic personal abilities may be limited, the possibility to improve and attain mastery is still probable. These opposing ways of thinking play a significant role in how they influence individuals to approach challenges, respond to failures and highlight opportunities for development. {{f}} === '''The role of neuroplasticity''' === [[File:Homosynaptic Plasticity-1.jpg|thumb|'''Figure 2'''''.'' Learning strengthens neural connectivity ]] From a biological perspective, the malleability of a human's mindset may not be limited to psychological explanation but can also be considered from the perspective of neural processes demonstrating that the human [[wikipedia:Brain|brain]] is capable of structural and functional change (Sarrasin et al., 2018). [[wikipedia:Neuroplasticity|Neuroplasticity]] refers to the brains {{g}} ability to alter, form new connections, and recognise pre-existing neural connections when responding to environmental demands, experience and [[wikipedia:Learning|learning]] (Ng, 2018). Meaning {{awkward}} that the brain is not a fixed structure, but with repeated learning and exposure to new experiences these practices can contribute to changes in skills and abilities. [[wikipedia:Synaptic_plasticity|Synaptic plasticity]], meaning the changes in strength between [[wikipedia:Neuron|neurons]], serving as the primary explanation for how [[wikipedia:Memory|memory]] and learning can become encoded and subsequent actions can be reproduced (see Figure 2) (Martin et al., 2000; Abbott & Nelson, 2000) {{g}}. When adopting a growth mindset, encouraging determination, seeking challenges, resilience when faced with failure, creates the conditions through which learning and synaptic plasticity transformation can occur. Neuroplasticity provides a plausible biological explanation for understanding how learning and repeated experiences, components accompanying a growth mindset, can contribute to changes in neural functioning (Sarrasin et al., 2018) {{g}}. However, though neuroplasticity provides a biological basis for which learning can occur, it cannot be inferred that adopting a growth mindset directly alters our brains on a neurological level (Ng, 2018). But {{g}} rather as a supporting feature, through which repeated experiences and learning provide an important contribution to pursue development and to strengthen a growth mindset long-term.   == Goals create patterns in behaviour == It has been highlighted that the [[wikipedia:Goal|goals]] that are integrated within a individual’s [[wikipedia:Cognition|cognitions]], behaviour and affect create the framework in which guide them to interpret and respond to experiences and events that occur (Dweck & Leggett, 1988). A goal is a task that is yet to be accomplished but strives to attain (Reeve, 2024). Individuals who adopt incremental theory, also pursue stronger learning goals, adopt more positive beliefs about effort and engage in fewer "helpless" behaviours (Blackwell et al., 2007). Those that have a well-defined, difficult, and congruent goal are more likely to direct their [[wikipedia:Mental_energy|energy]] and sustain [[wikipedia:Motivation|motivation]] to complete an objective than those without (Reeve, 2024). When adopting differing types of [[wikipedia:Goal_orientation|goal-orientations]], an individual is likely to respond and enact behavioural patterns accordingly within the bounds and purposes of the intended goal (Dweck & Leggett, 1988). When manipulating children's perceived ability of a task, Elliott and Dweck (1988) established that the goal an individual may strive for may directly alter the patterns of achievement-related behaviour, with mastery or performance-oriented behaviour being associated with subsequent achievement. Understanding the adopted goal an individual accepts provides significant importance in ensuring the maintenance of a growth mindset, as varying goal orientations predict the approaches individuals partake in when [[wikipedia:Learning|learning]], persistence in the face of challenge and responses to failure (Elliott & Dweck, 1988).   === '''Performance and Mastery Goals''' === Elliot and Dweck (1988) proposed that in order to engage in achievement behaviour, it must first be distinguished what goal an individuals {{g}} adopts when approaching a task. There are two types of goals an individual may adopt {{g}} performance or mastery goals. The approach to defining these goals is based on the perspective an individual takes when evaluating their competence and skills. Meaning {{g}} that those who take on varying goals intuitively adopt varying perceptions and reactions in different circumstances (Dweck & Leggett, 1988). Those that {{g}} adopt performance goals are those that {{g}} seek perform actions that are a testament to their competence rather than their incompetence. Those that {{g}} pursue performance goals are likely to interpret difficult circumstances as a measure of their proficiency and [[wikipedia:Aptitude|aptitude]]. Meaning {{g}} that when faced with challenges, performance-oriented individuals may experience apprehensive of negative judgement from others, with their actions seeking to demonstrate high ability, and competence in pursuit of positive evaluation (Dweck & Leggett, 1988). Within this framework, when failure threatens these goals, the likelihood that [[wikipedia:Learned_helplessness|learned helplessness]] response patterns emerge, characterised by negative self-affect and impaired performance (Seligman & Maier, 1967) {{g}}. In contrast to performance driven goals, those that {{g}} adopt [[wikipedia:Learning|learning]] or mastery-oriented goals perceive challenge as a means to seek [[wikipedia:Feedback|feedback]] and persist in tasks when they become difficult. Those that pursue mastery goals seek not to reach a goal due to external evaluation, but to acquire new skills to extend their mastery through effort and determination (Dweck & Leggett, 1988; Elliott & Dweck, 1988). Confronting failure, when enacting a mastery-oriented approach, individuals exhibit constructive self-feedback and positive self-affect. Consequently, the distinction between a mastery-oriented and performance-oriented goals is significant to understanding and informing the pattern of behaviour that may be exercised in goal pursuit to either hinder or encourage the sustainability of a growth mindset. Mastery-oriented goals provide the possibility to fuel personal beliefs that an individual is capable of learning and improvement, supportive components to prolonging the cultivation of a growth mindset. The motivational consequences of these goal orientations differ if seeking either avoidance or improvement (Elliott & Dweck, 1988), a discrepancy that is further explored in the 2 x 2 achievement goal framework (Elliot & McGregor, 2001). '''Table 1.''' ''Comparing implicit theory of intelligence, goal orientation and predicted behavioural patterns'' (Dweck & Leggett, 1988) {| class="wikitable" |+ ! !Implicit theory of intelligence !Goal Orientation !Behavioural patterns |- |'''Growth Mindset''' |Incremental theorist |Performance goal orientation (Objective is to be positively appraised on skill/s) Performance approach OR Performance avoidance goals (Elliot & McGregor, 2001) |Learned helplessness, avoidant to challenge |- |'''Fixed Mindset''' |Entity theorist |Learning/Mastery goal orientation (Objective is to develop and improve on a skill/s) Mastery approach OR mastery avoidant goals (Elliot & McGregor, 2001) |Seeks challenge, Mastery oriented, pursuing improvement on abilities |} Revising the achievement goal theory, Elliot (1999) proposed the idea of the Trichotomous Achievement Goal Framework in which suggests 3 goal types, mastery goals, performance-approach goals and performance-avoidance. Performance-approach goals involve striving to demonstrate competence relative or beyond others, whilst performance-avoidant goals are centred around refraining from being perceived as incompetent. Subsequently, Elliot & McGregor (2001) expanded on this with the 2 x 2 achievement goal framework, by differentiating mastery-approach goals and mastery-avoidance goals. Mastery-approach goals focus on developing competence, whereas mastery-avoidance goals is the pursuit to avoid task failure. Revised versions of subtypes of goals provide a nuanced perspective of how goal motivations are associated with cultivating and sustaining a growth mindset (see Table 1). For example, mastery goals allow individuals to engage in {{g}} with setbacks and to seek improvement supporting growth mindset beliefs (Elliot & McGregor, 2001; Dweck & Leggett, 1988). Goal types of individuals employ influence whether these growth mindset beliefs are translated into cognitive, behavioural and affect patterns in the context of learning and challenge. == Feedback and praise == [[wikipedia:Feedback|Feedback]] and [[wikipedia:Praise|praise]] for exhibiting high ability is often the frequent response in identifying the next actionable steps. However, praise for demonstrating competence may seem like the normalised response, this form of response may construe an individual's perception and motivation to succeed (Mueller & Dweck, 1998). Some individuals may perceive praising student [[wikipedia:Intelligence|intelligence]] as a source of motivation that enables students to pursue in learning as they internalise this feedback as true. However, the impact of feedback and praise is dependent on the information conveyed in the feedback along with how this is comprehended by the recipient (Mueller & Dweck, 1998; Haimovitz & Dweck, 2016; Gunderson et al., 2013). For instance, feedback that emphasises improvement may present the opportunity for individuals to perceive challenges as actionable potential to change, then feedback that criticises the self. === '''Person versus process feedback''' === Mueller and Dweck (1998) examined the way feedback is conveyed may lead to undermining an individuals {{g}} performance and further motivation {{rewrite}}. A seminal study by Kamins and Dweck (1999) explored how {{ic|how many?}} {{where}} children respond in scenarios involving success and failure in regard to receiving praise that was person versus process focused. Researchers had found that process-focused participants who were praised for efforts were more likely to sustain determination in the face of difficulty, respond constructively to criticism and engage in mastery behavioural patterns. Conversely, person-focused participants who received praise towards their fixed abilities showed increased likelihood to appraise failures as indicative of low ability and low self-esteem. In the way feedback may be conveyed lends itself useful in reinforcing determination and effective learning, supporting the possibility of growth (Mueller & Dweck, 1998). {{Robelbox|title=Person or process focused feedback?|theme=2}} Peter has recently finished his semester of at school. He received his report from his teacher at the end of the day with them commenting that he had “worked really hard this semester and stayed focused”. As Peter’s teacher praised his effort and concentration, they supported a growth mindset and person-focused feedback However, when he got home, he handed his academic report to his parents who said “good job! You must be so intelligent” after seeing he received straight A’s {{g}}. His parents unbeknown to person-focused feedback, supported a fixed mindset by praising his fixed ability (Mueller & Dweck, 1998). {{RoundBoxBottom}} === '''Interpreting and integrating criticism''' === [[File:Feedback-loop-general.svg|thumb|'''Figure 3.''' Feedback creates a cycle in which helps us to keep growing]] However, Haimotz and Dweck, (2016) outlined that the effect of feedback is still important but might be significantly more influential depending on how an individual interprets failure feedback. Importantly, London and Smither (2002) suggests that in regard to the maintenance of a growth mindset, feedback should not be perceived as a single event, but as a part of an ongoing reinforcement cycle (see Figure 3). The concept of positive feedback orientation refers to an individual's overall willingness to receive and respond to feedback. When given feedback, those that {{g}} perceive feedback as constructive information are more intuitively integrating this into their future behaviour and performance than those that {{g}} dismiss it (London & Smither, 2002). In experiencing ongoing feedback forming positive feedback cycles, individuals are more inclined to engage in mastery-oriented goals, adapt and apply new behaviours, accomplish tasks develop growth-oriented beliefs (Mueller & Dweck, 1988; London & Smither, 2002). However, this may only occur when external factors (e.g., [[wikipedia:Organizational_culture|workplace feedback culture]]) may provide feedback that is process-focused and clear task outcomes are establish {{g}}, in order for individuals to feel more receptive to feedback (Kamins & Dweck, 1999; London & Smither, 2002). == Impact from parents and teachers == In cultivating and sustaining a growth mindset, [[wikipedia:Parent|parents]] and [[wikipedia:Teacher|teachers]] have the important influence of creating a culture of feedback that guides the way individuals evaluate their abilities and sustain the motivation to develop (London & Smither, 2002). Haimovitz and Dweck (2016) explored how the visible mindset of a parent can be transmitted intergenerationally. Specifically, these findings demonstrate that for parents who modelled failure as a debilitating outcome, were more likely to endorse performance-oriented mindsets by acts of focusing on their children's lack of ability. Subsequently, these behaviours primed children's intelligence beliefs, by children interpreting parental failure mindsets (Haimovitz & Dweck, 2016). Furthermore, an observational study by Gunderson et al. (2018) showed that the effects of process-focused parent-child praise can have long-term implications in sustaining an incremental framework. Results indicated that process-focused praise in [[wikipedia:Toddler|toddler]] years (i.e., praise for effort and strategies) were indicative of informing growth mindsets in later childhood academic results (Gunderson et al., 2018). These findings may extend the concept of a growth mindset as not an immediate motivational response but remains relevant in altering the belief of intelligence and [[wikipedia:Academic_achievement|academic achievement]] (Macnamara & Burgoyne, 2022). [[File:05092012 - Oyster class visit Teacher Appreciation 237.jpg|thumb|'''Figure 4.''' Teachers can encourage students to engage in growth-oriented beliefs]] In addition to this, Yeager et al. (2022) suggested that the mindset-plus-supportive-context hypothesis, a theory that illustrates a student’s growth mindset can be sustained if environmental factors, such as teaching environments, allow those behaviours to flourish. Teachers have the role of conveying to student’s {{g}} corrective growth mindset attributes through modelling how mistakes can be learning opportunities and not signs of incompetence with evidence shown in reports and assignment evaluations to demonstrate continued improvement (see Figure 4) (Yeager et al., 2022). Blackwell et al. (2007) supported this through a growth mindset intervention study, investigating transitioning middle school students and the effects that an [[wikipedia:Education|education]] exercise teaching students about the malleability of intelligence and strategies for learning may demonstrate increase academic achievement in the [[wikipedia:Classroom|classroom]]. Results had shown that those students who had been taught the malleability of a mindset not only demonstrated an improvement in motivation and academic performance, but this improvement showed no decline post intervention and during challenge (Blackwell et al., 2007). Findings outline the importance that growth mindset interventions may have in the classroom. Findings suggest that teaching interventions may be particularly valuable in the longevity of a growth mindset by equipping students with practical application of growth mindset attributes derived from research and messaging.   Despite these promising findings, Macnamara and Burgoyne (2022) conducted a meta-analysis {{ic|of how many studies?}} interested in whether growth mindset interventions in educational setting legitimately increase academic achievement. Findings suggested that these growth mindset interventions did provide {{ic|how much?}} improvement to academic achievement, but it was important to consider alternative explanations for this. For example, the nature of growth mindset interventions may provide encouragement to “work harder” rather than integrating these changes into a student's mindset long-term (Macnamara & Burgoyne, 2022). Although teachers may contribute to influencing growth mindset beliefs by establishing supportive learning environments and interventions, further research is necessary in reliably understanding how these interventions translate into long-term outcomes on mindsets. ==Conclusion== Cultivating and sustaining a growth mindset extend beyond the initial adoption of the belief that intelligence and ability can be malleable (Dweck & Yeager, 2019). More though it seeks to represent the ongoing psychological processes and external environmental influences that shape how an individual approaches challenge, creates goals and reinforcing motivation to continue learning {{rewrite}}. Internally, goal orientation, interpretating and internalising feedback and failure inform the possibility of individuals engaging in growth-oriented beliefs in the face of challenge. The importance of mastery-oriented goals, and willingness to adopt learning strategies support the determination and the likelihood that ongoing improvement will be enacted. However, psychological processes do not operate independently, with environmental factors playing an important role in creating the conditions which guide individuals to engage in growth-oriented beliefs and behaviour. Environmental factors such as parental and educational influences help develop and maintain growth mindset attributes through feedback, process-focused praise with opportunities to apply this learning strategies in real-world scenarios. In bridging these insights, a growth mindset is not a perpetual state that remains stagnant once an outcome is achieved, but rather a position which requires continued cultivation, reinforcement and exercise to sustain human motivation. ==See also== * [[Motivation and emotion/Book/2019/Growth mindset development|Growth mindset development]] (Book chapter, 2019) * [[w:Mindset|Mindsets]] (Wikipedia) * [[wikipedia:Implicit_theories_of_intelligence|Implicit theory of intelligence]] (Wikipedia) * [[wikipedia:Goal_orientation|Goal orientation]] (Wikipedia) ==References== {{Hanging indent|1= Abbott, L. F., & Nelson, S. B. (2000). Synaptic plasticity: taming the beast. Nature Neuroscience, 3(S11), 1178–1183. https://doi.org/10.1038/81453 Blackwell, L. S., Trzesniewski, K. H., & Dweck, C. S. (2007). Implicit Theories of Intelligence Predict Achievement Across an Adolescent Transition: A Longitudinal Study and an Intervention. Child Development, 78(1), 246–263. https://doi.org/10.1111/j.1467-8624.2007.00995.x Dweck, C. S., & Leggett, E. L. (1988). A social-cognitive approach to motivation and personality. Psychological Review, 95(2), 256–273. https://doi.org/10.1037/0033-295X.95.2.256 Dweck, C. S., & Yeager, D. S. (2019). Mindsets: A View From Two Eras. Perspectives on Psychological Science, 14(3), 481–494. https://doi.org/10.1177/1745691618804166 Elliot, A. J. (1999). Approach and avoidance motivation and achievement goals. Educational Psychologist, 34(3), 169–189. https://doi.org/10.1207/s15326985ep3403_3 Elliot, A. J., & McGregor, H. A. (2001). A 2 X 2 achievement goal framework. Journal of Personality and Social Psychology, 80(3), 501–519. https://doi.org/10.1037/0022-3514.80.3.501 Elliott, E. S., & Dweck, C. S. (1988). Goals: An Approach to Motivation and Achievement. Journal of Personality and Social Psychology, 54(1), 5–12. https://doi.org/10.1037/0022-3514.54.1.5 Gunderson, E. A., Sorhagen, N. S., Gripshover, S. J., Dweck, C. S., Goldin-Meadow, S., & Levine, S. C. (2018). Parent praise to toddlers predicts fourth grade academic achievement via children’s incremental mindsets. Developmental Psychology, 54(3), 397–409. https://doi.org/10.1037/dev0000444 Haimovitz, K., & Dweck, C. S. (2016). What Predicts Children’s Fixed and Growth Intelligence Mind-Sets? Not Their Parents’ Views of Intelligence but Their Parents’ Views of Failure. Psychological Science, 27(6), 859–869. https://doi.org/10.1177/0956797616639727 London, M., & Smither, J. W. (2002). Feedback orientation, feedback culture, and the longitudinal performance management process. Human Resource Management Review, 12(1), 81–100. https://doi.org/10.1016/S1053-4822(01)00043-2 Macnamara, B. N., & Burgoyne, A. P. (2023). Do growth mindset interventions impact students’ academic achievement? A systematic review and meta-analysis with recommendations for best practices. Psychological Bulletin, 149(3–4), 133–173. https://doi.org/10.1037/bul0000352 Martin, S. J., Grimwood, P. D., & Morris, R. G. M. (2000). Synaptic Plasticity and Memory: An Evaluation of the Hypothesis. Annual Review of Neuroscience, 23(1), 649–711. https://doi.org/10.1146/annurev.neuro.23.1.649 Mesler, R. M., Corbin, C. M., & Martin, B. H. (2021). Teacher mindset is associated with development of students’ growth mindset. Journal of Applied Developmental Psychology, 76, 101299. https://doi.org/10.1016/j.appdev.2021.101299 Mueller, C. M., & Dweck, C. S. (1998). Praise for intelligence can undermine children’s motivation and performance. Journal of Personality and Social Psychology, 75(1), 33–52. https://doi.org/10.1037/0022-3514.75.1.33 Ng, B. (2018). The Neuroscience of Growth Mindset and Intrinsic Motivation. Brain Sciences, 8(2), 20. https://doi.org/10.3390/brainsci8020020 Reeve, J. (2024). Understanding Motivation and Emotion. (8th ed.). John Wiley & Sons, Incorporated. Sarrasin, J. B., Nenciovici, L., Foisy, L.-M. B., Allaire-Duquette, G., Riopel, M., & Masson, S. (2018). Effects of teaching the concept of neuroplasticity to induce a growth mindset on motivation, achievement, and brain activity: A meta-analysis. Trends in Neuroscience and Education, 12, 22–31. https://doi.org/10.1016/j.tine.2018.07.003 Seligman, M. E., & Maier, S. F. (1967). Failure to escape traumatic shock. Journal of Experimental Psychology, 74(1), 1–9. https://doi.org/10.1037/h0024514 Yeager, D. S., Carroll, J. M., Buontempo, J., Cimpian, A., Woody, S., Crosnoe, R., Muller, C., Murray, J., Mhatre, P., Kersting, N., Hulleman, C., Kudym, M., Murphy, M., Duckworth, A. L., Walton, G. M., & Dweck, C. S. (2022). Teacher Mindsets Help Explain Where a Growth-Mindset Intervention Does and Doesn’t Work. Psychological Science, 33(1), 18–32. https://doi.org/10.1177/09567976211028984 }} ==External links== * [https://open.uts.edu.au/insights/career-hacks/cultivating-a-growth-mindset/ Cultivating a Growth Mindset] (UTS) * [https://www.youtube.com/watch?v=hiiEeMN7vbQ Developing a Growth Mindset with Carol Dweck] (Youtube.com) * [https://www.youtube.com/watch?v=J-swZaKN2Ic The Power of Yet | Carol S Dweck | Tedx] (Youtube.com) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Mindset/Growth]] 12xh3wxzlygd9y8x39ayjif1iwnq749 0 331258 2842456 2842176 2026-10-24T20:33:22Z U3263365 3104938 maladaptive moral affect 2842456 wikitext text/x-wiki {{title|Moral emotions and ethical behaviour<br>How do moral emotions motivate ethical and prosocial action?}} __TOC__ ==Overview== {{RoundBoxTop|theme=7}} [[File:Begging man Harmony Lot 27 Elliot Street downtown Brattleboro VT July 2025 02.jpg|right|thumb|150px|'''Figure 1'''. Homeless man begging]] ;Scenario Imagine that you are doing your weekly shopping and you come across a homeless person who is asking for food or money. The person present in an unkempt state and initially you make sure you walk past, your stomach drops, and subconsciously wrinkle your nose and you decide to take a longer path to the shops in order to avoid them. When paying for your groceries, you realise that you don’t have quite enough to pay the full amount, instantly, your face gets hot as there is a line of people waiting behind you and it seems they are all staring at you, judging your error. Whilst you are talking to the register assistant, asking for some time to go through your groceries to determine which items are a priority, the person behind you offers to make up the difference. Suddenly, the hot feeling in your face becomes a warm feeling in your chest and you start profusely thanking the person and ask if there is anything you can do to make it up to them, they respond with “when you are in a position to pay it forward to some else, please do so.” {{RoundBoxBottom}} [[w:Moral_emotions|Moral emotions]] potentially set humans apart from other animals, by enabling people to react to social events that do not directly affect the self but concern the interests or welfare either of others and society as a whole (''Haidt, 2003).'' They arise when individuals evaluate actions, intentions, or social situations in terms of right or wrong, in accordance with social norms and moral standards. This guides behaviours that support cooperation, fairness and social harmony. This distinguishes moral emotions from [[Motivation and emotion/Lectures/Individual emotions|basic emotions]], which are a person's primary response to a situation. Basic emotions fast, short lived, whilst moral emotions, like [https://www.simplypsychology.org/primary-and-secondary-emotions.html secondary emotions], are more complex and require a cognitive evaluation of the self from learned social meaning. To understand how moral emotions motivate behaviour, examination of the psychological/ psychological mechanisms behind these emotions is required. Moral emotions involve appraisals of responsibility and social meaning, biological responses that prepare the body for action, and expressive behaviours that communicate intentions. These mechanisms create ''action readiness'', transforming moral evaluations into behavioural tendencies such as helping, repairing harm, confronting wrongdoing, or avoiding social violations (''Reeve'', 2023). Understanding moral emotions is essential to link how they motivate ethical and prosocial action. [[w:Self-conscious_emotions|Self-conscious moral emotions]] - such as guilt, shame, embarrassment, and authentic pride - occur when people evaluate their own behaviour against internalised moral standards. These emotions regulate the moral self by motivating reparative action, helping others, rule compliance, or withdrawal (''Tangney, Tracy, & Robins'', 2007). Importantly [[Motivation and emotion/Book/2018/Guilt and shame|guilt and shame]] differ in their internalised state, appraisal of self and motivation urge resulting in different behaviours (''Tangney, Stuewig, & Mashek'', 2007). Other‑evaluative moral emotions—including compassion, gratitude, elevation/ awe, along with contempt, disgust and moral outrage—arise when people respond to the experiences or actions of others. These emotions generate intuitive impulses to help, reciprocate kindness, emulate moral exemplars, confront wrongdoing or avoid social violations (''Haidt'', 2003). Together, these insights show how moral emotions serve as powerful motivational systems that shape ethical and prosocial behaviour in everyday life. {{RoundBoxTop|theme=5}}'''Focus questions''' {{ic|}} * What moral emotions and how do they differ from basic emotions? <br> * How are moral emotions felt and motivate action?<br> * How do self-conscious moral emotions influence ethical behaviour? * How to other-evaluating emotions motivate prosocial behaviour?{{RoundBoxBottom}} ==What are moral emotions?== Moral emotions, defined to [[w:Jonathan_Haidt|Jonathan Haidt]] (2003), are the complex cognitively enriched emotional systems that have origins to simpler evolutionary precursors and responses to changes affecting the self. The defining qualifier of moral emotions are those involving social appraisals, cultural meaning systems and norm-based judgements that elicit motivation to act in ways that are shaped by local values and knowledge, which in turn indirectly provide benefits to the self. E.g., contempt depends on understanding rank; guilt depends on communal relationships; elevation depends on recognising moral virtue. === Historical development === In 1932, [[w:Jean_Piaget|Jean Piaget]] developed his [[w:Piaget's_theory_of_cognitive_development|theory of Cognitive development]] and in 1965 published "The moral Judgement of the child" which explores how children develop moral reasoning, tracing the evolution of their understanding of rules, justice, justice and responsibility from early childhood to adolescence (Cherry, 2026). [[w:Lawrence_Kohlberg|Lawrence Kohlberg]] further explored upon Piaget's theory and outlined the individual [[w:Lawrence_Kohlberg's_stages_of_moral_development|stages of moral development]], indicating six stages and how moral decisions are made based on various factors; such as avoiding punishment, following laws or ethical principles. It depicts how morality evolves through their lives with not all stages being achieved (Cherry, 2026 and McLeod, 2026). Haidt emphasises that researchers historically overlooked moral emotions because they focused on basic, low‑cognition forms rather than the richer, socially embedded forms that drive human morality. Although moral emotions rely on cognitive processes such as appraising injustice, evaluating social hierarchy, or recognising moral excellence, Haidt does not classify them as cognitions; instead, he frames them as "emotion–cognition hybrids" that operate intuitively and motivate moral action more powerfully than reasoning, challenging both [https://www.youtube.com/watch?v=eJTIo_MhG3M Piaget] and [https://www.youtube.com/watch?v=f0axVjiTe9Q Kolberg] theories of cognition‑centered moral psychology. Contemporary affective science views moral emotions as cognitively enriched affective states that arise when people evaluate actions, intentions, or social situations in terms of right and wrong. Rather than treating emotion and reason as separate systems, modern theorists such as Zaki and Stellar argue that moral emotions emerge from the integration of basic affective processes with higher‑order appraisals of harm, fairness, responsibility, and cultural norms. This framework emphasises that moral emotions serve adaptive social functions—motivating care, cooperation, and norm enforcement—while remaining malleable through development, cultural learning, and deliberate practices such as compassion training. In this view, moral emotions are not distinct biological modules but constructed, socially tuned emotional responses that provide functional roles that motivate ethical behaviour in complex social environments (Zaki & Stella, 2025). === Basic emotions build moral ones === [[File:Four Components of Emotion.jpg|thumb|379x379px|Figure 2. Four components of emotion]] Reeve explains emotions as innate, universal patterns of feeling and bodily responses that arise automatically in response to significant life events which motivate the body for adaptive action, using expressive behaviours by individuals to navigate the environment around them. Reeve identifies the [[Motivation and emotion/Book/2010/Ekman's basic emotions|basic emotions]] (or fundamental) as fear, anger, disgust, sadness, interest, joy and contempt. He reconciles the biological/ cognitive perspectives by stating that each basic emotion, is the foundation of a family related characteristics that are the same in each of the four components (figure 2), and distinguishes first order and second order stages, that are rooted in evolutionary adaptation (Reeve, 2022). Moral emotions are second-order emotions that are cognitively complex variants with Haidt treating them as evolved emotions families that motivate responses to violations of autonomy, community and divinity (Haidt, 2003). Additionally, moral emotions emerge when basic affective emotions are enriched by cognitive appraisals of harm, fairness transforming universal emotional responses into context sensitive judgments of right and wrong (Zaki &Stella, 2025) == How moral emotions arise and motivate action == Moral emotions develop when emotional reactions—such as anger, disgust, empathy, and admiration—are shaped by cognitive appraisals of harm, fairness, responsibility, and social norms. These appraisals give universal emotional responses moral meaning and turn them into motivations for ethical action (see table 1). As Haidt emphasises that moral emotions can be distinguished by two prototypical features; ''elicitors'' and ''action tendencies''. (Haidt, 2003). Moral emotions arise when the basic emotions families are extrapolated into social-moral context and are grouped into four distinct families (Haidt, 2003). Contemporary views highlight that they bridge feeling and doing, explaining how emotional experiences motivate ethical action and shape moral behaviour in everyday life (Zaki & Stellar, 2025). === Prototypical features === Haidt (2003) argues that moral emotions can be identified by two prototypical features that distinguish them from basic emotions. First, they are triggered by ''disinterested elicitors''—events that do not directly affect the self but carry moral significance, such as witnessing injustice, cruelty, kindness, or moral excellence. Second, they generate ''prosocial action tendencies,'' motivating behaviours that uphold or restore the moral order, including helping, comforting, punishing wrongdoing, repairing harm, or aspiring to moral ideals. These features show how moral emotions transform basic affective reactions into ethical motivations, making them central drivers of moral behaviour. === Four families of moral emotions === Haidt (2003) groups moral emotions arise when the basic emotions are extrapolated into social- moral context that can be grouped into four cognitively complex functional families that * '''Self‑conscious emotions:''' guilt, shame, and embarrassment. * '''Other‑condemning emotions:''' anger, contempt, and disgust. * '''Other‑praising emotions:''' awe, gratitude and elevation. * '''Other‑suffering emotions:''' compassion. === How moral emotions are felt and the action they motivate === {| class="wikitable" |+Table 1: emotional |'''Emotion type''' |'''Emotion''' |'''Bodily response''' |'''Motivational direction''' |- |Self-conscious |Guilt |Stomach discomfort, tension, lowered gaze |Repair harm, apologise, confess |- |Self-conscious |Shame |Collapsed posture, avoidance gaze, heat in face |Withdraw, hide, conform to norms |- |Self-conscious |Embarrassment |Blushing, nervous laughter, gaze aversion |Appease others, restore social acceptance |- |Other-focused |Empathy/Compassion |Softened facial muscles, warm affect, parasympathetic activation |Help, comfort, support |- |Other-focused |Moral Outrage |Increased heart rate, clenched jaw, sympathetic arousal |Punish, protest, advocate |- |Other-focused |Gratitude |Relaxed posture, warmth, prosocial activation |Reciprocate, strengthen bonds |- |Other-focused |Moral Elevation |Warmth in chest, tears, uplifted posture |Emulate virtue, act altruistically |} === Moral identity === [[w:Moral_identity|Moral identity]] refers to how central morality is to one’s self‑concept. Lefebvre et al.’s (2019) meta‑analysis shows that individuals with a strong moral identity experience '''more intense and more frequent moral emotions''', particularly guilt, compassion, and gratitude. Moral identity acts as an '''amplifier''' of moral emotions by: * increasing sensitivity to moral cues, * strengthening appraisals of responsibility and fairness, * enhancing prosocial action tendencies such as helping, repairing harm, and reciprocating kindness. Thus, moral identity shapes not only ''how'' moral emotions arise but also ''how strongly'' they motivate behaviour. === Moral emotion development === Moral emotions develop through a gradual integration of children’s emerging affective capacities with increasingly sophisticated social‑cognitive skills. Early in life, infants display basic precursors of empathy, but young children often struggle to coordinate their own perspective with that of others, leading to patterns such as the “[[w:Happy_victimizing|happy victimizer]]” phenomenon, where they may feel good after wrongdoing because they do not yet fully appreciate the victim’s emotional experience (Malti & Dys, 2013). As children’s [[w:Theory_of_mind|theory of mind]] and emotion understanding advance, they become better able to interpret intentions, consequences, and others’ internal states, supporting the emergence of more complex moral emotions such as guilt, sympathy, and compassion (Lane et al., 2010). By adolescence, these capacities enable meta‑perspective taking—reflecting on moral ideals, trust, and social expectations—which contributes to more stable anticipatory moral emotions, including guilt and pride, and a stronger integration of moral emotions with moral reasoning and identity (Malti & Dys, 2013). == Self-conscious moral emotions and ethical behaviour == === Guilt === Explore the following ''"Guilt is recognized as a social and moral emotion that drives prosocial behaviour, typically through actions to repair relationships with victims, often at a personal cost" .('' Kinosada, Y., et al, 2026, p.1) === Shame === Traditionally shame has been used to punish and reform behaviour, but does it actually reduce criminality? Discuss. **one reference still waiting on88 cannot confirm if it is applicable. Is shame an effective emotion now or redundant and causes more harm than good? evolutionary, it was esentail for survival, in a modern society is it causing prevalence of mental health disorders. [https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2013.00310/full#S7 Frontiers | The Role of Self-Blaming Moral Emotions in Major Depression and Their Impact on Social-Economical Decision Making] Punishment and Shame: A Philosophical Study - a North Carolina Agricultural and Technical State University, Greensboro, NC, United States == Other- focused moral emotions and prosocial behaviour == === Elevation/ Gratitude - === Discuss how positive moral emotions (Elevation and gratitude) can contribute to moral flourishing and improve patient care, (linking back to the scenario). (Bai,C., Liu, R., and Zhang, H. 2026) [https://pmc.ncbi.nlm.nih.gov/articles/PMC2689844/ Witnessing excellence in action: the ‘other-praising’ emotions of elevation, gratitude, and admiration - PMC] {{RoundBoxTop|theme=7}} ;Let's look back at the scenario You walk back to the car and are amazed at the kindness of strangers and so grateful that when you come across the person who was asking for money again. This time, you decide that even though you don’t have much in the way of money, you want to offer the person one of the ready meals in your shopping and wish the person a good day. Walking away, you feel a warm sensation in your chest and your back straightened and you find you are smiling to yourself. {{RoundBoxBottom}} == Maladaptive moral affect == While moral emotions regulate ethical and prosocial behaviour, it is important to recognise that distortions of these emotions can develop via adverse childhood experiences and trauma predict antisocial outcomes, with maladaptive forms of guilt and shame linked to [[Motivation and emotion/Book/2020/Shame, guilt, and recidivism|recidivism]] Tangney, J.,P., et all (2007) review establishes the foundational distinction between adaptive guilt and maladaptive shame, showing that guilt is behaviour‑focused and reparative, whereas shame involves global negative self‑evaluation, withdrawal, anger, and externalisation of blame. Adverse childhood experiences undermine the development of moral-motivational systems, pushing individuals towards shame based identity, interpersonal conflict and disengagement rather than guilt based repair or prosocial action (Wojcik, K., D., 2019). Furthermore, childhood abuse and neglect foster chronic, identity‑based shame, with dissociation partially mediating the trauma–shame link by fragmenting self‑representations and preventing integration of painful moral emotions (Santoro, G., et al., 2025) . Dissociation thereby reinforces shame‑laden self‑schemas and disrupts adaptive moral regulation, aligning with Tangney’s account of shame’s tendency to derail empathy and promote defensive avoidance rather than moral repair (Tangey, J., et al., 2007) === Moral disengagement === [[w:Moral_disengagement|Moral disengagement]] occurs as a way to justify unethical behaviour when the moral emotion appraisal process is distorted to not be applied to oneself. Research shows that individuals cognitively neutralise guilt and shame both before wrongdoing (anticipatory disengagement) and after wrongdoing (post‑hoc disengagement), using mechanisms such as minimising harm, diffusing responsibility, or dehumanising victims to protect the self from moral emotions that would otherwise motivate repair (Tillman, C.,J., et al., 2018) . Learning about the real consequences of one’s harmful actions intensifies guilt and shame, prompting further disengagement as a coping mechanism, illustrating how maladaptive moral affect can be sustained through cognitive distancing rather than resolved through reparative action. Developmental findings from bullying contexts further illustrate how moral emotions shape behaviour: primary‑school aggressors consistently attribute less shame and more indifference or pride to victims, reflecting entrenched disengagement and reduced empathic concern (Romera E.,M., et al., 2019). In contrast, children who experience or attribute guilt show greater moral sensitivity and prosocial tendencies, consistent with Tangney’s (2007) claim that guilt fosters empathy and constructive action . === Mitigation === === Environmental influences === * Adverse Childhood experiences - (Katharine D. Wojcik, Daniel W. Cox, David Kealy, Adverse childhood experiences and shame- and guilt-proneness: Examining the mediating roles of interpersonal problems in a community sample, Child Abuse & Neglect, Volume 98, 2019, 104233, ISSN 0145-2134, <nowiki>https://doi.org/10.1016/j.chiabu.2019.104233</nowiki> .) * '''Eisenberg, N., Spinrad, T. L., & Morris, A. S. (2014).''' ''Empathy-related responding in children.'' Supports environmental influences . *positive environmental influence ==Figures== [[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]] * For the book chapter, use several figures to illustrate concepts, add interest, and to serve as examples * For the topic development, use at least one figure (even if not ideal) to show that you know how to embed, caption, and cite a figure * Figures can show any kind of media such as photos, diagrams, graphs, video, audio, and so on * Embed figures throughout the chapter, starting with the scenario in the Overview section * Provide descriptive figure captions (use '''Figure #''' and explain the relevance of the image to the text) * Images must be embedded from [[commons:|Wikimedia Commons]] * Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Cite each figure at least once in the main text (e.g., see Figure 2) ==Learning features== Learning features help to bring book chapters to life and can be embedded throughout the chapter. Here are some options: {{anchor|Scenarios}} ;Scenarios * Scenarios, case studies, or examples that illustrate concepts in action * Present using [[#Feature boxes|feature boxes]]; can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages) * Can be real or fictional; if real, provide citation(s) {{anchor|Feature box}} ;Feature boxes * Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect * Consider using feature boxes for: ** [[#Scenarios|Scenarios]], case studies, or examples ** Focus questions ** Tips ** Quiz questions ** Take-home messages ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") *[[w:Moral_emotions|Moral emotions]] include [[Motivation and emotion/Book/2018/Guilt and shame|guilt, shame]], embarrassment, {{anchor|Tables}} ;Tables * Use tables to organise and summarise information * Cite each table at least once in the main text (e.g., see Table 1) * Tables should be captioned * [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted '''Table 1''' ''Bodily Responses to Moral emotions'' {| class="wikitable" !Strategy !Bodily responses |- |Embarrassment | |- |Guilt | |- |Shame | |- |Pride | |- |Righteous anger | |- |Comptempt | |- |Disgust | |- |Elevation | |- |Gratitude | |} ;Quizzes * One or two quiz questions for each main section is better than a long quiz at the end * Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as the take-home messages * Ask easy rather than hard questions * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display="simple"> {Unlike guilt, which pertains to specific actions, shame involves a global negative evaluation of the self. |type="()"} + True - False {Self-conscious emotions include righteous anger, contempt, disgust, elevation and gratitude |type="()"} - True + False </quiz> ==Conclusion== * Moral emotions motivate ethical decision and behaviour * Childhood trauma can disrupt the development of moral emotions, * All is not lost - protective factors can mitigate the negative impact of shame. * Provide at least three bullet-points for this section even at the topic development stage, based on preliminary thinking * For the book chapter, develop clear take-home message(s) that address the focus questions based on psychological theory and research * Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing this problem * Recommended length: 150 to 330 words {{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== [[Help:Contents/Links#Interwiki_links|Internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity) * {{tip|Suggestions for this section: * Link to the most relevant internal resources about the topic * Include the source in parentheses }} ==References== Provide the references for all citations in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. Alternatively, you can use wiki style (as used on Wikipedia), as long as the information is complete and the formatting is consistent. {{Hanging indent|1= Bai,C., Liu, R., and Zhang, H. (2026) Moral elevation, gratitude, and meaning in nurses’ positive mental health. ''Nursing Ethics'' Vol. 33(4) 1078–1093 [https://journals.sagepub.com/doi/10.1177/09697330261424358 https://doi.org/10.1177/09697330261424358] Kinosada, Y., Furukawa, Y., Wakai, T., & Nakashima, K. (2026). Does guilt motivate prosocial behaviour at the expense of others? Preregistered replications and exploratory statistical modelling. ''Asian Journal of Social Psychology'', 29, e70099. https://doi-org.ezproxy.canberra.edu.au/10.1111/ajsp.70099 Santoro, G., Sideli, L., Musetti, A., & Schimmenti, A. (2025) The Relationship Between Childhood Trauma and Shame: The Mediating Role of Dissociation. ''European Journal of Investigation in Health, Psychology and Education''. ;15(8):151. [[doi:10.3390/ejihpe15080151|http://doi.org/10.3390/ejihpe15080151]] Romera EM, Ortega-Ruiz R, Rodríguez-Barbero S and Falla D (2019) How Do You Think the Victims of Bullying Feel? A Study of Moral Emotions in Primary School. ''Frontiers in Psychology, Educational Psychology.'' 10:1753. https://doi.org/10.3389/fpsyg.2019.01753 [https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2019.01753/full l] Tangney, J. P., Stuewig, J., & Mashek, D., J., (2007). Moral Emotions and Moral Behavior. ''Annual Review of Psychology'' 58:345-372. https://doi.org/10.1146/annurev.psych.56.091103.070145 [https://doi.org/10.1146/annurev.psych.56.091103.070145] Turner, J., H., & Stets, J., E (2006). Moral Emotions. In ''Handbook of the Sociology of Emotions (''pp.544 - 566). Spinger. [https://link.springer.com/book/10.1007/978-0-387-30715-2 https://doi.org/10.1007/978-0-387-30715-2] . Tillman CJ, Gonzalez K, Whitman MV, Crawford WS and Hood AC (2018) A Multi-Functional View of Moral Disengagement: Exploring the Effects of Learning the Consequences. ''Frontiers in Psychology, Personality and Social Psychology''. 8:2286. [https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2017.02286/full#F2 doi: 10.3389/fpsyg.2017.02286] Wojcik, K., D., Cox, D., W., & Kealy,D. (2019) Adverse childhood experiences and shame- and guilt-proneness: Examining the mediating roles of interpersonal problems in a community sample, ''Child Abuse & Neglec''t, Volume 98, 2019, 104233, ISSN 0145-2134, https://doi.org/10.1016/j.chiabu.2019.104233 }} APA style example: {{Hanging indent|1= Rosenberg, B. D., & Siegel, J. T. (2018). A 50-year review of psychological reactance theory: Do not read this article. ''Motivation Science'', ''4''(4), 281–300. https://doi.org/10.1037/mot0000091 Sacks, O. (1985). ''The man who mistook his wife for a hat and other clinical tales''. Harper & Row. }} {{tip|Suggestions for this section: * Important aspects of APA referencing style ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case (except first letter and proper names), ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** doi as a URL which is a working hyperlink (i.e., clickable) ** Hanging indent: Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]: Use "Edit source" <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** dois which aren't clickable as working hyperlinks ** Citing sources that haven't been consulted }} ==External links== [https://www.ted.com/talks/asha_curran_how_acts_of_kindness_sparked_a_global_movement Ideas change everything] (TED) [https://www.youtube.com/watch?v=b4CfyLe0IlQ Prosocial Behavior | Ethics Defined] (Youtube) [[Help:Contents/Links#External_links|External links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Link to the most relevant external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Emotion]] [[Category:Motivation and emotion/Book/Morality]] teqjkcyif1jn67dwt66vw7n3pydctxl 2842561 2842456 2026-10-25T11:18:10Z U3263365 3104938 /* Environmental influences */ finalised multiple sections 2842561 wikitext text/x-wiki {{title|Moral emotions and ethical behaviour<br>How do moral emotions motivate ethical and prosocial action?}} __TOC__ ==Overview== {{RoundBoxTop|theme=7}} [[File:Begging man Harmony Lot 27 Elliot Street downtown Brattleboro VT July 2025 02.jpg|right|thumb|150px|'''Figure 1'''. Homeless man begging]] ;Scenario Imagine that you are doing your weekly shopping and you come across a homeless person who is asking for food or money. The person present in an unkempt state and initially you make sure you walk past, your stomach drops, and subconsciously wrinkle your nose and you decide to take a longer path to the shops in order to avoid them. When paying for your groceries, you realise that you don’t have quite enough to pay the full amount, instantly, your face gets hot as there is a line of people waiting behind you and it seems they are all staring at you, judging your error. Whilst you are talking to the register assistant, asking for some time to go through your groceries to determine which items are a priority, the person behind you offers to make up the difference. Suddenly, the hot feeling in your face becomes a warm feeling in your chest and you start profusely thanking the person and ask if there is anything you can do to make it up to them, they respond with “when you are in a position to pay it forward to some else, please do so.” {{RoundBoxBottom}} [[w:Moral_emotions|Moral emotions]] potentially set humans apart from other animals, by enabling people to react to social events that do not directly affect the self but concern the interests or welfare either of others and society as a whole (''Haidt, 2003).'' They arise when individuals evaluate actions, intentions, or social situations in terms of right or wrong, in accordance with social norms and moral standards. This guides behaviours that support cooperation, fairness and social harmony. This distinguishes moral emotions from [[Motivation and emotion/Lectures/Individual emotions|basic emotions]], which are a person's primary response to a situation. Basic emotions fast, short lived, whilst moral emotions, like [https://www.simplypsychology.org/primary-and-secondary-emotions.html secondary emotions], are more complex and require a cognitive evaluation of the self from learned social meaning. To understand how moral emotions motivate behaviour, examination of the psychological/ psychological mechanisms behind these emotions is required. Moral emotions involve appraisals of responsibility and social meaning, biological responses that prepare the body for action, and expressive behaviours that communicate intentions. These mechanisms create ''action readiness'', transforming moral evaluations into behavioural tendencies such as helping, repairing harm, confronting wrongdoing, or avoiding social violations (''Reeve'', 2023). Understanding moral emotions is essential to link how they motivate ethical and prosocial action. [[w:Self-conscious_emotions|Self-conscious moral emotions]] - such as guilt, shame, embarrassment, and authentic pride - occur when people evaluate their own behaviour against internalised moral standards. These emotions regulate the moral self by motivating reparative action, helping others, rule compliance, or withdrawal (''Tangney, Tracy, & Robins'', 2007). Importantly [[Motivation and emotion/Book/2018/Guilt and shame|guilt and shame]] differ in their internalised state, appraisal of self and motivation urge resulting in different behaviours (''Tangney, Stuewig, & Mashek'', 2007). Other‑evaluative moral emotions—including compassion, gratitude, elevation/ awe, along with contempt, disgust and moral outrage—arise when people respond to the experiences or actions of others. These emotions generate intuitive impulses to help, reciprocate kindness, emulate moral exemplars, confront wrongdoing or avoid social violations (''Haidt'', 2003). Together, these insights show how moral emotions serve as powerful motivational systems that shape ethical and prosocial behaviour in everyday life. {{RoundBoxTop|theme=5}}'''Focus questions''' {{ic|}} * What moral emotions and how do they differ from basic emotions? <br> * How are moral emotions felt and motivate action?<br> * How do self-conscious moral emotions influence ethical behaviour? * How to other-evaluating emotions motivate prosocial behaviour?{{RoundBoxBottom}} ==What are moral emotions?== Moral emotions, defined to [[w:Jonathan_Haidt|Jonathan Haidt]] (2003), are the complex cognitively enriched emotional systems that have origins to simpler evolutionary precursors and responses to changes affecting the self. The defining qualifier of moral emotions are those involving social appraisals, cultural meaning systems and norm-based judgements that elicit motivation to act in ways that are shaped by local values and knowledge, which in turn indirectly provide benefits to the self. E.g., contempt depends on understanding rank; guilt depends on communal relationships; elevation depends on recognising moral virtue. === Historical development === In 1932, [[w:Jean_Piaget|Jean Piaget]] developed his [[w:Piaget's_theory_of_cognitive_development|theory of Cognitive development]] and in 1965 published "The moral Judgement of the child" which explores how children develop moral reasoning, tracing the evolution of their understanding of rules, justice, justice and responsibility from early childhood to adolescence (Cherry, 2026). [[w:Lawrence_Kohlberg|Lawrence Kohlberg]] further explored upon Piaget's theory and outlined the individual [[w:Lawrence_Kohlberg's_stages_of_moral_development|stages of moral development]], indicating six stages and how moral decisions are made based on various factors; such as avoiding punishment, following laws or ethical principles. It depicts how morality evolves through their lives with not all stages being achieved (Cherry, 2026 and McLeod, 2026). Haidt emphasises that researchers historically overlooked moral emotions because they focused on basic, low‑cognition forms rather than the richer, socially embedded forms that drive human morality. Although moral emotions rely on cognitive processes such as appraising injustice, evaluating social hierarchy, or recognising moral excellence, Haidt does not classify them as cognitions; instead, he frames them as "emotion–cognition hybrids" that operate intuitively and motivate moral action more powerfully than reasoning, challenging both [https://www.youtube.com/watch?v=eJTIo_MhG3M Piaget] and [https://www.youtube.com/watch?v=f0axVjiTe9Q Kolberg] theories of cognition‑centered moral psychology. Contemporary affective science views moral emotions as cognitively enriched affective states that arise when people evaluate actions, intentions, or social situations in terms of right and wrong. Rather than treating emotion and reason as separate systems, modern theorists such as Zaki and Stellar argue that moral emotions emerge from the integration of basic affective processes with higher‑order appraisals of harm, fairness, responsibility, and cultural norms. This framework emphasises that moral emotions serve adaptive social functions—motivating care, cooperation, and norm enforcement—while remaining malleable through development, cultural learning, and deliberate practices such as compassion training. In this view, moral emotions are not distinct biological modules but constructed, socially tuned emotional responses that provide functional roles that motivate ethical behaviour in complex social environments (Zaki & Stella, 2025). === Basic emotions build moral ones === [[File:Four Components of Emotion.jpg|thumb|379x379px|Figure 2. Four components of emotion]] Reeve explains emotions as innate, universal patterns of feeling and bodily responses that arise automatically in response to significant life events which motivate the body for adaptive action, using expressive behaviours by individuals to navigate the environment around them. Reeve identifies the [[Motivation and emotion/Book/2010/Ekman's basic emotions|basic emotions]] (or fundamental) as fear, anger, disgust, sadness, interest, joy and contempt. He reconciles the biological/ cognitive perspectives by stating that each basic emotion, is the foundation of a family related characteristics that are the same in each of the four components (figure 2), and distinguishes first order and second order stages, that are rooted in evolutionary adaptation (Reeve, 2022). Moral emotions are second-order emotions that are cognitively complex variants with Haidt treating them as evolved emotions families that motivate responses to violations of autonomy, community and divinity (Haidt, 2003). Additionally, moral emotions emerge when basic affective emotions are enriched by cognitive appraisals of harm, fairness transforming universal emotional responses into context sensitive judgments of right and wrong (Zaki &Stella, 2025) == How moral emotions arise and motivate action == Moral emotions develop when emotional reactions—such as anger, disgust, empathy, and admiration—are shaped by cognitive appraisals of harm, fairness, responsibility, and social norms. These appraisals give universal emotional responses moral meaning and turn them into motivations for ethical action (see table 1). As Haidt emphasises that moral emotions can be distinguished by two prototypical features; ''elicitors'' and ''action tendencies''. (Haidt, 2003). Moral emotions arise when the basic emotions families are extrapolated into social-moral context and are grouped into four distinct families (Haidt, 2003). Contemporary views highlight that they bridge feeling and doing, explaining how emotional experiences motivate ethical action and shape moral behaviour in everyday life (Zaki & Stellar, 2025). === Prototypical features === Haidt (2003) argues that moral emotions can be identified by two prototypical features that distinguish them from basic emotions. First, they are triggered by ''disinterested elicitors''—events that do not directly affect the self but carry moral significance, such as witnessing injustice, cruelty, kindness, or moral excellence. Second, they generate ''prosocial action tendencies,'' motivating behaviours that uphold or restore the moral order, including helping, comforting, punishing wrongdoing, repairing harm, or aspiring to moral ideals. These features show how moral emotions transform basic affective reactions into ethical motivations, making them central drivers of moral behaviour. === Four families of moral emotions === Haidt (2003) groups moral emotions arise when the basic emotions are extrapolated into social- moral context that can be grouped into four cognitively complex functional families that * '''Self‑conscious emotions:''' guilt, shame, and embarrassment. * '''Other‑condemning emotions:''' anger, contempt, and disgust. * '''Other‑praising emotions:''' awe, gratitude and elevation. * '''Other‑suffering emotions:''' compassion. === How moral emotions are felt and the action they motivate === {| class="wikitable" |+Table 1: emotional |'''Emotion type''' |'''Emotion''' |'''Bodily response''' |'''Motivational direction''' |- |Self-conscious |Guilt |Stomach discomfort, tension, lowered gaze |Repair harm, apologise, confess |- |Self-conscious |Shame |Collapsed posture, avoidance gaze, heat in face |Withdraw, hide, conform to norms |- |Self-conscious |Embarrassment |Blushing, nervous laughter, gaze aversion |Appease others, restore social acceptance |- |Other-focused |Empathy/Compassion |Softened facial muscles, warm affect, parasympathetic activation |Help, comfort, support |- |Other-focused |Moral Outrage |Increased heart rate, clenched jaw, sympathetic arousal |Punish, protest, advocate |- |Other-focused |Gratitude |Relaxed posture, warmth, prosocial activation |Reciprocate, strengthen bonds |- |Other-focused |Moral Elevation |Warmth in chest, tears, uplifted posture |Emulate virtue, act altruistically |} === Moral identity === [[w:Moral_identity|Moral identity]] refers to how central morality is to one’s self‑concept. Lefebvre et al.’s (2019) meta‑analysis shows that individuals with a strong moral identity experience '''more intense and more frequent moral emotions''', particularly guilt, compassion, and gratitude. Moral identity acts as an '''amplifier''' of moral emotions by: * increasing sensitivity to moral cues, * strengthening appraisals of responsibility and fairness, * enhancing prosocial action tendencies such as helping, repairing harm, and reciprocating kindness. Thus, moral identity shapes not only ''how'' moral emotions arise but also ''how strongly'' they motivate behaviour. === Moral emotion development === Moral emotions develop through a gradual integration of children’s emerging affective capacities with increasingly sophisticated social‑cognitive skills. Early in life, infants display basic precursors of empathy, but young children often struggle to coordinate their own perspective with that of others, leading to patterns such as the “[[w:Happy_victimizing|happy victimizer]]” phenomenon, where they may feel good after wrongdoing because they do not yet fully appreciate the victim’s emotional experience (Malti & Dys, 2013). As children’s [[w:Theory_of_mind|theory of mind]] and emotion understanding advance, they become better able to interpret intentions, consequences, and others’ internal states, supporting the emergence of more complex moral emotions such as guilt, sympathy, and compassion (Lane et al., 2010). By adolescence, these capacities enable meta‑perspective taking—reflecting on moral ideals, trust, and social expectations—which contributes to more stable anticipatory moral emotions, including guilt and pride, and a stronger integration of moral emotions with moral reasoning and identity (Malti & Dys, 2013). === Environmental influences === Costa Martins, M., et al. (2021) concluded that secure attachment is consistently linked to healthier moral‑emotion development, including empathy, guilt, forgiveness, and emotion regulation. Secure attachment fosters the emotional competencies necessary for prosocial behaviour, while insecure attachment increases vulnerability to maladaptive moral affect (e.g., shame, emotional withdrawal). Understanding these pathways is essential for supporting children’s emotional, relational, and academic wellbeing. [[w:Adverse_childhood_experiences|Adverse childhood experiences]] (ACE) undermine the development of moral-motivational systems, pushing individuals towards shame based identity, interpersonal conflict and disengagement rather than guilt based repair or prosocial action (Wojcik, K., D., 2019). Furthermore, childhood abuse and neglect foster chronic, identity‑based shame, with dissociation partially mediating the trauma–shame link by fragmenting self‑representations and preventing integration of painful moral emotions (Santoro, G., et al., 2025) . Dissociation thereby reinforces shame‑laden self‑schemas and disrupts adaptive moral regulation, aligning with Tangney’s account of shame’s tendency to derail empathy and promote defensive avoidance rather than moral repair (Tangey, J., et al., 2007) == Self-conscious moral emotions and ethical behaviour == === Guilt === Explore the following ''"Guilt is recognized as a social and moral emotion that drives prosocial behaviour, typically through actions to repair relationships with victims, often at a personal cost" .('' Kinosada, Y., et al, 2026, p.1) === Shame === Traditionally shame has been used to punish and reform behaviour, but does it actually reduce criminality? Discuss. **one reference still waiting on88 cannot confirm if it is applicable. Is shame an effective emotion now or redundant and causes more harm than good? evolutionary, it was esentail for survival, in a modern society is it causing prevalence of mental health disorders. [https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2013.00310/full#S7 Frontiers | The Role of Self-Blaming Moral Emotions in Major Depression and Their Impact on Social-Economical Decision Making] Punishment and Shame: A Philosophical Study - a North Carolina Agricultural and Technical State University, Greensboro, NC, United States === Quiz === <quiz display="simple"> {Unlike guilt, which pertains to specific actions, shame involves a global negative evaluation of the self. |type="()"} + True - False {Self-conscious emotions include righteous anger, contempt, disgust, elevation and gratitude |type="()"} - True + False </quiz> == Other- focused moral emotions and prosocial behaviour == === Elevation/ Gratitude - === Discuss how positive moral emotions (Elevation and gratitude) can contribute to moral flourishing and improve patient care, (linking back to the scenario). (Bai,C., Liu, R., and Zhang, H. 2026) [https://pmc.ncbi.nlm.nih.gov/articles/PMC2689844/ Witnessing excellence in action: the ‘other-praising’ emotions of elevation, gratitude, and admiration - PMC] {{RoundBoxTop|theme=7}} ;Let's look back at the scenario You walk back to the car and are amazed at the kindness of strangers and so grateful that when you come across the person who was asking for money again. This time, you decide that even though you don’t have much in the way of money, you want to offer the person one of the ready meals in your shopping and wish the person a good day. Walking away, you feel a warm sensation in your chest and your back straightened and you find you are smiling to yourself. {{RoundBoxBottom}} == Maladaptive moral affect == While moral emotions regulate ethical and prosocial behaviour, it is important to recognise that distortions of these emotions can develop via adverse childhood experiences and trauma predict antisocial outcomes, with maladaptive forms of guilt and shame linked to [[Motivation and emotion/Book/2020/Shame, guilt, and recidivism|recidivism]]. Costa Martins, M., et al. (2021) found that attachment insecurity in children was linked with poor [[Motivation and emotion/Book/2018/Emotion regulation in children|emotional regulation]] and predicted maladaptive moral affect. Avoidant attachment is associated with higher shame and lower empathy. For example, avoidant children show significantly higher shame scores, in adolescence there was less empathic concern and lower perspective taking. This aligns with broader findings that insecure attachment disrupts the development of adaptive guilt and empathy. (Costa Martins et al 2021). Furthermore, Tangney, J.,P., et all (2007) review establishes the foundational distinction between adaptive guilt and maladaptive shame, showing that guilt is behaviour‑focused and reparative, whereas shame involves global negative self‑evaluation, withdrawal, anger, and externalisation of blame. === Moral disengagement === [[w:Moral_disengagement|Moral disengagement]] occurs as a way to justify unethical behaviour when the moral emotion appraisal process is distorted to not be applied to oneself. Research shows that individuals cognitively neutralise guilt and shame both before wrongdoing (anticipatory disengagement) and after wrongdoing (post‑hoc disengagement), using mechanisms such as minimising harm, diffusing responsibility, or dehumanising victims to protect the self from moral emotions that would otherwise motivate repair (Tillman, C.,J., et al., 2018) . Learning about the real consequences of one’s harmful actions intensifies guilt and shame, prompting further disengagement as a coping mechanism, illustrating how maladaptive moral affect can be sustained through cognitive distancing rather than resolved through reparative action. Developmental findings from bullying contexts further illustrate how moral emotions shape behaviour: primary‑school aggressors consistently attribute less shame and more indifference or pride to victims, reflecting entrenched disengagement and reduced empathic concern (Romera E.,M., et al., 2019). In contrast, children who experience or attribute guilt show greater moral sensitivity and prosocial tendencies, consistent with Tangney’s (2007) claim that guilt fosters empathy and constructive action . === Mitigation === {| class="wikitable" |+ !Maladaptive concern !Maladaptive development pathway !Mitigation strategies !Reference |- |ACEs → shame/guilt proneness |Interpersonal sensitivity → shame; interpersonal aggression → guilt |Improve interpersonal functioning; attachment repair; social‑emotional skills; reduce interpersonal sensitivity/aggression |Wojcik, K.,D., et al (2019) |- |Moral disengagement as avoidance of guilt/shame |People neutralise guilt/shame before & after wrongdoing |Increase consequence awareness; empathy training; highlight victim impact; reduce cognitive distancing |Tillman et al (2018) |- |Aggressors show low shame, high indifference/pride |Moral disengagement in children; lack of victim perspective |Moral‑emotion education; victim perspective‑taking; empathy development; encourage guilt‑based repair |Romera et al (2019) |- |Shame maladaptive; guilt adaptive |Shame → withdrawal, defensiveness; guilt → reparative action |Shift shame → guilt; promote responsibility‑taking; encourage reparative behaviours; reduce global self‑condemnation |Tangey et al (2007) |- |Insecure attachment → shame, anger, disengagement |Attachment insecurity distorts moral‑emotion development |Strengthen attachment security; caregiver sensitivity; relational repair; promote empathy and guilt development |Costa Martins  et at (2021) |- |Developmental pathways of moral emotions |Moral emotions shaped by socialisation, cognition, empathy |Developmental scaffolding; emotion coaching; moral reasoning support; structured opportunities for prosocial action |Malti & Dys (2015) |} ==Figures== [[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]] * For the book chapter, use several figures to illustrate concepts, add interest, and to serve as examples * For the topic development, use at least one figure (even if not ideal) to show that you know how to embed, caption, and cite a figure * Figures can show any kind of media such as photos, diagrams, graphs, video, audio, and so on * Embed figures throughout the chapter, starting with the scenario in the Overview section * Provide descriptive figure captions (use '''Figure #''' and explain the relevance of the image to the text) * Images must be embedded from [[commons:|Wikimedia Commons]] * Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Cite each figure at least once in the main text (e.g., see Figure 2) ==Learning features== Learning features help to bring book chapters to life and can be embedded throughout the chapter. Here are some options: {{anchor|Scenarios}} ;Scenarios * Scenarios, case studies, or examples that illustrate concepts in action * Present using [[#Feature boxes|feature boxes]]; can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages) * Can be real or fictional; if real, provide citation(s) {{anchor|Feature box}} ;Feature boxes * Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect * Consider using feature boxes for: ** [[#Scenarios|Scenarios]], case studies, or examples ** Focus questions ** Tips ** Quiz questions ** Take-home messages ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") *[[w:Moral_emotions|Moral emotions]] include [[Motivation and emotion/Book/2018/Guilt and shame|guilt, shame]], embarrassment, {{anchor|Tables}} ;Tables * Use tables to organise and summarise information * Cite each table at least once in the main text (e.g., see Table 1) * Tables should be captioned * [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted '''Table 1''' ''Bodily Responses to Moral emotions'' {| class="wikitable" !Strategy !Bodily responses |- |Embarrassment | |- |Guilt | |- |Shame | |- |Pride | |- |Righteous anger | |- |Comptempt | |- |Disgust | |- |Elevation | |- |Gratitude | |} ;Quizzes * One or two quiz questions for each main section is better than a long quiz at the end * Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as the take-home messages * Ask easy rather than hard questions * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": ==Conclusion== * Moral emotions motivate ethical decision and behaviour * Childhood trauma can disrupt the development of moral emotions, * All is not lost - protective factors can mitigate the negative impact of shame. * Provide at least three bullet-points for this section even at the topic development stage, based on preliminary thinking * For the book chapter, develop clear take-home message(s) that address the focus questions based on psychological theory and research * Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing this problem * Recommended length: 150 to 330 words {{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== [[Help:Contents/Links#Interwiki_links|Internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity) * {{tip|Suggestions for this section: * Link to the most relevant internal resources about the topic * Include the source in parentheses }} ==References== Provide the references for all citations in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. Alternatively, you can use wiki style (as used on Wikipedia), as long as the information is complete and the formatting is consistent. {{Hanging indent|1= Bai,C., Liu, R., and Zhang, H. (2026) Moral elevation, gratitude, and meaning in nurses’ positive mental health. ''Nursing Ethics'' Vol. 33(4) 1078–1093 [https://journals.sagepub.com/doi/10.1177/09697330261424358 https://doi.org/10.1177/09697330261424358] Costa Martins, M., Santos, A. F., Fernandes, M., & Veríssimo, M. (2021). Attachment and the Development of Moral Emotions in Children and Adolescents: A Systematic Review. Children, 8(10), 915. https://doi.org/10.3390/children8100915 Kinosada, Y., Furukawa, Y., Wakai, T., & Nakashima, K. (2026). Does guilt motivate prosocial behaviour at the expense of others? Preregistered replications and exploratory statistical modelling. ''Asian Journal of Social Psychology'', 29, e70099. https://doi-org.ezproxy.canberra.edu.au/10.1111/ajsp.70099 Santoro, G., Sideli, L., Musetti, A., & Schimmenti, A. (2025) The Relationship Between Childhood Trauma and Shame: The Mediating Role of Dissociation. ''European Journal of Investigation in Health, Psychology and Education''. ;15(8):151. [[doi:10.3390/ejihpe15080151|http://doi.org/10.3390/ejihpe15080151]] Romera EM, Ortega-Ruiz R, Rodríguez-Barbero S and Falla D (2019) How Do You Think the Victims of Bullying Feel? A Study of Moral Emotions in Primary School. ''Frontiers in Psychology, Educational Psychology.'' 10:1753. https://doi.org/10.3389/fpsyg.2019.01753 [https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2019.01753/full l] Tangney, J. P., Stuewig, J., & Mashek, D., J., (2007). Moral Emotions and Moral Behavior. ''Annual Review of Psychology'' 58:345-372. https://doi.org/10.1146/annurev.psych.56.091103.070145 [https://doi.org/10.1146/annurev.psych.56.091103.070145] Turner, J., H., & Stets, J., E (2006). Moral Emotions. In ''Handbook of the Sociology of Emotions (''pp.544 - 566). Spinger. [https://link.springer.com/book/10.1007/978-0-387-30715-2 https://doi.org/10.1007/978-0-387-30715-2] . Tillman CJ, Gonzalez K, Whitman MV, Crawford WS and Hood AC (2018) A Multi-Functional View of Moral Disengagement: Exploring the Effects of Learning the Consequences. ''Frontiers in Psychology, Personality and Social Psychology''. 8:2286. [https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2017.02286/full#F2 doi: 10.3389/fpsyg.2017.02286] Wojcik, K., D., Cox, D., W., & Kealy,D. (2019) Adverse childhood experiences and shame- and guilt-proneness: Examining the mediating roles of interpersonal problems in a community sample, ''Child Abuse & Neglec''t, Volume 98, 2019, 104233, ISSN 0145-2134, https://doi.org/10.1016/j.chiabu.2019.104233 }} APA style example: {{Hanging indent|1= Rosenberg, B. D., & Siegel, J. T. (2018). A 50-year review of psychological reactance theory: Do not read this article. ''Motivation Science'', ''4''(4), 281–300. https://doi.org/10.1037/mot0000091 Sacks, O. (1985). ''The man who mistook his wife for a hat and other clinical tales''. Harper & Row. }} {{tip|Suggestions for this section: * Important aspects of APA referencing style ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case (except first letter and proper names), ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** doi as a URL which is a working hyperlink (i.e., clickable) ** Hanging indent: Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]: Use "Edit source" <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** dois which aren't clickable as working hyperlinks ** Citing sources that haven't been consulted }} ==External links== [https://www.ted.com/talks/asha_curran_how_acts_of_kindness_sparked_a_global_movement Ideas change everything] (TED) [https://www.youtube.com/watch?v=b4CfyLe0IlQ Prosocial Behavior | Ethics Defined] (Youtube) [[Help:Contents/Links#External_links|External links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Link to the most relevant external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Emotion]] [[Category:Motivation and emotion/Book/Morality]] 6utwezviouvaqvwgn9665uz8b22amb7 0 331447 2842553 2841193 2026-10-25T08:27:47Z Jtneill 10242 Copyediting 2842553 wikitext text/x-wiki {{title|Gender and achievement motivation:<br>How does gender shape where, how, and under what conditions achievement motivation is expressed?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}} [[File:Liitle girl cleaning a table.jpg|thumb|236x236px|'''Figure 1:''' Young Girl Learning To Clean]] ;Imagine this... A young girls {{g}} go to school every day and when it's time to pack up the classroom the boys are instructed to move the heavy chairs on top of the tables, and the girls are told to tidy up the smaller messes around the classroom, putting books and pencils away, and picking up toys off the floor, as seen in Figure 1 {{ic|Split this opening sentence into two sentences}}. As the years go on {{g}} this physical strength is continually rewarded in boys, while organisation and cleaning up others' messes is praised in girls. As these girls grow older they tend to be hesitant when improving stereotypical male traits, {{g}} because these traits have never been encouraged or praised when exhibited, thus this motivation is lower. {{RoundBoxBottom}} This idea of achievement motivation applies to a number {{vague}} of scenarios, where attributes and skills that are highly regarded are praised and pushed in boys, while girls are not encouraged, and are often punished for engaging in. {{ic|Does it also work the other way? i.e., boys are not encouraged in some skill areas?}} This difference in socialisation can lead to a clear split in how and where each gender is motivated to perform well within the attributes that society establishes as valuable. Understanding how gender begins to affect people's achievement motivation is important for minimising the obstacles that hold people back from working towards goals. {{RoundBoxTop|theme=3}} ;Focus questions * What {{vague}} differences can we see in Achievement Motivation? * Where do we see these differences most often exhibited {{vague}}? * How does gender account for these {{what}} differences? {{RoundBoxBottom}} == Differentiating Achievement Motivations == Achievement motivation is the focus of motivation on performance in tasks with a clear measurement of success and failure (Eccles et al. 1983). Early studies of motivation recognised the importance of the effect of motivation on outcomes, but lacked clear values that motivation could be measured by. In the 1960s, David McClelland initially developed three key needs that drive motivation, and in 1975 John Atkinson further developed the understanding of achievement motivation and how it shaped behaviour, resulting in Expectancy Value Theory {{f}}. The three needs that McClelland (1965) found to drive motivation were: * Need for achievement, where people are motivated by a desire to be better simply for the sake of it * Need for power, a motivation to achieve using other people's success and one's ability to surpass it as a metric of success * Need for affiliation, where people are motivated by social connections and community acceptance These initial measures of motivation were developed for use in a corporate setting with the intention to find what factors of motivation make an effective manager, where McCelland and Burnham came to the conclusion that power was the strongest predicting factor of motivation in good managers (McClelland & Burnham, 1995). Subsequent study has seen achievement motivation further defined, with an emphasis on approach tendencies, where people take active steps towards achieving their desired outcome, rather than avoidance tendencies, where actions are taken in order to avoid failure, {{g}} these avoidance tendencies are not considered as a demonstration of achievement motivation (Pang, 2010). <quiz display=simple> {Which of these are not one of McClelland's drive motivations {{g}} |type="[]"} + Need for Stability - Need for Power - Need for Achievement - Need for Affiliation </quiz> == How Gender Shapes Motivation Theory == We often believe that motivation is derived from one of two places, inside or outside ourselves, and as our understanding of motivations grows we find new ways to define and measure these sources of motivation and to what degree they affect our actions {{ic|Rewrite in 3rd person perspective}}. Along with our growing understanding of motivation is our increasingly nuanced understanding of gender, which, once seen as a strict set of behaviours and attributes, has been foundational for developing our understanding of self as fluid and changing through different ages, cultures, and situations. {{f}} {{vague}} === Attribution theory === Early {{what}} research found that females are more likely to be extrinsic motivated than males, and thus are more vulnerable to external influence, which may be a contributing factor for the discrepancy between academic results {{where}} (Boggiano et al., 1991)(Baer & Kaufman, 2008). As students move through school and onto more difficult work, females are more likely to be affected by negative feedback and a lack of high success, this lessening of positive, extrinsic feedback results in a reduced motivation to excel academically. Boggiano’s study suggests that as girls move through school and are increasingly exposed to significant external feedback from teachers, even girls who initially identity as intrinsically motivated may transition to a more extrinsically motivated mindset, making them more vulnerable to both external validation, and commonly expressed expectations of what areas they should be most driven to excel at (Deci & Ryan, 1987). {{ic|Where were these studies conducted? They are over 30 years old; how relevant are they?}} More recent research has found the opposite to be true, with female students {{where}} exhibit {{g}} significantly higher levels of intrinsic motivation, while male students display slightly higher levels of extrinsic motivation. (Hung et al., 2025). This change in attribution of motivation between genders is not largely studied, but may be due to a changing understanding of how intrinsic and extrinsic are defined. Extrinsic motivation is defined as the attribution of motivation to external sources, like a goal imposed by someone else, while intrinsic motivation is a desire to achieve for one's own sake (Bandura, 1977), but as extrinsic motivation sources increase, people may begin to misattribute their intrinsic motivation to the numerous available extrinsic motivation sources (Frey, 1997). {{example}} The increase in extrinsic motivation may also be due to an increased presence of external locus of control, as the two have been found to be linked, and thus a simultaneous increase of both factors can be expected (Ryan & Deci, 2000).  External locus of control is seen when a person views their actions as having little to do with the achievement or reward that they may gain, while internal locus of control is demonstrated when individuals view their own actions and efforts as being the main contributing force for their outcomes and achievements (Rotter, 1954). People with Internal locus of control are found to be more motivated to perform well at work than those with an external locus of control in the workplace, due to their understanding that their actions do have an impact on their work and that they are in control of their ability to achieve any rewards and opportunities available (Kamdron, 2015). {{ic|How is this related to gender?}} Internal locus of control are {{g}} found to be associated with higher academic achievement, and historically locus of control was a much stronger predictor of academic achievement for boys than girls (Findley & Cooper, 1983), which would indicate that boys were more prone to internal locus of control than girls. However {{g}} this was thought to largely be due to the way that boys and girls were socialied, rather than an innate biological inclination to either locus of control, with girls historically receiving less pressure to perform academically, thus not equating their academic performance to their personal abilities (Sherman et al., 1997). As views change and the way that girls and boys are socialised through childhood changes, it is likely that this tendency for boys to be prone to internal locus of control would lessen and the gender differences between internal and external control would lessen, {{g}} this may be contributing to the recent increase in the average females {{g}} academic score (Ullah & Ullah, 2019). === Expectancy-Value Theory === When people believe that they will do well at a task, and that the successful completion of a task will result in substantial gains for themselves, they are more motivated to work hard at and complete the task according to Expectancy-Value Theory (Wigfield & Eccles, 2000). The metric for success can differ between individuals, with the measure for expected success being often measured by two metrics, one's perception of their competency, focusing on the individual's skill level, and the expected degree of success, how successful these skills can be applied to specific tasks {{f}}. These two metrics can often be difficult to distinguish from one another when empirically measured, meanwhile the expected value of completing tasks is easily defined, and commonly divided into, attainment value, how important it is that the task is well performed, intrinsic value, often measured by enjoyment of the task, utility value, where value is measured against how useful it will be going forward, and cost to the individual, including both physical and emotional resources {{f}}. Gender differences in both task value and expected competency have been examined since the early 1970s, but the outcomes of the studies have been changing as more modern studies are published. Early examination of students' academic expectations found that boys had much higher educational expectations (Dweck & Gilliard, 1975) but these initial studies viewed academies as monolithic, and once research began to examine the different areas of academia and the expected performance outcomes, more nuanced divisions were found. Boys were found to have much higher perceived abilities in maths, and sports, while girls had higher expected outcomes in music and reading across initial years of school (Eccles et al., 1993). However {{g}} these differences were found to reduce as students aged, with boys' belief in their abilities falling at a much faster rate than girls did, which eventually results in a similar expected ability across multiple subjects by the end of secondary school (Jacobs et al., 2002). <quiz display=simple> {How might gender stereotypes influence students' achievement motivation? |type="()"} - Gender stereotypes have no influence on students' academic motivation. + Gender stereotypes may influence students' beliefs about their abilities and the value they place on different subjects. - Gender stereotypes determine students' abilities regardless of their learning experiences. </quiz> Expectancy value is found to positively predict students {{g}} academic outcomes (Plante et al., 2012), and while belief in abilities may fall as students age, the closer they get to graduation the more utility and attainment value successful academic performance will provide. When students enter college or university, their academic outcomes are determined by the perceived value of the service they would provide with their degrees (Moely & Ilustre, 2016), demonstrating a less self-interested view of value than that seen in younger school children. A significant difference level of empathy found between genders (Rochat, 2022) results in degrees and future careers holding varying values, this results in women being more motivated to enter into healthcare, education and social services because of their increased perceived values in the importance of services that directly provide community benefit, while men dominate manual and technical fields {{f}}. == What contexts cultivate achievement motivations == {{expand}} === Motivation development in educational setting === [[File:The Motivational Theory of Role Modeling.png|thumb|663x663px|'''Figure 2'''. The Motivational Theory of Role Modeling {{f}}, demonstrating the cyclical nature of goal aspiration and goal attainment]] Despite an increasing push to equalise the ratio of men and women in many fields, with a particular emphasis on STEM fields, many stereotypically feminine jobs, like healthcare and early education, still remain predominantly female, while typically masculine jobs like construction and transport are primarily occupied by men {{f}}. As academic achievement equalises between females and males, males continue to have higher perceived math skill and math oriented motivation when compared to girls (Skaalvik & Rankin, 1994), {{g}} this difference in perceived capabilities has nothing to do with skill, and yet girls in school find themselves less motivated to achieve in math {{f}}. When viewing this date {{sp}} through the lens of expectancy-value theory, a key factor in explaining this difference in motivation lies in the expected value that girls believe they will achieve through academic performance {{f}}. Studies show that even when girls have a desire to go into male dominated fields of work, many change course of study or jobs due to the accommodating nature of these fields when concerning women {{f}}. Male dominated fields tend to lack the flexibility that women dominated fields offer, which many girls and women who plan to start a family consider a strong deciding factor when considering job opportunities, which steers young women away from fields that they may be passionate about and skills in (Frome et al., 2006). Women who did enter male dominated jobs find that success is more important to them than women in female dominated fields did, and held strong views about the importance of their work, these factors may mitigate the effects previously mentioned that typically turn women away from male dominated fields {{f}}. === Gender performance and achievement motivation === Role models play an important part in how people develop both their presentation of gender and the goals that they work towards. In Morgenroth’s study of role models and goal development they find that three key attributes, goal embodiment, attainability and desirability, are influential in how people develop role models, and that alignment between the aspirant and the model on these attributes influences the strength of learning and goal development that the aspirant obtains from the role model (Morgenroth et al., 2015). This development of goal adoption results in a further alignment of goal perception and goal adaption, which in turn creates a further and stronger alignment between the person developing goals and their role model, which can result in a cyclical nature of continuing the strength of alignment with a role models goals and attributes {{f}}, as is demonstrated in '''Figure 2'''. This goal development influences things such as academic pursuits, gender performance expectations and career choices, and choosing role models that align with one's own characteristics often results in people, especially children (Drury et al., 2011), choosing role models of the same gender. This results in children striving for careers and academic achievement similar to those that are already typical in society, meaning that young girls who don't see other women being successful in STEM fields, and young boys who dont see men being prevalent in humanities fields, are less likely to see these fields as desirable or find acquisition of skills in these fields as important to work towards. Much of the recent study on role models and their influence uses Bandura {{g}} early work on social learning as a foundation for the understanding of how we develop role models and how their actions act as a model for our own, and this development of role models is not determined by gender, but by many other factors, such as race and culture {{f}}. Guerzoni et al.’s (2024) study of the influence of both gender and ethnicity of role model development found that while ethnicity and abilities were important factors {{where}}, gender is a much stronger deciding variable. This is possibly because the study focused on Aboriginal and Torres Strait Islander children, who have fewer ethnically similar role models outside of their direct community, and ethnicity may prove to be a stronger factor in caucasian children's development of role models due to their increased prevalence in media and Australian society {{explain}} {{f}}. == Conclusion == It is clear that there are real and significant differences in how each gender expresses Achievement Motivation, but the consistent change in how and when those motivations are expressed, both through the individuals live and over the course of multiple generations of people, show that it is not the biological nature of gender that shapes achievement motivation, but the difference in socialisation that children experience that determines how achievement motivation is expressed. One can assume that if people were to grow up in a world where gender differences held less significance in the wider zeitgeist than these gender differences would be further reduced, and possibly even dissolve entirely. Males and females are already more alike than different on a variety of  psychological variables (Hyde, 2005), and theories like Atkinson's Expectancy-Value Theory and Morgenroth’s study of how we are shaped by role models show that if the communities surrounding people treated gender as a less defining trait, than gender would cease being a defining characteristic in how achievement motivation is expressed. == See also == * [[wikipedia:Gender|Gender]] (Wikipedia) * [[iarchive:introductiontomo0000john|An introduction to motivation]] (Atkinson, 1964) * [[Motivation and emotion/Book/2024/Gender differences in sexual motivation|Gender differences in sexual motivation]] (Book chapter, 2024) * [[Motivation and emotion/Book/2023/Grit and achievement|Grit and achievement]] (Book chapter, 2023) == References == {{Hanging indent|1= Atkinson, J. W. (1964). ''An introduction to motivation''. Van Nostrand Reinhold Company, Cop. Baer, J., & Kaufman, J. C. (2008). Gender differences in creativity. ''The Journal of Creative Behavior'', ''42''(2), 75–105. https://doi.org/10.1002/j.2162-6057.2008.tb01289.x Bandura, A. (1977). Self-efficacy: toward a unifying theory of behavioral change. ''Psychological Review'', ''84''(2), 191–215. https://doi.org/10.1037/0033-295X.84.2.191 Bandura, A. (1977). ''Social learning theory''. Prentice-Hall. Boggiano, A. K., Main, D. S., & Katz, P. (1991). Mastery motivation in boys and girls: The Role Of Intrinsic Versus Extrinsic Motivation. ''Sex Roles'', ''25''(9–10), 511–520. https://doi.org/10.1007/bf00290060 Damon, W., & Lerner, R. M. (2008). ''Child and adolescent development : an advanced course''. (pp. 406–413) John Wiley & Sons. De Brabander, C. J., & Martens, R. L. (2014). Towards a unified theory of task-specific motivation. ''Educational Research Review'', ''11'', 27–44. https://doi.org/10.1016/j.edurev.2013.11.001 Deci, E. L., & Ryan, R. M. (1987). The support of autonomy and the control of behavior. ''Journal of Personality and Social Psychology'', ''53''(6), 1024–1037. Drury, B. J., Siy, J. O., & Cheryan, S. (2011). When do female role models benefit women? The importance of differentiating recruitment from retention in stem. ''Psychological Inquiry'', ''22''(4), 265–269. https://doi.org/10.1080/1047840x.2011.620935 Dweck, C. S., & Gilliard, D. (1975). Expectancy statements as determinants of reactions to failure: Sex differences in persistence and expectancy change. ''Journal of Personality and Social Psychology'', ''32''(6), 1077–1084. https://doi.org/10.1037/0022-3514.32.6.1077 Eccles-Parsons JS, Adler TF, Futterman R, Goff SB, Kaczala CM, et al. (1983).. Expectancies, values, and academic behaviors. . In ''Achievement and Achievement Motivation'', ed. JT Spence , pp. 75–146. San Francisco:: W.H. Freeman Eccles, J., Wigfield, A., Harold, R. D., & Blumenfeld, P. (1993). Age and gender differences in children’s self- and task perceptions during elementary school. ''Child Development'', ''64''(3), 830. https://doi.org/10.2307/1131221 Farmer, H. S. (1987). A multivariate model for explaining gender differences in career and achievement motivation. ''Educational Researcher'', ''16''(2), 5–9. https://doi.org/10.3102/0013189x016002005 Findley, M. J., & Cooper, H. M. (1983). Locus of control and academic achievement: A literature review. ''Journal of Personality and Social Psychology,'' 44(2), 419–427. https://doi.org/10.1037/0022-3514.44.2.419 Frey, B. S. (1997). ''Not Just for the Money''. Edward Elgar Publishing. Frome, P. M., Alfeld, C. J., Eccles, J. S., & Barber, B. L. (2006). Why don’t they want a male-dominated job? An investigation of young women who changed their occupational aspirations. ''Educational Research and Evaluation'', ''12''(4), 359–372. https://doi.org/10.1080/13803610600765786 Guerzoni, M. A., Prehn, J., & Peacock, H. (2024). Mothers and sportsmen: The gendered and racialised nature of role model selection for Aboriginal and/or Torres Strait Islander youths. ''Australian Journal of Social Issues,'' ''59(''2), 281–296. https://doi.org/10.1002/ajs4.31 Hare, A. P., Koenigs, R. J., & Hare, S. E. (1997). Perceptions of observed and model values of male and female managers. ''Journal of Organizational Behavior,'' ''18''(5), 437–447. Hung, C., Ibrahim, S. T., & Katapally, T. R. (2025). Exploring ethical monitoring of physical activity behaviors among adults: A smart platform study operationalizing digital citizen science. ''PeerJ'', ''13'', e19793. https://doi.org/10.7717/peerj.19793 Hyde, J. S. (2005). The gender similarities hypothesis. ''The American Psychologist'', ''60''(6), 581–592. https://doi.org/10.1037/0003-066X.60.6.581 Jacobs, J. E., Lanza, S., Osgood, D. W., Eccles, J. S., & Wigfield, A. (2002). Changes in children’s self-competence and values: Gender and domain differences across grades one through twelve. ''Child Development'', ''73''(2), 509–527. https://doi.org/10.1111/1467-8624.00421 Kamdron, T. (2015). Work motivation: Relationships with job satisfaction, locus of control and motivation orientation. ''International Journal of Liberal Arts and Social Science'', ''3''(6), 125-148. Kim, H., & Kang, T. H. (2019). Effects of supervisor gender on promotability of female managers. ''Asia Pacific Journal of Human Resources, 58''(1), 85–106. https://doi.org/10.1111/1744-7941.12224 McClelland, D. C. (1965). Toward a theory of motive acquisition. ''American Psychologist, 20''(5), 321–333. https://doi.org/10.1037/h0022225 McClelland, D. C., & Burnham, D. H. (1995). Power is the great motivator. ''Harvard Business Review, 73'', 126-139. Moely, B. E., & Ilustre, V. (2016). Pre-college factors influencing college students’ civic attitudes: The importance of familial and community service experiences. ''International Journal of Research on Service-Learning and Community Engagement, 4''(1). doi.org/10.37333/001c.29625 Morgenroth, T., Ryan, M. K., & Peters, K. (2015). The motivational theory of role modeling: how role models influence role aspirants’ goals. ''Review of General Psychology, 19''(4), 465–483. doi.org/10.1037/gpr0000059 Pang, J. S. (2010). The achievement motive: A review of theory and assessment of achievement, hope of success, and fear of failure. ''Implicit motives, 1'', 30-71. Plante, I., O’Keefe, P. A., & Théorêt, M. (2012). The relation between achievement goal and expectancy-value theories in predicting achievement-related outcomes: A test of four theoretical conceptions. ''Motivation and Emotion, 37''(1), 65–78. https://doi.org/10.1007/s11031-012-9282-9 Rochat, M. J. (2022). Sex and gender differences in the development of empathy. ''Journal of Neuroscience Research, 101''(5). doi.org/10.1002/jnr.25009 Rotter, J. B. (1954). Social learning and clinical psychology. Prentice-Hall, Inc. https://doi.org/10.1037/10788-000 Ryan, R. M., & Deci, E. L. (2000). Intrinsic and extrinsic motivations: Classic Definitions and New Directions. ''Contemporary Educational Psychology, 25(''1), 54–67. Sherman, A. C., Higgs, G. E., & Williams, R. L. (1997). Gender differences in the locus of control construct. ''Psychology & Health, 12''(2), 239–248. https://doi.org/10.1080/08870449708407402 Singh, K. (2011). Study of achievement motivation in relation to academic achievement of students. ''International Journal of Educational Planning & Administration, 1''(2), 161-171. Skaalvik, E. M., & Rankin, R. J. (1994). Gender differences in mathematics and verbal achievement, self-perception and motivation. ''British Journal of Educational Psychology, 64''(3), 419–428. doi.org/10.1111/j.2044-8279.1994.tb01113.x Steinmayr, R., & Spinath, B. (2008). Sex differences in school achievement: Ahat are the roles of personality and achievement motivation? ''European Journal of Personality, 22''(3), 185–209. https://doi.org/10.1002/per.676 Ullah, R., & Ullah, H. (2019). Boys versus girls’ educational performance: Empirical evidences from global north and global south. ''African Educational Research Journal, 7''(4), 163–167. https://doi.org/10.30918/aerj.74.19.036 Weiner, B. (1985). An attributional theory of achievement motivation and emotion. ''Psychological Review, 92''(4), 548–573. doi.org/10.1037/0033-295x.92.4.548 Wigfield, A., & Eccles, J. S. (2000). Expectancy–value theory of achievement motivation. ''Contemporary Educational Psychology, 25(''1), 68–81. https://doi.org/10.1006/ceps.1999.1015 }} ==External links== * [https://www.mindtools.com/aznjntj/mcclellands-human-motivation-theory/ Simplified McClellans Motivation Theory] * [https://www.wgea.gov.au/publications/gender-segregation-in-australias-workforce Australian Workforce Statistics] * [https://www.youtube.com/watch?v=9hdSLiHaJz8&t=489s Motivation Explanation Simplified] [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Achievement motivation]] [[Category:Motivation and emotion/Book/Gender]] k9wj303mhntqgyyf0wy6dvhj1mabyes 0 331455 2842524 2839342 2026-10-25T05:04:54Z Jtneill 10242 Copyediting 2842524 wikitext text/x-wiki {{title|Game of dice task and decision-making:<br>What does the game of dice task reveal about risk-based decision-making?}} __TOC__ ==Overview== {{RoundBoxTop|theme=2}} [[File:Gaming dice multisided blue.jpg|right|thumb|200px|'''Figure 1'''. Dice have clearly defined odds to roll a specific number, but how Emile decides to bet changes his chances of winning or losing.]] ; Imagine this... Your friend Emile has money to spend on a night out and decides the two of you should go to a casino. At a dice table, a card in front of you lists the rules: * The dealer will roll one die with six faces. * Players can bet on one specific number and double their money, but they have to put in at least $50. * Players can also bet on four numbers at once, winning if the die lands on just one of the four. This also doubles their money, but they can only put in up to $5. The odds of winning are not shown, but they can be worked out (see Figure 1). For the first few rolls, Emile spends his money on cheap bets. After a couple of small wins, he switches to the single number bet. He loses. He goes for the single number again. He loses again. By the end of the night, despite clear odds the whole time, Emile had lost almost all of his money on the riskiest, least likely bets. ''Why do you think Emile made risky choices? How would you choose to bet?'' {{RoundBoxBottom}} Some choices carry calculable probabilities, like the chances of [[wikipedia:Side_effect|side-effects]] from medical treatments, the probability of developing health issues from [[wikipedia:Smoking|smoking]] or [[wikipedia:Alcohol_abuse|drinking]], or indeed, the odds of winning at a [[wikipedia:Gambling|gambling]] game (Brand et al., 2006). In theory, such choices can be calculated, guiding people towards [[wikipedia:Optimal_decision|optimal decisions]]. In reality, people often choose to make [[wikipedia:Risk|risky]] decisions even when they know their chances of success are low (Schiebener et al., 2014). If the odds are known, why do people still choose bets they are likely to lose? The Game of Dice Task (GDT) was designed to study exactly this kind of decision-making. By making the rules clear and the probabilities calculable, it removes ignorance as a reason for poor choices (Schiebener et al., 2014). This chapter examines theory and research on the GDT, which suggests that [[wikipedia:Executive_functions|executive functions]] (EFs) protect people from the [[wikipedia:Motivation|motivational]] pulls of risky [[Motivation and emotion/Book/2014/Theory of goal setting and task performance|goals]], rewards, and losses. {{RoundBoxTop|theme=4}} '''Focus questions''' * What is the Game of Dice Task? * How do self-regulation and executive functioning guide risk-based decisions? * What motivational influences pull people towards risky decisions and how do these relate to the Game of Dice Task? * What are the limits of the Game of Dice Task? * How can research on the Game of Dice Task be used to improve risk-based decision-making? {{RoundBoxBottom}} == The Game of Dice Task == The GDT is a computerised gambling [[wikipedia:Psychological_testing|task]] in which players try to maximise a [[wikipedia:Experimental_economics|fictional capital]] by betting on 18 rolls of a single die (Brand et al., 2005). It measures decision-making under risk, where an option can turn out worse than hoped for and the decision maker knows, or could know, how likely that is (Schiebener & Brand, 2015). This contrasts with the [[wikipedia:Iowa_gambling_task|Iowa Gambling Task]] (IGT), for instance, where players are not given clear odds and must learn which choices are advantageous as they play (Bechara & Damasio, 2005). Before each roll, players bet on one, two, three, or four numbers. If the number rolled is in a player's bet, they win the associated amount. Otherwise, they lose that same amount. These rules are explained at the start of the game and never change. Players complete the GDT on a single computer screen (see Figure 2). After a bet is selected, a die rolling video is played. Feedback is given by displaying the amount the player has just won or lost, adjusting the total [[wikipedia:Balance_(accounting)|balance]] number and chart, and playing an associated win or loss sound. {{f}} [[File:GDT_Schematic_Mockup.svg|center|thumb|800px|'''Figure 2'''. The typical presentation of the GDT, adapted from Brand et al. (2005). Clockwise from top left are: Die rolling video, most recent gain/loss with current total balance below, balance chart for a visual representation of current balance, round number, quit button, gains/losses associated with the possible bets, and possible bet choices.]] The one and two number bets are risky and disadvantageous, since they win less than half the time, while the three and four number bets are advantageous, since they win half the time or more (Brand et al., 2005). Riskier bets win or lose larger amounts (see Table 1). The probabilities are not stated, but can be calculated from the rules. Overall score is not a player's final balance, but the number of advantageous bets minus the number of disadvantageous bets, ranging from -18 to +18. This makes results comparable regardless of luck. <div style="display: flex; justify-content: center;"> {| class="wikitable" |+'''Table 1'''. The four bets in the GDT and their associated odds. Expected value calculated using the amounts in Brand et al. (2005). !Bet !Chance of winning !Amount lost/gained !Expected value !Category |- |1 number |1 in 6 (16.67%) |$1,000 | - 667 |Disadvantageous, most risky |- |2 numbers |2 in 6 (33.33%) |$500 | - 167 |Disadvantageous, risky |- |3 numbers |3 in 6 (50%) |$200 |0 |Advantageous, less risky |- |4 numbers |4 in 6 (66.67%) |$100 | + 33 |Advantageous, least risky |} </div> == Self-regulation and executive functions == The GDT is primarily intended to measure how EFs impact risk-based decision-making. Hofmann et al. (2012) describe three EFs that are core to maintaining control over behaviour, or [[wikipedia:Self-regulation_theory|self-regulating]]: updating, which holds and refreshes information in [[wikipedia:Working_memory|working memory]]; [[wikipedia:Inhibitory_control|inhibition]], which overrides automatic responses; and [[wikipedia:Cognitive_flexibility|shifting]], which moves attention between tasks and rules. Self-regulation requires goals and the executive capacity to pursue them. This [[Motivation and emotion/Book/2019/Executive functioning and emotion regulation|executive capacity]] may be weakened by stress or distractions, leading behaviour to follow impulses rather than rationality (Hofmann et al., 2012). === From somatic markers to a dual-process model === Assuming that people will always make the best choice when the odds are known or calculable is assuming that they have "unlimited knowledge, time, and information-processing power" (Bechara & Damasio, 2005, p. 336). The [[wikipedia:Somatic_marker_hypothesis|somatic marker hypothesis]] proposes that automatic bodily signals like a quickening heart rate guide decisions before they are reasoned through, particularly when the odds are hidden (Bechara & Damasio, 2005). Brand et al. (2006) proposed that under explicit risk, EFs hold payoffs and probabilities in working memory, categorise the options by risk, and form a strategy. Somatic markers may be helpful, but are supposedly unnecessary {{explain}}. [[File:Schiebener and Brand's (2015) risk decision model, simplified.svg|thumb|'''Figure 3'''. A simplified version of the dual-process model of decision-making under risk, in which feedback from each decision can change whether the impulsive or reflective system leads. Adapted from Schiebener and Brand (2015).]] Schiebener and Brand (2015) later revised this into a [[wikipedia:Dual_process_theory|dual-process]] model in which an impulsive system of emotions and automatic responses runs alongside a reflective system of EFs and reasoning, with feedback changing which leads (see Figure 3). The model was developed as a [[wikipedia:Post_hoc|post hoc]] explanation of how choices in explicit gambling tasks like the GDT happen, limiting its predictive validity. === Evidence for EFs in the GDT === Higher GDT performance has been found in healthy adults who are better at inhibition and shifting, although all the studies reviewed were conducted in Germany (Colautti et al., 2022). People with [[wikipedia:Parkinson's_disease|Parkinson's disease]] made riskier choices than controls, with self-regulatory EFs linked to GDT performance in two out of three studies (Colautti et al., 2023). Other groups with executive impairments, such as [[wikipedia:Korsakoff_syndrome|Korsakoff syndrome]] and preadolescent [[wikipedia:Attention_deficit_hyperactivity_disorder|ADHD]], also make more risky choices despite normal [[wikipedia:Intelligence_quotient|IQ]] (Schiebener & Brand, 2015). Among 152 adults, general executive control was the only direct predictor of performance, but other EFs contributed through it (Schiebener et al., 2014). In a study where 30 participants played the GDT with no feedback elements, they chose the riskiest bet more often and switched between risky and safe bets about twice as often as in the standard GDT (Brand, 2008). A purely calculating player would bet the same way in both versions, so players appear to use feedback to check and settle on a strategy. Further GDT research suggests this depends on EFs, though this may be because the GDT was originally designed to examine EFs. People with Korsakoff syndrome played equally badly with or without feedback, and after losing on a risky bet, only a third switched to a safer bet, compared with two thirds of healthy players (Brand et al., 2009). Impairing EF also affects healthy players, where those who completed a [[wikipedia:N-back|2-back]] working memory task while playing the GDT made more risky choices than those doing a 1-back task or just the GDT (Starcke et al., 2011). Conversely, calculating the odds before playing was correlated with safer bets (Pertl et al., 2017), possibly since options and strategies could be evaluated outside of feedback influences. Self-regulation therefore appears necessary for advantageous play. == Motivational pulls to risk in the GDT == If self-regulation is necessary to hold back impulses, why are risky choices creating these motivations in the first place? Three theories can help explain what pulls players towards risky bets: the feelings associated with reward, sensitivity to rewards and punishments, and reference points and goals used to judge outcomes. === Risk-as-feelings === If people only weighed the outcomes by which was more likely, players of the GDT would only pick the four number bet. The [[Motivation and emotion/Book/2022/Risk-as-feelings|risk-as-feelings]] hypothesis {{f}} offers an explanation for why rewards make risky choices appealing. When outcomes carry strong feelings, people are more likely to compare the outcomes alone and disregard their probabilities (Pachur et al., 2014). The $1,000 reward {{explain}} in the GDT may therefore feel appealing to people who are excited by the large payout, no matter its odds, aligning with Brand et al.’s (2006) suggestion that pathological gamblers found the large payouts of risky bets too appealing to resist. However, because larger payouts in the GDT come with low odds, no study has directly tested the pull of reward size in the task. === Reinforcement sensitivity === [[wikipedia:Reinforcement_sensitivity_theory|Reinforcement sensitivity theory]] proposes that the behavioural approach system (BAS), activated by signs of rewards, produces desire and excitement. The behavioural inhibition system (BIS), activated by conflict within the BAS, produces caution and risk assessment (Corr et al., 2013). Bets in the GDT could be seen as creating conflict, since they have a prize to approach but an equal loss to avoid. People with a strong BAS should be drawn to the risky bets, especially after wins, while a strong BIS should make them pick the safer bets. Unlike Schiebener and Brand's (2015) impulsive-reflective model, reinforcement sensitivity theory describes [[wikipedia:Personality|personality]] components, and the BIS detects conflict and produces anxiety automatically rather than disrupting impulsive behaviours as a result of EFs (Corr et al., 2013). In a sample of 83 students who played a modified GDT that produced winning or losing streaks, the 20 participants high in reward sensitivity and low in punishment sensitivity made riskier choices after a winning streak, while other participants became safer. The 19 participants with low reward and high punishment sensitivity did the opposite, making safer choices during a losing streak, while other participants became riskier (Kim & Lee, 2011). This aligns with the predictions that a high BAS draws people's motivations to rewards and away from probability. However, the effects and sample were small, and were contradicted in a sample of 1,002 older adults, where higher reward sensitivity predicted less risky betting in the GDT (Sinclair et al., 2021). Common questionnaires such as the BIS/BAS scales may not be reliable at separating these systems (Corr et al., 2013) and Sinclair et al. (2021) suggest reward sensitivity may appear in the GDT as exploring options and seeking feedback rather than going after rewards. Links between personality, motivation and explicit gambling tasks are rare and inconsistent, with only some impulsive traits showing associations (Schiebener & Brand, 2015). === Prospect theory === [[wikipedia:Prospect_theory|Prospect theory]] suggests that people judge outcomes as gains or losses compared with a [[wikipedia:Reference_dependence|reference point]], rather than by their final total {{f}}. It also proposes that people [[Motivation and emotion/Book/2018/Loss aversion|loss averse]], feeling loss more intensely than gain, and [[wikipedia:Framing_effect_(psychology)|framing effects]] influence people to change their behaviours based on how a problem is presented (Tversky & Kahneman, 1981). Pabst et al. (2013) tested this with gain-only and loss-only versions of the GDT. In gain-only, players started at zero and rolls either won money or nothing, whereas in loss-only players started at $1,000 and either lost money or nothing. Betting on one number had the lowest chance of winning in both, but in loss-only, the single number had the best expected value (see Table 2). Participants facing losses made more disadvantageous choices than those facing gains, suggesting they were focused on avoiding loss instead of calculating the values. Each group's starting reference points may have also changed their betting patterns, since any loss was falling short of the reference point, while any gain was exceeding it. This also aligns with risk-as-feelings, where the more emotionally impactful effects of the loss led to disregarding the probabilities (Pachur et al., 2014). <div style="display: flex; justify-content: center;"> {| class="wikitable" |+'''Table 2'''. The odds and values in the gain-only (+) and loss-only (-) GDTs (Pabst et al., 2013). !Bet !Chance of winning / losing !Amount gained / lost !Expected value (+ and -) !Category |- |1 number | + 1 in 6 (16.67%) - 5 in 6 (83.33%) | + $600 - $120 |$100 | + Disadvantageous - Advantageous |- |2 numbers | + 2 in 6 (33.33%) - 4 in 6 (66.67%) | + $400 - $200 |$133 | + Disadvantageous - Advantageous |- |3 numbers | + 3 in 6 (50%) - 3 in 6 (50%) | + $333 - $333 |$166 | + Advantageous - Disadvantageous |- |4 numbers | + 4 in 6 (66.67%) - 2 in 6 (33.34%) | + $300 - $600 |$200 | + Advantageous - Disadvantageous |} </div> Goals may also act as reference points, where outcomes below a goal feel like losses. In the GDT, showing players a fake leaderboard increased risky betting despite heavy losses (Schiebener et al., 2012). The leaderboard showed scores up to $14,500, far above the real best of $3,500. Players who set higher goals made more single-number bets since they were the only way to approach such scores. Setting goals against the leaderboard reduced risk in players with strong EFs and increased it in those with weaker EFs, possibly because stronger EFs help revise unrealistic goals. Falling behind may also explain why [[Motivation and emotion/Book/2018/Gambling addiction motivation|pathological gamblers]] become riskier late in the game, when the riskiest bets were the only way to make up for earlier losses (Brand et al., 2006). {{RoundBoxTop}} '''Check your understanding''' <quiz display=simple> {The Game of Dice Task measures decision-making under risk, which is calculable probability. |type="()"} + True - False {According to Schiebener and Brand's (2015) dual-process model, what helps players avoid risky bets? |type="()"} - Somatic markers signalling danger + The reflective system, which relies on executive functions to hold and apply a strategy - Reward sensitivity {Whether people receive feedback has nothing to do with how they make decisions while playing the GDT. |type="()"} - True + False {Why did players make more single-number bets after seeing a leaderboard with very high scores? |type="()"} - The leaderboard explained the odds of each bet - High scores reduced their motivation to win + The scores acted as a reference point, so falling short felt like a loss that only risky bets could make up </quiz> {{RoundBoxBottom}} == Limits of the Game of Dice Task == Several limits apply to the GDT. The GDT was designed primarily as a measure of EFs under explicit risk, so there is little research directly linking it to risk-based decision-making and motivation. Within its design, riskier bets in the GDT always have worse average values, so the task cannot separate a player who seeks risk from one who has not recognised the advantageous bets. The standard GDT also uses fictional money, which may weaken the reward and punishment signals the motivational theories rely on and encourage calculation (Pachur et al., 2014), so the task may understate how feelings drive real gambles. Further to this, the rolls are [[wikipedia:Pseudorandomness|pseudorandomised]] so that each number appears exactly three times (Brand et al., 2005). Once a number is rolled, its chance of appearing next drops from 1 in 6 to 2 in 17, so the [[wikipedia:Gambler's_fallacy|gambler's fallacy]] becomes an advantageous strategy in later trials. Finally, most evidence comes from small, mostly German samples (Colautti et al., 2022), and the task predicts real behaviour poorly. Among 226 adolescent gamblers, GDT scores did not distinguish the problem gamblers from others (Donati et al., 2019). == Improving risk-based decisions == Despite these limitations, GDT research suggests several strategies for decisions where the odds are known, such as managing money, choosing medical treatments, or gambling. Working out the odds before deciding, and deciding away from distractions, reduces the load on self-regulation (Pertl et al., 2017; Starcke et al., 2011). Setting realistic goals rather than comparing against other people's results or earlier losses may reduce the drive for achievement that makes rewarding risks appealing (Schiebener et al., 2012). These findings may be relevant for people with conditions that affect EFs, such as ADHD or Parkinson's disease (Colautti et al., 2023; Schiebener & Brand, 2015). Because several groups who struggled with the GDT had normal general intelligence, risky choices may reflect difficulty holding a strategy against the pull of rewards and losses, rather than a lack of intelligence or a personal failing. Identifying choices and thinking through their possible outcomes before a decision may help. == Conclusion == The GDT was designed to test whether self-regulation guides decisions once the odds are known. Reviews and experiments suggest that it does, since people with stronger EFs make fewer risky bets, and loading working memory makes choices riskier. However, there is a tension between the reflective system of cognition and the impulsive system of feelings described in Schiebener and Brand's (2015) model. The effects of EF, framing, and goals have the strongest evidence in the GDT, while reward sensitivity evidence is conflicting. The main takeaway from the GDT is that people are not purely probabilistic decision-makers. Even when the odds are knowable, risk-based decisions depend on people's goals, their past experiences, their responses to feedback, and their capacity to maintain an effective strategy. Recognising these pulls can help {{missing}} ==See also== * [[Motivation and emotion/Book/2026/Adolescent risk-taking and reward-system development|Adolescent risk-taking and reward-system development]] (Book chapter, 2026) * [[wikipedia:Decision-making|Decision-making]] (Wikipedia) * [[Motivation and emotion/Book/2024/Dopamine and decision making|Dopamine and decision making]] (Book chapter, 2024) * [[Motivation and emotion/Book/2018/Gambling addiction motivation|Gambling addiction motivation]] (Book chapter, 2018) * [[Motivation and emotion/Book/2026/Impulsivity versus sensation-seeking|Impulsivity versus sensation-seeking]] (Book chapter, 2026) * [[Motivation and emotion/Book/2018/Loss aversion|Loss aversion]] (Book chapter, 2018) * [[wikipedia:Risk|Risk]] (Wikipedia) ==References== {{Hanging indent|1= Bechara, A., & Damasio, A. R. (2005). The somatic marker hypothesis: A neural theory of economic decision. ''Games and Economic Behavior'', ''52''(2), 336–372. https://doi.org/10.1016/j.geb.2004.06.010 Brand, M. (2008). Does the feedback from previous trials influence current decisions? A study on the role of feedback processing in making decisions under explicit risk conditions. ''Journal of Neuropsychology'', ''2''(2), 431–443. https://doi.org/10.1348/174866407X220607 Brand, M., Fujiwara, E., Borsutzky, S., Kalbe, E., Kessler, J., & Markowitsch, H. J. (2005). Decision-making deficits of Korsakoff patients in a new gambling task with explicit rules: Associations with executive functions. ''Neuropsychology'', ''19''(3), 267–277. https://doi.org/10.1037/0894-4105.19.3.267 Brand, M., Labudda, K., & Markowitsch, H. J. (2006). Neuropsychological correlates of decision-making in ambiguous and risky situations. ''Neural Networks'', ''19''(8), 1266–1276. https://doi.org/10.1016/j.neunet.2006.03.001 Brand, M., Pawlikowski, M., Labudda, K., Laier, C., von Rothkirch, N., & Markowitsch, H. J. (2009). Do amnesic patients with Korsakoff’s syndrome use feedback when making decisions under risky conditions? An experimental investigation with the Game of Dice Task with and without feedback. ''Brain and Cognition'', ''69''(2), 279–290. https://doi.org/10.1016/j.bandc.2008.08.003 Colautti, L., Antonietti, A., & Iannello, P. (2022). Executive functions in decision making under ambiguity and risk in healthy adults: A scoping review adopting the hot and cold executive functions perspective. ''Brain Sciences'', ''12''(10). https://doi.org/10.3390/brainsci12101335 Colautti, L., Iannello, P., Silveri, M. C., & Antonietti, A. (2023). Decision-making under ambiguity and risk and executive functions in Parkinson’s disease patients: A scoping review of the studies investigating the Iowa Gambling Task and the Game of Dice. ''Cognitive, Affective, & Behavioral Neuroscience'', ''23''(5), 1225–1243. https://doi.org/10.3758/s13415-023-01106-3 Corr, P. J., DeYoung, C. G., & McNaughton, N. (2013). Motivation and personality: A neuropsychological perspective. ''Social and Personality Psychology Compass'', ''7''(3), 158–175. https://doi.org/10.1111/spc3.12016 Donati, M. A., Frosini, A., Izzo, V. A., & Primi, C. (2019). The effectiveness of the Game of Dice Task in predicting at-risk and problem gambling among adolescents: The contribution of the neural networks. ''Journal of Gambling Studies'', ''35''(1), 1–14. https://doi.org/10.1007/s10899-018-9796-5 Hofmann, W., Schmeichel, B. J., & Baddeley, A. D. (2012). Executive functions and self-regulation. ''Trends in Cognitive Sciences'', ''16''(3), 174–180. https://doi.org/10.1016/j.tics.2012.01.006 Kim, D-Y., & Lee, J-H. (2011). Effects of the BAS and BIS on decision-making in a gambling task. ''Personality and Individual Differences'', ''50''(7), 1131–1135. https://doi.org/10.1016/j.paid.2011.01.041 Pabst, S., Brand, M., & Wolf, O. T. (2013). Stress effects on framed decisions: There are differences for gains and losses. ''Frontiers in Behavioral Neuroscience'', ''7'', Article 142. https://doi.org/10.3389/fnbeh.2013.00142 Pachur, T., Hertwig, R., & Wolkewitz, R. (2014). The affect gap in risky choice: Affect-rich outcomes attenuate attention to probability information. ''Decision'', ''1''(1), 64–78. https://doi.org/10.1037/dec0000006 Pertl, M-T., Zamarian, L., & Delazer, M. (2017). Reasoning and mathematical skills contribute to normatively superior decision making under risk: Evidence from the game of dice task. ''Cognitive Processing'', ''18''(3), 249–260. https://doi.org/10.1007/s10339-017-0813-x Schiebener, J., & Brand, M. (2015). Decision making under objective risk conditions—a review of cognitive and emotional correlates, strategies, feedback processing, and external influences. ''Neuropsychology Review'', ''25''(2), 171–198. https://doi.org/10.1007/s11065-015-9285-x Schiebener, J., Wegmann, E., Gathmann, B., Laier, C., Pawlikowski, M., & Brand, M. (2014). Among three different executive functions, general executive control ability is a key predictor of decision making under objective risk. ''Frontiers in Psychology'', ''5'', Article 1386. https://doi.org/10.3389/fpsyg.2014.01386 Schiebener, J., Wegmann, E., Pawlikowski, M., & Brand, M. (2012). Anchor effects in decision making can be reduced by the interaction between goal monitoring and the level of the decision maker’s executive functions. ''Cognitive Processing'', ''13''(4), 321–332. https://doi.org/10.1007/s10339-012-0522-4 Sinclair, C., Eramudugolla, R., Brady, B., Cherbuin, N., & Anstey, K. J. (2021). The role of cognition and reinforcement sensitivity in older adult decision-making under explicit risk conditions. J''ournal of Clinical and Experimental Neuropsychology'', ''43''(3), 238–254. https://doi.org/10.1080/13803395.2021.1909709 Starcke, K., Pawlikowski, M., Wolf, O. T., Altstötter-Gleich, C., & Brand, M. (2011). Decision-making under risk conditions is susceptible to interference by a secondary executive task. ''Cognitive Processing'', ''12''(2), 177–182. https://doi.org/10.1007/s10339-010-0387-3 Tversky, A., & Kahneman, D. (1981). The framing of decisions and the psychology of choice. ''Science'', ''211''(4481), 453–458. https://doi.org/10.1126/science.7455683 }} ==External links== * [https://www.psychologytoday.com/au/blog/hovercraft-full-eels/202012/risky-gamble-or-uncertain-future A risky gamble or an uncertain future?] (Psychology Today) * [https://www.gamcare.org.uk/self-help/links-to-other-support-agencies/international-support-contacts/ International Support Contacts] (GamCare) * [https://www.millisecond.com/library/gameofdicetask Game of dice task] (Millisecond Software) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Decision making]] [[Category:Motivation and emotion/Book/Personal control]] jzrqm73n8d0la0hhlqi4cl370h6h8ko 1 331559 2842525 2837092 2026-10-25T05:05:39Z Jtneill 10242 Book chapter feedback 2842525 wikitext text/x-wiki == Feedback and Suggestions == Hello! I really enjoyed reading what you have so far. I think the opening scenario makes the Game of Dice Task easy to understand and gives a clear example as to why people may take risky decisions even when the odds were obvious. I think it would be interesting to add a learning feature such as a questionnaire that asks the reader what decision they'd make in a similar situation as your scenario described. I think it would make the page more interactive and allow the reader to have a GDT experience of their own. Just a suggestion! Thanks! - Amy :) [[User:Amyuniversity|Amyuniversity]] ([[User talk:Amyuniversity|discuss]] • [[Special:Contributions/Amyuniversity|contribs]]) 16:33, 26 August 2026 (UTC) <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]] |2= <!-- Headings --> # See earlier comment about [[#heading casing|heading casing]] <!-- Heading structure --> # Basic [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] – could benefit from further development (expand) <!-- 1-level --> # Promising [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development (e.g., consider using subheadings) <!-- Alignment with focus questions --> # Reasonably good alignment between focus questions and heading structure, but aim for closer alignment |3= <!-- Overview--> <!-- Scenario --> # A scenario or case study is presented in a feature box with an image at the start of this section <!-- Description --> # A promising description of the problem/topic is planned or presented <!-- Focus questions --> # Develop closer alignment between the sub-title, focus questions, and top-level headings |4= <!-- Key points--> <!-- Overall --> # Promising development # Connect to motivation # Highlight the most relevant theories and synthesise the best research on the topic # Do these key points include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. <!-- Conclusion --> # Conclusion is underway # What are the practical, take-home messages? (address the focus questions) |5= <!-- Figure --> # Relevant figure(s) presented, captioned, and cited <!-- Learning feature --> <!-- Interwiki links ---> # Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Scenarios/examples/case studies --> # Consider use of more scenarios/examples/case studies <!-- Quiz --> # Promising use of quiz question(s) <!-- Tables --> # Also consider using tables to summarise key information |7= <!-- References --> <!-- Overall --> # Excellent <!-- Systematic reviews --> # What are the most relevant systematic reviews/meta-analyses about this topic? <!-- Citations --> # Only include references which have been accessed and read |8= <!-- Resources --> <!-- See also --> # See also ## Excellent <!-- External links --> # External links ## Excellent |9= <!-- User page --> # Used effectively <!-- Description about self --> # Description about self provided <!-- Links to profile(s) --> # Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks. <!-- Link to book chapter --> # A link to the book chapter is provided |10= <!-- Social contribution --> # Excellent – at least three different types of contributions with direct link(s) to evidence }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 08:44, 31 August 2026 (UTC) == Recommendations: Game of dice task == Hello, To start, I'd like to highlight how interesting your overview was and how visually appealing the mass of your content is, which made it really easy to stay engaged and carry on reading. Well done. My main point is, as someone who doesn't really know much about the Game of Dice task, I was a bit confused by the explanation. As your topic is centering on a very specific task to relate to decision making, I think it would strengthen your book chapter to clearly clarify how the game is played in terms that new students or readers can find easy to understand. Or linking specific terms (such as fictive capital) to wikipages to increase accessability for readers who may not have as much context in the field as you do. Making sure they understand the rules, objectives and features of the game will really contribute to strengthening the discussion you have after. Good work [[User:Monuc9|Monuc9]] ([[User talk:Monuc9|discuss]] • [[Special:Contributions/Monuc9|contribs]]) 02:17, 9 October 2026 (UTC) <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is an excellent chapter that successfully uses psychological theory and research to address a practical, real-world phenomenon or problem <!-- Overall - Edit summaries --> #* Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits <!-- Overall – Citations --> # Excellent use of academic, peer-reviewed citations to support claims <!-- Overall – Copyedits --> # For additional feedback, see the following comments and [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FGame_of_dice_task_and_decision-making&diff=2842524&oldid=2839342 these copyedits] |2= <!-- Overview – Comments... --> # Excellent <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box <!-- Overview – Explains problem --> # Clearly explains the psychological problem or phenomenon <!-- Overview – Focus questions --> # The focus questions are excellent (clear and relevant) # The focus questions could be improved by: ## being [[w:Open-ended question|open-ended]] rather than [[w:Closed-ended question|closed-ended]] (fixed) |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Excellent—key theories are very well explained and applied <!-- Theory – Builds on --> # Builds exceptionally well on [[w:|Wikipedia]] articles and related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Insightfu depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Excellent use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # Key citations are well used <!-- Theory – Examples --> # Excellent use of examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Excellent/ review of relevant research # What are the most relevant systematic reviews and/or meta-analyses? # Claims are well referenced <!-- Research – Critical thinking --> # Excellent[[w:Critical thinking|critical thinking]] about relevant research is evident |5= <!-- Integration comments... --> # Excellent integration between the most relevant theory(ies) and the best research |6= <!-- Conclusion comments... --> # Excellent summary and conclusion # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are well summarised <!-- Conclusion – Focus questions --> # The focus questions are addressed <!-- Conclusion – Take-home messages --> # Clear take-home message(s) |7= <!-- Written expression – Style comments... --> <!-- Written expression – Written expression --> # Written expression ## The quality of written expression is excellent <!-- Written expression – Layout --> # Layout ## The chapter is well structured, with major sections using sub-sections ## Make consistent use of abbreviations in the headings (GDT) <!-- Written expression – Grammar --> # Grammar and spelling are excellent ## Use past tense when describing research studies, although implications of findings could be in the present tense <!-- Written expression – APA style --> # APA style ## Use [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies sentence casing (not capitals) for disorders, therapies, theories, etc.] <!-- Written expression – Figures --> ## Figures ### Very well captioned <!-- Written expression – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style <!-- Written expression – Tables --> ## Tables ### Very well ### Use an APA style caption for each table ([[Motivation and emotion/Assessment/Chapter/Tables|see example]]) <!-- Written expression – Tables - Citations --> ### Each Table is referred to at least once within the main text using APA style <!-- Written expression – Citations --> # Citations use excellent [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] <!-- Written expression – References --> ## References use excellent APA style: |8= <!-- Learning features comments... --> # Excellent use of learning features <!-- Learning features – Wikipedia embedded links --> # Excellent use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles <!-- Learning features – Wikiversity embedded links --> # Very good use of embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Excellent use of figure(s) # Excellent use of table(s) # Reasonably good use of feature box(es) # Excellent use of scenarios, case studies, or examples <!-- Learning features – Quizzes --> # Excellent use of quiz(zes) and/or reflection question(s) <!-- Learning features – See also --> # Excellent use of the [[{{PAGENAME}}#See also|See also]] section <!-- Learning features – External links --> # Excellent use of the [[{{PAGENAME}}#External links|External links]] section |9= <!-- Social contribution comments... --> # ~7 logged, useful, mostly major contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] # Thanks for the Wiki Commons uploads }} dgasoma8855dgsyj8m3zipq529v3ij4 0 331569 2842515 2842181 2026-10-25T02:20:49Z Amirrorslens 3106278 brought in relevant top line headings, realigned focus questions, organised subheadings into table in attempt to organise information and incorporate learning features 2842515 wikitext text/x-wiki {{title|Awe and the diminished self:<br>How does awe diminish the self and how can this be applied?}} __TOC__ ==Overview== {{RoundBoxTop|theme=2}} [[File:NASA-HS201427a-HubbleUltraDeepField2014-20140603.jpg|right|thumb|200px|'''Figure 1'''. NASA Hubble Ultra Deep Field]] ; Scenario Leo walks into the desert late at night and is taken aback at the night sky above him. Being alone in the vast and flat plane lets him really perceive the vastness of the stars and leaves him feeling very present and aware of his smallness in comparison. This overwhelming feeling leaves Leo seeking connection with others, and a new found sense of purpose. * [Author Note: REPLACE WITH A REAL CASE STUDY - such as near death experience (The Good Place coded) (Might modify scenario to incude this aspect similar to the purple feature box on [[Motivation and emotion/Book/2025/Motivation for using AI companions|this page]])] {{RoundBoxBottom}} '''''Author Note: KEY PSYCH SCIENCE''''' {{ic|Move this detail into subsequent sections}} * ''Maslow and Carl Rogers -> self-concept and peak experiences'' * ''KEY PIECE OF LITERATURE: Keltner and Haidt: Awe 2003'' *''Jean Piaget - Cognitive Development'' * Awe is considered a [[wikipedia:Peak_experience|peak experience]] according to Maslow [x] {{ic|Also consider awe as an emotion}}, (see [[Motivation and emotion/Book/2017/Awe and well-being|Awe and well-being]]) ** Experiencing awe = indicator of reaching highest level of self-actualisation and self-transcendence) * Awe can have positive individual and community effects (pro-social behaviour) {{ic|The topic is not just about awe - it is about the effect of awe on self-perception}} {{RoundBoxTop|theme=3}} '''Focus questions''' * What is awe as an emotion? > Heading: Awe as an emotion * What is humanistic psychology and theory of needs? > Heading: Humanistic Psychology, Maslow Theory of Needs, Self Actualisation (Cut Self) * What is the role of awe in Maslow's peak experiences? * How can we apply awe as peak experiences for individual good? * > Heading: Awe as peak experiences * How does the experience of awe diminish the self? * What is the effect of a diminished sense of self? * How can awe be integrated into modern life? i.e. How can we trigger or experience everyday moments of awe?{{RoundBoxBottom}} {{RoundBoxTop|theme=1}} '''QUESTIONS AND NOTES FOR SELF''' * Where do I fit discussions of the Self in here? * Should I get rid of the physical and conceptual distinction for the headings? Rework to the five appraisals from that one article - [Author's Notes] {{RoundBoxBottom}} == Awe as an emotion == [[File:Plutchik-wheel.svg|thumb|Figure 2: Plutchik's wheel of emotions, where awe is modelled as a combination of suprise and fear.]] [[wikipedia:Awe|Awe]] is an emotion generally described as [insert synonyms]. In Robert Plutchik's wheel of emotion as seen in Figure 2, awe is modelled as a combination of suprise and fear (see [[Motivation and emotion/Book/2016/Awe as an emotion|Awe as an emotion]]). ''- KEY REFERENCE: Keltner and Haidt, 2003:'' * ''Additional appraisals account for variation in the hedoic (pleasure) tone of awe experiences'' ** ''threat, beauty, exceptional ability, virtue, and the supernatural'' * ''Edwards et al. (2026) define awe to be an emotion elicited from experiencing great vastness that dramaticallt expands a person's frame of reference.'' ** ''see [[Motivation and emotion/Book/2016/Awe as an emotion|Awe as an emotion]]'' In clear cases, two appraisals are central to the experience of awe as an emotion; perceived vastness, and a following need for accommodation (Keltner and Haidt, 2003). ==== Vastness ==== Obscurity is another propertry that produces the 'sublime experience' (Burke, 1990, as cited in Keltner and Hiadt., 2014). * "Although the term “vastness” implies great physical size, in this usage “vast” describes any stimulus that challenges one's accustomed frame of reference in some dimension ** A stimulus may convey vastness in physical space, in time, in number, in complexity of detail, in ability, even in volume of human experience. *** For example, one may experience a sense of vastness in a mathematical equation, not because the equation is literally long, but because of the vast number of observed physical processes it is able to explain and predict. ** What is critical is that the stimulus dramatically expands the observer's usual frame of reference in some dimension or domain." (Shiota et al., 2007) ===== Physical vastness ===== ''[Author's Note: Nature, Space, might be wise to mention depth perception studies here. Why particuarly these things. Work in thesis on space video here.]'' * The overview effect = witnessing Earth from the atmosphere = life-altering degree of awe -> transcendent -> complete breakdown of perception of borders and countries = feeling of connectedness (Yaden et al., 2016) * Eclipses: study that used twitter data (over 2 million users) -> individuals exhibited more awe and expressed less self focused and more pro social, affiliative, humble and collective language * Witnessing natural disasters may induce a state of negatively valenced awe (Chaudhury et al., 2022; Piff et al., 2015) ** Link to heading section 'Effect of awe on the self' -> threat-based awe (Keltner and Haidt, 2003). ===== Conceptual vastness ===== Witnessing other people or learning things, coming of age moments, witnessing things for the first time theories here. Music and virtues act of skills. * Acts of Virtue * Acts of skills * Learning things ** A stimulus may convey vastness in physical space, in time, in number, in complexity of detail, in ability, even in volume of human experience. *** For example, one may experience a sense of vastness in a mathematical equation, not because the equation is literally long, but because of the vast number of observed physical processes it is able to explain and predict. ** What is critical is that the stimulus dramatically expands the observer's usual frame of reference in some dimension or domain." (Shiota et al., 2007) ==== Psychedelics ==== ''[Author's Note: I feel psychedelics needs its own subsection here as it is particuarly relevant to my applications, and provides a good segue to need for accommodations.]'' * psychedielic-occassioned mystical experience is characterised by profound awe ** small self -> construct that in the extreme is analogous to those of unitive experience and ego dissolution (diminishing of self) (Hendricks, 2018) * psychedelics have been shown to produce enduring positive change across a range of psychological and behavioural outcomes (Hendricks, 2018) via ''(Griffiths et al., 2018; Griffiths, Richards, Johnson, McCann, & Jesse, 2008, Griffiths et al., 2011; Griffiths, Richards, McCann, & Jesse, 2006; Lebedev et al., 2016; MacLean, Johnson, & Griffiths, 2011)'' as well as treatment of depression. == Need for accommodation == * Awe is defined by Keltner and Haidt as an inability to assimilate an experience into current mental structures. (Keltner and Haidt, 2003) * Accommodation is one of the two ongoing processes (assimilation being the other) in [[wikipedia:Piaget's_theory_of_cognitive_development|Piaget's theory of cognitive development]], where existing [[wikipedia:Schema_(psychology)|schemas]] are modified, or even created to deal with a new event or experience (Jean Piaget, {find specific reference}). ** See also: [[wikipedia:Schema_(psychology)|Schemas]] * ''[Authors Note: Insert table of epistimec emotions such as wonder, elevation, sublime, admiration (Keltner and Haidt), not sure of relevance or format]'' ''- [Authors Note: unsure whether I should spend a sub section on schemas or not, or how much explanation should be done for these sorts of fundamental but still unfamiliar to reader concepts'' - Discussion of what it means to require the need for knowledge structures to be updated. == The role of awe in Maslow's peak experiences == - {| class="wikitable" |+ !Features of peak experience !Example of study |- |disorientation in time and space | |- |ego transcendece & self-forgetfulness |ego death |- |self-forgetfulness |meditation? |- |perception that the world is good, beautiful and desirable |is there a study that supports this resulting notion, |- |feeling passive, receptive and humble |non-attachment?, radical acceptable therapy, meditation/mindfulness |- |sense that polarities and dichotomies have been transcended or resolved |no borders or boundaries, the out of orbit // observer effect |- |feelings of being lucky, fortunate, or graced |gratitude practice |} ====== <s>Self</s> ====== * <s>Self in philosophy = individual's own being, knowledge and values (and relationship between these attributes ##CUT##</s> ** <s>self is distingushable from others</s> *** <s>[Authors Note: have a section later in connectedness how when the self is diminished, the concept of otherness is also diminished -> oneness]</s> === Self-perception === [[wikipedia:Self-concept|Self-concept]] = made up of one's self-schemas *possible selves ===== Ego-death ===== * small self -> construct that in the extreme is analogous to those of unitive experience and ego dissolution (diminishing of self) (Hendricks, 2018) * See also [[wikipedia:Ego_death|Ego death]] = complete loss of subjective self concept * ## MOVE TO PSYCHEDLICS SECTION OR MOVE PSYCHEDELICS HERE AND REFERENCE HEADING IN THAT PARAGRAPH === Oneness / Seperation === See Buddhism, see psychedelics, see Connectedness. [''Author's Note: unsure if this should be it's own section, it feels like it should be in the affect of awe on the self section, after speaking on the feeling of connectedness].'' == Effect of awe on the self == ''[Authors Note: This is where we will have a short section on the self (what it means, collection of experiences and stories we tell ourselves, superceding that, buddhist definitions of the self and awareness and appreciating life as tenets, lojong etc.)]'' === Key feelings and effects === * Awe diminishes the sense of self and shifts attention away from individual interests and concerns (Bai et al., 2017) * Differences among individuas for experiencing awe: ** higher in openness tend to experience greater levels of awe (SK Chen, 2021) -> Shiota et al., 2006; Silvia et al., 2015 ** study found that individuals from higher classes reported less frequently experiencing awe compared to individuals from lower social classes (Piff & Moskowitz, 2017). [''Author's Note: I really like this fact and want to include it somewhere but not sure where to include].'' ** not only religious but also non-religious individuals report experiencing awe; *** however, religious onementions explicitly religious or life-or-death events while non-religious one talks about the wonder of nature (Preston & Shin, 2017). **** simulation of viewing Earth from the space, individuals who experienced awe were non-religious (Reinerman-Jones, Sollins, Gallagher, & Janz, 2013). {| class="wikitable" |+ !Key feelings and effects ! |- |Smallness | |- |Profoundness | |- |Connectedness | |- |Perception of the self | |- |Pro-social behaviour (b) | |} ==== Smallness ==== * study found that smallness was experienced but not insignificance (Edwards, et al. 2025) ** awe shrinks the self without making the individual feel insignificant ==== Profoundness ==== * [x] discuss significance of emotion * the present moment ** forced presence *time dilation (what article was that??) ==== Connectedness ==== * seeking connection after feelings of smallness * higher in openness tend to experience greater levels of awe (SK Chen, 2021) -> Shiota et al., 2006; Silvia et al., 2015 ===== Perception of the self ===== * presenting more balanced view of personal strengths and weaknesses * increased acknowledgment of outside forces in one's accomplishments (Stellar et al., 2018) * epistemic humility = measures of wisdom and moral character * positive self-transcendent experiences can be utilised as driving force for better understanding one's decision and perspectives impact others (Kim et al., 2023) === Behavioural changes === * Link back to need for accommodation and changes [x]. Link back to motivation content [x] ==== Pro-social behaviour ==== [[wikipedia:Prosocial_behavior|Prosocial behaviour]] refers to social behaviours that benefit everyone. * How does this relate back to the need for accommodation? [x] === Failures to accommodate === * Awe is more specifically defined in (Shiota et al. 2007) as an emotional response to perceptually vast stimuli that overwhelm current mental structures, yet facilitate ''attempts'' at accommodation. ** Segue intro into failures to accommodate, threat-based awe, etc ==== Fear ==== * Keltner and Haidt (2003) argued that awe is situated on the boundary of fear * Threat-based awe (Keltner and Haidt) - flavour of the stimulus * See Nihilism ==== Trauma ==== [x] * failure to accommodate threat-based awe experienes can result in trauma == Current applications == * experiencing awe can be regulatory, used for finding balance, and coping with stress (Chen & Mongrain, 2020) * (Rudd et al. 2012) found evidence that experience awe may alletive time scarcity feeling in modern life ''[Authors Note:'' (''link back to appraisal from K&H of time distortion)]'' === Wellbeing === ''[Author's Note: find additional references for five processes and explain in paragraph]'' * experiences in nature, moved by mysic, psychedelics, spiritual contemplation promote mental health? via awe (Monroy and Keltner, 2023) ** awe engages five processes *** shifts in neurophysiology *** diminished focus on self *** increased prosocail realtionality *** greater social integration *** heightened sense of meaning (Monroy and Keltner, 2023) === Therapies === *intro [x] ==== Existential Therapy ==== * death is a main consideration in existential thought (Yalom, 2008) * death awareness reults in anxiety which can be experienced in two different ways: ** a) an overwhelming anxiety (dread or depression) resulting in various dysfunctional behaviors and moodstates associated with the choice of escape from it ** b) an energizing source that leads to live life fully as much as possible. (Furman, 2003) ** awe can be considered as a possible way of dealing with death anxiety (Schenider, 2021) ===== Gestalt ===== * here and now * reliving past experiences in the present moment to reexperience and attempt to accommodate * (Corey, 2009) ==== Psychedelic therapies ==== * evidence [x] * how related to awe [x] == Conclusion == * Awe diminishes the self via the small self affect etc. * Awe requires perceived vastness (stimulus from nature, or learning or challenge to mental schema), and a need for accommodation * Diminishing the self results in pro-social behaviour * Awe and its resulting experience of self-transcendent emotions can be utilised for therapies and healing * Awe promotes wellbeing and underscores other positive stimuli such as psychedelics or spiritual contemplation and learning * Focus Question: [What is awe as an emotion and experience?] ... Prelimary answer: [awe is made up of perceived vastness and a need for an accommodation that often leaves individuals with a smaller sense of self, connectedness and purpose/increased meaning] * Focus Question: [How does the experience of awe diminish the self?] ... Prelimary answer: [awe diminishes the self mostly through the small-self construct, as well as diminishing the ego and seperatedness feeling] * Focus Question: [What is the effect of a diminished sense of self?] ... Prelimary answer: [diminished sense of self results in prosocial behaviour (broad) need to specify what behaviours] * Focus Question: [How can awe be integrated into modern life? i.e. How can we trigger or experience everyday moments of awe?] ** Prelimary answer: [experiencing awe can be regulatory, used for finding balance, and coping with stress (Chen & Mongrain, 2020) ** (Rudd et al. 2012) found evidence that experience awe may alletive time scarcity feeling in modern life ''[Authors Note:'' (''link back to appraisal from K&H of time distortion)]]'' * Focus Question: [How can the experience of awe be applied for social and individual good?] ... Prelimary answer: [therapy, wellbeing, psycotherapy, deliberating triggering awe experiences for healing] * ''[Authors Note: summarise topic, significance, how psych science contributes to understanding and addressing topic, practical take home messages, 150-330 words, mirror in [[#Overview|Overview]]]'' ==See also== * [[wikipedia:Awe|Awe]] (Wikipedia) * [[Motivation and emotion/Book/2016/Awe as an emotion|Awe as an emotion]] (Book chapter, 2016) * [[wikipedia:Ego_death|Ego death]] (Wikipedia) * [[wikipedia:Piaget's_theory_of_cognitive_development|Piaget's theory of cognitive development]] (Wikipedia) * [[wikipedia:Prosocial_behavior|Prosocial behaviour]] (Wikipedia) * [[wikipedia:Schema_(psychology)|Schemas]] (Wikipedia) * [[wikipedia:Self|Self]] (Wikipedia) * [[wikipedia:Self|Self-concept]] (Wikipedia) ==References== {{Hanging indent|1= Arvidson, P. S. (2022). Reverent Awe and the Field of Consciousness. Human Studies, 45(3), 397-416. Bai, Y., Maruskin, L. A., Chen, S., Gordon, A. M., Stellar, J. E., McNeil, G. D., Peng, K., & Keltner, D. (2017). Awe, the diminished self, and collective engagement: Universals and cultural variations in the small self. Journal of personality and social psychology, 113(2), 185. Cavallaro, R. M., Church, M. H., Giddens, B. J., & Rivera, G. N. (2025). Complex effects of awe on meaning in life and true self-knowledge. Self and Identity, 24(3), 189-216. Chen, S. K., & Mongrain, M. (2021). Awe and the interconnected self. The Journal of Positive Psychology, 16(6), 770-778. Edwards, M. E., Mendenhall, K., Sanders, C., & King, L. A. (2026). Small but still significant: Awe and the self. Personality and Social Psychology Bulletin, 52(9), 3032-3045. Furman, R. (2003). Poetry therapy and existential practice. The Arts in Psychotherapy, 30(4), 195-200. Gao, R. (2026). From awe to pro-environmental behavior-a meta-analysis of the relationship between awe and pro-environmental behavior. The Journal of Positive Psychology, 21(5), 958-970. Goldy, S. P., Jones, N. M., & Piff, P. K. (2022). The social effects of an awesome solar eclipse. Psychological Science, 33(9), 1452-1462. Hendricks, P. S. (2018). Awe: a putative mechanism underlying the effects of classic psychedelic-assisted psychotherapy. International Review of Psychiatry, 30(4), 331-342. Jiang, T., Hicks, J. A., Yuan, W., Yin, Y., Needy, L., & Vess, M. (2024). The unique nature and psychosocial implications of awe. Nature Reviews Psychology, 3(7), 475-488. Jiang, T., & Sedikides, C. (2022). Awe motivates authentic-self pursuit via self-transcendence: Implications for prosociality. Journal of personality and social psychology, 123(3), 576. Keltner, D., & Haidt, J. (2003). Approaching awe, a moral, spiritual, and aesthetic emotion. Cognition and emotion, 17(2), 297-314. Kim, Y., Nusbaum, H. C., & Yang, F. (2023). Going beyond ourselves: The role of self-transcendent experiences in wisdom. Cognition and Emotion, 37(1), 98-116. Li, R., Hou, Z., Zhang, C., Xu, Q., & Nie, A. (2024). A meta-analysis examining the relationship between awe and prosocial behavior. Current Psychology, 43(29), 24702-24711. Lucht, A., & van Schie, H. T. (2024). The evolutionary function of awe: A review and integrated model of seven theoretical perspectives. Emotion Review, 16(1), 46-63. Maslow, A. H. (1964). Religions, values, and peak-experiences. Columbus, OH: Ohio State University Press. McMann, G. B. (2025). Role of Vastness and Accommodation of Awe in Personal Change and Growth Grand Canyon University]. Perez, K. A., Lench, H. C., Thompson, C. G., & North, S. (2023). Experimental elicitations of awe: A meta-analysis. Cognition and Emotion, 37(1), 18-33. Piff, P. K., Dietze, P., Feinberg, M., Stancato, D. M., & Keltner, D. (2015). Awe, the small self, and prosocial behavior. Journal of personality and social psychology, 108(6), 883. Piff, P. K., Singhal, I., & Bai, Y. (2025). Bridging me to we: Awe is a conduit to cohesive collectives. Current Opinion in Psychology, 62, 101979. Rivera, G. N., Vess, M., Hicks, J. A., & Routledge, C. (2020). Awe and meaning: Elucidating complex effects of awe experiences on meaning in life. European Journal of Social Psychology, 50(2), 392-405. Shiota, M. N., Keltner, D., & Mossman, A. (2007). The nature of awe: Elicitors, appraisals, and effects on self-concept. Cognition and Emotion, 21(5), 944-963. Shukla, S. (2021). Indian classical music, awe, and healing. The Humanistic Psychologist, 49(3), 355. Stellar, J. E., Gordon, A., Anderson, C. L., Piff, P. K., McNeil, G. D., & Keltner, D. (2018). Awe and humility. Journal of personality and social psychology, 114(2), 258. Sturm, V. E., Datta, S., Roy, A. R., Sible, I. J., Kosik, E. L., Veziris, C. R., Chow, T. E., Morris, N. A., Neuhaus, J., & Kramer, J. H. (2022). Big smile, small self: Awe walks promote prosocial positive emotions in older adults. Emotion, 22(5), 1044. Tanhan, F., DENİZ, M., & Akgün, G. (2022). Awe Therapy: Fundamentals, Formulation and Usage Area Awe Terapi: Temelleri, Formülasyonu ve Kullanım Alanları. Turkish Psychological Counseling and Guidance Journal, 12(65). Tanhan, F., Deniz, M. E., & Akgün, G. E. (2022). Awe Therapy: Fundamentals, formulation and usage area. Turkish Psychological Counseling and Guidance Journal, 12(65), 300-319. Yaden, D. B., Iwry, J., Slack, K. J., Eichstaedt, J. C., Zhao, Y., Vaillant, G. E., & Newberg, A. B. (2016). The overview effect: Awe and self-transcendent experience in space flight. Psychology of Consciousness: Theory, Research, and Practice, 3(1), 1. Yalom, I. D. (2008). Staring at the sun: Overcoming the terror of death. The Humanistic Psychologist, 36(3-4), 283. }} ''[Authors Note: what about italicisation for the references / article titles (APA?]'' ==External links== * [https://self-transcendence.org/theory Self-transendence] (Self-transcendence.org) * [https://fitmind.org/ Ancient wisdom + modern science] (Fitmind.org) * [https://therapyintampa.com/the-neuroscience-of-awe-how-shared-wonder-supports-mental-health/ The Neuroscience of Awe: How Shared Wonder Supports Mental Health] (North Star Therapy) * Ted Talk Neuroscience video [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Awe]] [[Category:Motivation and emotion/Book/Self]] 7obg3chbcbnlmtx3k6dah2973x6u68i 2842518 2842515 2026-10-25T03:06:50Z Amirrorslens 3106278 added vastness sentence 2842518 wikitext text/x-wiki {{title|Awe and the diminished self:<br>How does awe diminish the self and how can this be applied?}} __TOC__ ==Overview== {{RoundBoxTop|theme=2}} [[File:NASA-HS201427a-HubbleUltraDeepField2014-20140603.jpg|right|thumb|200px|'''Figure 1'''. NASA Hubble Ultra Deep Field]] ; Scenario Leo walks into the desert late at night and is taken aback at the night sky above him. Being alone in the vast and flat plane lets him really perceive the vastness of the stars and leaves him feeling very present and aware of his smallness in comparison. This overwhelming feeling leaves Leo seeking connection with others, and a new found sense of purpose. * [Author Note: REPLACE WITH A REAL CASE STUDY - such as near death experience (The Good Place coded) (Might modify scenario to incude this aspect similar to the purple feature box on [[Motivation and emotion/Book/2025/Motivation for using AI companions|this page]])] {{RoundBoxBottom}} '''''Author Note: KEY PSYCH SCIENCE''''' {{ic|Move this detail into subsequent sections}} * ''Maslow and Carl Rogers -> self-concept and peak experiences'' * ''KEY PIECE OF LITERATURE: Keltner and Haidt: Awe 2003'' *''Jean Piaget - Cognitive Development'' * Awe is considered a [[wikipedia:Peak_experience|peak experience]] according to Maslow [x] {{ic|Also consider awe as an emotion}}, (see [[Motivation and emotion/Book/2017/Awe and well-being|Awe and well-being]]) ** Experiencing awe = indicator of reaching highest level of self-actualisation and self-transcendence) * Awe can have positive individual and community effects (pro-social behaviour) {{ic|The topic is not just about awe - it is about the effect of awe on self-perception}} {{RoundBoxTop|theme=3}} '''Focus questions''' * What is awe as an emotion? > Heading: Awe as an emotion * What is humanistic psychology and theory of needs? > Heading: Humanistic Psychology, Maslow Theory of Needs, Self Actualisation (Cut Self) * What is the role of awe in Maslow's peak experiences? * How can we apply awe as peak experiences for individual good? * > Heading: Awe as peak experiences * How does the experience of awe diminish the self? * What is the effect of a diminished sense of self? * How can awe be integrated into modern life? i.e. How can we trigger or experience everyday moments of awe?{{RoundBoxBottom}} {{RoundBoxTop|theme=1}} '''QUESTIONS AND NOTES FOR SELF''' * Where do I fit discussions of the Self in here? * Should I get rid of the physical and conceptual distinction for the headings? Rework to the five appraisals from that one article - [Author's Notes] {{RoundBoxBottom}} == Awe as an emotion == [[File:Plutchik-wheel.svg|thumb|Figure 2: Plutchik's wheel of emotions, where awe is modelled as a combination of suprise and fear.]] [[wikipedia:Awe|Awe]] is an emotion generally described as [insert synonyms]. In Robert Plutchik's wheel of emotion as seen in Figure 2, awe is modelled as a combination of suprise and fear (see [[Motivation and emotion/Book/2016/Awe as an emotion|Awe as an emotion]]). ''- KEY REFERENCE: Keltner and Haidt, 2003:'' * ''Additional appraisals account for variation in the hedoic (pleasure) tone of awe experiences'' ** ''threat, beauty, exceptional ability, virtue, and the supernatural'' * ''Edwards et al. (2026) define awe to be an emotion elicited from experiencing great vastness that dramaticallt expands a person's frame of reference.'' ** ''see [[Motivation and emotion/Book/2016/Awe as an emotion|Awe as an emotion]]'' In the prototypical perspectice of awe offered by Keltner and Haidt, (2003), two features are central to the experience of awe as an emotion; perceived vastness, and a following need for accommodation. ==== Vastness ==== Vastness refers to experiences that are perceived as being much larger than the self or their ordinary frame of reference. (Keltner and Haidt, 2003). - power Obscurity is another propertry that produces the 'sublime experience' (Burke, 1990, as cited in Keltner and Hiadt., 2014). * "Although the term “vastness” implies great physical size, in this usage “vast” describes any stimulus that challenges one's accustomed frame of reference in some dimension ** A stimulus may convey vastness in physical space, in time, in number, in complexity of detail, in ability, even in volume of human experience. *** For example, one may experience a sense of vastness in a mathematical equation, not because the equation is literally long, but because of the vast number of observed physical processes it is able to explain and predict. ** What is critical is that the stimulus dramatically expands the observer's usual frame of reference in some dimension or domain." (Shiota et al., 2007) ===== Physical vastness ===== ''[Author's Note: Nature, Space, might be wise to mention depth perception studies here. Why particuarly these things. Work in thesis on space video here.]'' * The overview effect = witnessing Earth from the atmosphere = life-altering degree of awe -> transcendent -> complete breakdown of perception of borders and countries = feeling of connectedness (Yaden et al., 2016) * Eclipses: study that used twitter data (over 2 million users) -> individuals exhibited more awe and expressed less self focused and more pro social, affiliative, humble and collective language * Witnessing natural disasters may induce a state of negatively valenced awe (Chaudhury et al., 2022; Piff et al., 2015) ** Link to heading section 'Effect of awe on the self' -> threat-based awe (Keltner and Haidt, 2003). ===== Conceptual vastness ===== Witnessing other people or learning things, coming of age moments, witnessing things for the first time theories here. Music and virtues act of skills. * Acts of Virtue * Acts of skills * Learning things ** A stimulus may convey vastness in physical space, in time, in number, in complexity of detail, in ability, even in volume of human experience. *** For example, one may experience a sense of vastness in a mathematical equation, not because the equation is literally long, but because of the vast number of observed physical processes it is able to explain and predict. ** What is critical is that the stimulus dramatically expands the observer's usual frame of reference in some dimension or domain." (Shiota et al., 2007) == Need for accommodation == * Awe is defined by Keltner and Haidt as an inability to assimilate an experience into current mental structures. (Keltner and Haidt, 2003) * Accommodation is one of the two ongoing processes (assimilation being the other) in [[wikipedia:Piaget's_theory_of_cognitive_development|Piaget's theory of cognitive development]], where existing [[wikipedia:Schema_(psychology)|schemas]] are modified, or even created to deal with a new event or experience (Jean Piaget, {find specific reference}). ** See also: [[wikipedia:Schema_(psychology)|Schemas]] * ''[Authors Note: Insert table of epistimec emotions such as wonder, elevation, sublime, admiration (Keltner and Haidt), not sure of relevance or format]'' ''- [Authors Note: unsure whether I should spend a sub section on schemas or not, or how much explanation should be done for these sorts of fundamental but still unfamiliar to reader concepts'' - Discussion of what it means to require the need for knowledge structures to be updated. == The role of awe in Maslow's peak experiences == - {| class="wikitable" |+ !Features of peak experience !Example of study |- |disorientation in time and space | |- |ego transcendece & self-forgetfulness |ego death |- |self-forgetfulness |meditation? |- |perception that the world is good, beautiful and desirable |is there a study that supports this resulting notion, |- |feeling passive, receptive and humble |non-attachment?, radical acceptable therapy, meditation/mindfulness |- |sense that polarities and dichotomies have been transcended or resolved |no borders or boundaries, the out of orbit // observer effect |- |feelings of being lucky, fortunate, or graced |gratitude practice |} ====== <s>Self</s> ====== * <s>Self in philosophy = individual's own being, knowledge and values (and relationship between these attributes ##CUT##</s> ** <s>self is distingushable from others</s> *** <s>[Authors Note: have a section later in connectedness how when the self is diminished, the concept of otherness is also diminished -> oneness]</s> === Self-perception === [[wikipedia:Self-concept|Self-concept]] = made up of one's self-schemas *possible selves ===== Ego-death ===== * small self -> construct that in the extreme is analogous to those of unitive experience and ego dissolution (diminishing of self) (Hendricks, 2018) * See also [[wikipedia:Ego_death|Ego death]] = complete loss of subjective self concept * ''%% MOVE TO PSYCHEDLICS SECTION OR MOVE PSYCHEDELICS HERE AND REFERENCE HEADING IN THAT PARAGRAPH %%'' ==== Psychedelics ==== ''[Author's Note: I feel psychedelics needs its own subsection here as it is particuarly relevant to my applications, and provides a good segue to need for accommodations.]'' * psychedielic-occassioned mystical experience is characterised by profound awe ** small self -> construct that in the extreme is analogous to those of unitive experience and ego dissolution (diminishing of self) (Hendricks, 2018) * psychedelics have been shown to produce enduring positive change across a range of psychological and behavioural outcomes (Hendricks, 2018) via ''(Griffiths et al., 2018; Griffiths, Richards, Johnson, McCann, & Jesse, 2008, Griffiths et al., 2011; Griffiths, Richards, McCann, & Jesse, 2006; Lebedev et al., 2016; MacLean, Johnson, & Griffiths, 2011)'' as well as treatment of depression. === Oneness / Seperation === See Buddhism, see psychedelics, see Connectedness. [''Author's Note: unsure if this should be it's own section, it feels like it should be in the affect of awe on the self section, after speaking on the feeling of connectedness].'' == Effect of awe on the self == ''[Authors Note: This is where we will have a short section on the self (what it means, collection of experiences and stories we tell ourselves, superceding that, buddhist definitions of the self and awareness and appreciating life as tenets, lojong etc.)]'' === Key feelings and effects === * Awe diminishes the sense of self and shifts attention away from individual interests and concerns (Bai et al., 2017) * Differences among individuas for experiencing awe: ** higher in openness tend to experience greater levels of awe (SK Chen, 2021) -> Shiota et al., 2006; Silvia et al., 2015 ** study found that individuals from higher classes reported less frequently experiencing awe compared to individuals from lower social classes (Piff & Moskowitz, 2017). [''Author's Note: I really like this fact and want to include it somewhere but not sure where to include].'' ** not only religious but also non-religious individuals report experiencing awe; *** however, religious onementions explicitly religious or life-or-death events while non-religious one talks about the wonder of nature (Preston & Shin, 2017). **** simulation of viewing Earth from the space, individuals who experienced awe were non-religious (Reinerman-Jones, Sollins, Gallagher, & Janz, 2013). {| class="wikitable" |+ !Key feelings and effects ! |- |Smallness | |- |Profoundness | |- |Connectedness | |- |Perception of the self | |- |Pro-social behaviour (b) | |} ==== Smallness ==== * study found that smallness was experienced but not insignificance (Edwards, et al. 2025) ** awe shrinks the self without making the individual feel insignificant ==== Profoundness ==== * [x] discuss significance of emotion * the present moment ** forced presence *time dilation (what article was that??) ==== Connectedness ==== * seeking connection after feelings of smallness * higher in openness tend to experience greater levels of awe (SK Chen, 2021) -> Shiota et al., 2006; Silvia et al., 2015 ===== Perception of the self ===== * presenting more balanced view of personal strengths and weaknesses * increased acknowledgment of outside forces in one's accomplishments (Stellar et al., 2018) * epistemic humility = measures of wisdom and moral character * positive self-transcendent experiences can be utilised as driving force for better understanding one's decision and perspectives impact others (Kim et al., 2023) === Behavioural changes === * Link back to need for accommodation and changes [x]. Link back to motivation content [x] ==== Pro-social behaviour ==== [[wikipedia:Prosocial_behavior|Prosocial behaviour]] refers to social behaviours that benefit everyone. * How does this relate back to the need for accommodation? [x] === Failures to accommodate === * Awe is more specifically defined in (Shiota et al. 2007) as an emotional response to perceptually vast stimuli that overwhelm current mental structures, yet facilitate ''attempts'' at accommodation. ** Segue intro into failures to accommodate, threat-based awe, etc ==== Fear ==== * Keltner and Haidt (2003) argued that awe is situated on the boundary of fear * Threat-based awe (Keltner and Haidt) - flavour of the stimulus * See Nihilism ==== Trauma ==== [x] * failure to accommodate threat-based awe experienes can result in trauma == Current applications == * experiencing awe can be regulatory, used for finding balance, and coping with stress (Chen & Mongrain, 2020) * (Rudd et al. 2012) found evidence that experience awe may alletive time scarcity feeling in modern life ''[Authors Note:'' (''link back to appraisal from K&H of time distortion)]'' === Wellbeing === ''[Author's Note: find additional references for five processes and explain in paragraph]'' * experiences in nature, moved by mysic, psychedelics, spiritual contemplation promote mental health? via awe (Monroy and Keltner, 2023) ** awe engages five processes *** shifts in neurophysiology *** diminished focus on self *** increased prosocail realtionality *** greater social integration *** heightened sense of meaning (Monroy and Keltner, 2023) === Therapies === *intro [x] ==== Existential Therapy ==== * death is a main consideration in existential thought (Yalom, 2008) * death awareness reults in anxiety which can be experienced in two different ways: ** a) an overwhelming anxiety (dread or depression) resulting in various dysfunctional behaviors and moodstates associated with the choice of escape from it ** b) an energizing source that leads to live life fully as much as possible. (Furman, 2003) ** awe can be considered as a possible way of dealing with death anxiety (Schenider, 2021) ===== Gestalt ===== * here and now * reliving past experiences in the present moment to reexperience and attempt to accommodate * (Corey, 2009) ==== Psychedelic therapies ==== * evidence [x] * how related to awe [x] == Conclusion == * Awe diminishes the self via the small self affect etc. * Awe requires perceived vastness (stimulus from nature, or learning or challenge to mental schema), and a need for accommodation * Diminishing the self results in pro-social behaviour * Awe and its resulting experience of self-transcendent emotions can be utilised for therapies and healing * Awe promotes wellbeing and underscores other positive stimuli such as psychedelics or spiritual contemplation and learning * Focus Question: [What is awe as an emotion and experience?] ... Prelimary answer: [awe is made up of perceived vastness and a need for an accommodation that often leaves individuals with a smaller sense of self, connectedness and purpose/increased meaning] * Focus Question: [How does the experience of awe diminish the self?] ... Prelimary answer: [awe diminishes the self mostly through the small-self construct, as well as diminishing the ego and seperatedness feeling] * Focus Question: [What is the effect of a diminished sense of self?] ... Prelimary answer: [diminished sense of self results in prosocial behaviour (broad) need to specify what behaviours] * Focus Question: [How can awe be integrated into modern life? i.e. How can we trigger or experience everyday moments of awe?] ** Prelimary answer: [experiencing awe can be regulatory, used for finding balance, and coping with stress (Chen & Mongrain, 2020) ** (Rudd et al. 2012) found evidence that experience awe may alletive time scarcity feeling in modern life ''[Authors Note:'' (''link back to appraisal from K&H of time distortion)]]'' * Focus Question: [How can the experience of awe be applied for social and individual good?] ... Prelimary answer: [therapy, wellbeing, psycotherapy, deliberating triggering awe experiences for healing] * ''[Authors Note: summarise topic, significance, how psych science contributes to understanding and addressing topic, practical take home messages, 150-330 words, mirror in [[#Overview|Overview]]]'' ==See also== * [[wikipedia:Awe|Awe]] (Wikipedia) * [[Motivation and emotion/Book/2016/Awe as an emotion|Awe as an emotion]] (Book chapter, 2016) * [[wikipedia:Ego_death|Ego death]] (Wikipedia) * [[wikipedia:Piaget's_theory_of_cognitive_development|Piaget's theory of cognitive development]] (Wikipedia) * [[wikipedia:Prosocial_behavior|Prosocial behaviour]] (Wikipedia) * [[wikipedia:Schema_(psychology)|Schemas]] (Wikipedia) * [[wikipedia:Self|Self]] (Wikipedia) * [[wikipedia:Self|Self-concept]] (Wikipedia) ==References== {{Hanging indent|1= Arvidson, P. S. (2022). Reverent Awe and the Field of Consciousness. Human Studies, 45(3), 397-416. Bai, Y., Maruskin, L. A., Chen, S., Gordon, A. M., Stellar, J. E., McNeil, G. D., Peng, K., & Keltner, D. (2017). Awe, the diminished self, and collective engagement: Universals and cultural variations in the small self. Journal of personality and social psychology, 113(2), 185. Cavallaro, R. M., Church, M. H., Giddens, B. J., & Rivera, G. N. (2025). Complex effects of awe on meaning in life and true self-knowledge. Self and Identity, 24(3), 189-216. Chen, S. K., & Mongrain, M. (2021). Awe and the interconnected self. The Journal of Positive Psychology, 16(6), 770-778. Edwards, M. E., Mendenhall, K., Sanders, C., & King, L. A. (2026). Small but still significant: Awe and the self. Personality and Social Psychology Bulletin, 52(9), 3032-3045. Furman, R. (2003). Poetry therapy and existential practice. The Arts in Psychotherapy, 30(4), 195-200. Gao, R. (2026). From awe to pro-environmental behavior-a meta-analysis of the relationship between awe and pro-environmental behavior. The Journal of Positive Psychology, 21(5), 958-970. Goldy, S. P., Jones, N. M., & Piff, P. K. (2022). The social effects of an awesome solar eclipse. Psychological Science, 33(9), 1452-1462. Hendricks, P. S. (2018). Awe: a putative mechanism underlying the effects of classic psychedelic-assisted psychotherapy. International Review of Psychiatry, 30(4), 331-342. Jiang, T., Hicks, J. A., Yuan, W., Yin, Y., Needy, L., & Vess, M. (2024). The unique nature and psychosocial implications of awe. Nature Reviews Psychology, 3(7), 475-488. Jiang, T., & Sedikides, C. (2022). Awe motivates authentic-self pursuit via self-transcendence: Implications for prosociality. Journal of personality and social psychology, 123(3), 576. Keltner, D., & Haidt, J. (2003). Approaching awe, a moral, spiritual, and aesthetic emotion. Cognition and emotion, 17(2), 297-314. Kim, Y., Nusbaum, H. C., & Yang, F. (2023). Going beyond ourselves: The role of self-transcendent experiences in wisdom. Cognition and Emotion, 37(1), 98-116. Li, R., Hou, Z., Zhang, C., Xu, Q., & Nie, A. (2024). A meta-analysis examining the relationship between awe and prosocial behavior. Current Psychology, 43(29), 24702-24711. Lucht, A., & van Schie, H. T. (2024). The evolutionary function of awe: A review and integrated model of seven theoretical perspectives. Emotion Review, 16(1), 46-63. Maslow, A. H. (1964). Religions, values, and peak-experiences. Columbus, OH: Ohio State University Press. McMann, G. B. (2025). Role of Vastness and Accommodation of Awe in Personal Change and Growth Grand Canyon University]. Perez, K. A., Lench, H. C., Thompson, C. G., & North, S. (2023). Experimental elicitations of awe: A meta-analysis. Cognition and Emotion, 37(1), 18-33. Piff, P. K., Dietze, P., Feinberg, M., Stancato, D. M., & Keltner, D. (2015). Awe, the small self, and prosocial behavior. Journal of personality and social psychology, 108(6), 883. Piff, P. K., Singhal, I., & Bai, Y. (2025). Bridging me to we: Awe is a conduit to cohesive collectives. Current Opinion in Psychology, 62, 101979. Rivera, G. N., Vess, M., Hicks, J. A., & Routledge, C. (2020). Awe and meaning: Elucidating complex effects of awe experiences on meaning in life. European Journal of Social Psychology, 50(2), 392-405. Shiota, M. N., Keltner, D., & Mossman, A. (2007). The nature of awe: Elicitors, appraisals, and effects on self-concept. Cognition and Emotion, 21(5), 944-963. Shukla, S. (2021). Indian classical music, awe, and healing. The Humanistic Psychologist, 49(3), 355. Stellar, J. E., Gordon, A., Anderson, C. L., Piff, P. K., McNeil, G. D., & Keltner, D. (2018). Awe and humility. Journal of personality and social psychology, 114(2), 258. Sturm, V. E., Datta, S., Roy, A. R., Sible, I. J., Kosik, E. L., Veziris, C. R., Chow, T. E., Morris, N. A., Neuhaus, J., & Kramer, J. H. (2022). Big smile, small self: Awe walks promote prosocial positive emotions in older adults. Emotion, 22(5), 1044. Tanhan, F., DENİZ, M., & Akgün, G. (2022). Awe Therapy: Fundamentals, Formulation and Usage Area Awe Terapi: Temelleri, Formülasyonu ve Kullanım Alanları. Turkish Psychological Counseling and Guidance Journal, 12(65). Tanhan, F., Deniz, M. E., & Akgün, G. E. (2022). Awe Therapy: Fundamentals, formulation and usage area. Turkish Psychological Counseling and Guidance Journal, 12(65), 300-319. Yaden, D. B., Iwry, J., Slack, K. J., Eichstaedt, J. C., Zhao, Y., Vaillant, G. E., & Newberg, A. B. (2016). The overview effect: Awe and self-transcendent experience in space flight. Psychology of Consciousness: Theory, Research, and Practice, 3(1), 1. Yalom, I. D. (2008). Staring at the sun: Overcoming the terror of death. The Humanistic Psychologist, 36(3-4), 283. }} ''[Authors Note: what about italicisation for the references / article titles (APA?]'' ==External links== * [https://self-transcendence.org/theory Self-transendence] (Self-transcendence.org) * [https://fitmind.org/ Ancient wisdom + modern science] (Fitmind.org) * [https://therapyintampa.com/the-neuroscience-of-awe-how-shared-wonder-supports-mental-health/ The Neuroscience of Awe: How Shared Wonder Supports Mental Health] (North Star Therapy) * Ted Talk Neuroscience video [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Awe]] [[Category:Motivation and emotion/Book/Self]] 0xxwj4jz1rial6ozjm8100joac9ni0l 2842521 2842518 2026-10-25T03:24:42Z Amirrorslens 3106278 moved failure to accommodate section, wondering if i should consider the prototype perspective as part of the awe as an emotion section or if i should include it in the theory section 2842521 wikitext text/x-wiki {{title|Awe and the diminished self:<br>How does awe diminish the self and how can this be applied?}} __TOC__ ==Overview== {{RoundBoxTop|theme=2}} [[File:NASA-HS201427a-HubbleUltraDeepField2014-20140603.jpg|right|thumb|200px|'''Figure 1'''. NASA Hubble Ultra Deep Field]] ; Scenario Leo walks into the desert late at night and is taken aback at the night sky above him. Being alone in the vast and flat plane lets him really perceive the vastness of the stars and leaves him feeling very present and aware of his smallness in comparison. This overwhelming feeling leaves Leo seeking connection with others, and a new found sense of purpose. * [Author Note: REPLACE WITH A REAL CASE STUDY - such as near death experience (The Good Place coded) (Might modify scenario to incude this aspect similar to the purple feature box on [[Motivation and emotion/Book/2025/Motivation for using AI companions|this page]])] {{RoundBoxBottom}} * * ''KEY PIECE OF LITERATURE: Keltner and Haidt: Awe 2003'' *''Jean Piaget - Cognitive Development'' * Awe is considered a [[wikipedia:Peak_experience|peak experience]] according to Maslow [x] {{ic|Also consider awe as an emotion}}, (see [[Motivation and emotion/Book/2017/Awe and well-being|Awe and well-being]]) ** Experiencing awe = indicator of reaching highest level of self-actualisation and self-transcendence) * Awe can have positive individual and community effects (pro-social behaviour) {{ic|The topic is not just about awe - it is about the effect of awe on self-perception}} {{RoundBoxTop|theme=3}} '''Focus questions''' * What is awe as an emotion? * What is humanistic psychology and theory of needs? * What is the role of awe in Maslow's peak experiences? * How can we apply awe as peak experiences for individual good? * How does the experience of awe diminish the self? * What is the effect of a diminished sense of self? * How can awe be integrated into modern life? i.e. How can we produce the conditions for everyday and intentional moments of awe?{{RoundBoxBottom}} == Awe as an emotion == [[File:Plutchik-wheel.svg|thumb|Figure 2: Plutchik's wheel of emotions, where awe is modelled as a combination of suprise and fear.]] [[wikipedia:Awe|Awe]] is an emotion generally described as [insert synonyms]. In Robert Plutchik's wheel of emotion as seen in Figure 2, awe is modelled as a combination of suprise and fear (see [[Motivation and emotion/Book/2016/Awe as an emotion|Awe as an emotion]]). ''- KEY REFERENCE: Keltner and Haidt, 2003:'' * ''Additional appraisals account for variation in the hedoic (pleasure) tone of awe experiences'' ** ''threat, beauty, exceptional ability, virtue, and the supernatural'' ** ''%% turn this into a table %%'' ** ''%% should i do a table of definitions %% what other learnings features are there'' - %% we use the experience of awe to define the eliciting emotion . perception of something vast -> triggers awe -> requires need for accomodation In the prototypical perspectice of awe offered by Keltner and Haidt, (2003), two features are central to the experience of awe as an emotion; perceived vastness, and a following need for accommodation. ==== Vastness ==== Vastness refers to experiences that are perceived as being much larger than the self or their ordinary frame of reference. (Keltner and Haidt, 2003). - power Obscurity is another propertry that produces the 'sublime experience' (Burke, 1990, as cited in Keltner and Hiadt., 2014). * "Although the term “vastness” implies great physical size, in this usage “vast” describes any stimulus that challenges one's accustomed frame of reference in some dimension ** A stimulus may convey vastness in physical space, in time, in number, in complexity of detail, in ability, even in volume of human experience. *** For example, one may experience a sense of vastness in a mathematical equation, not because the equation is literally long, but because of the vast number of observed physical processes it is able to explain and predict. ** What is critical is that the stimulus dramatically expands the observer's usual frame of reference in some dimension or domain." (Shiota et al., 2007) ===== Physical vastness ===== ''[Author's Note: Nature, Space, might be wise to mention depth perception studies here. Why particuarly these things. Work in thesis on space video here.]'' * The overview effect = witnessing Earth from the atmosphere = life-altering degree of awe -> transcendent -> complete breakdown of perception of borders and countries = feeling of connectedness (Yaden et al., 2016) * Eclipses: study that used twitter data (over 2 million users) -> individuals exhibited more awe and expressed less self focused and more pro social, affiliative, humble and collective language * Witnessing natural disasters may induce a state of negatively valenced awe (Chaudhury et al., 2022; Piff et al., 2015) ** Link to heading section 'Effect of awe on the self' -> threat-based awe (Keltner and Haidt, 2003). ===== Conceptual vastness ===== Witnessing other people or learning things, coming of age moments, witnessing things for the first time theories here. Music and virtues act of skills. * Acts of Virtue * Acts of skills * Learning things ** A stimulus may convey vastness in physical space, in time, in number, in complexity of detail, in ability, even in volume of human experience. *** For example, one may experience a sense of vastness in a mathematical equation, not because the equation is literally long, but because of the vast number of observed physical processes it is able to explain and predict. ** What is critical is that the stimulus dramatically expands the observer's usual frame of reference in some dimension or domain." (Shiota et al., 2007) == Need for accommodation == * Awe is defined by Keltner and Haidt as an inability to assimilate an experience into current mental structures. (Keltner and Haidt, 2003) * Accommodation is one of the two ongoing processes (assimilation being the other) in [[wikipedia:Piaget's_theory_of_cognitive_development|Piaget's theory of cognitive development]], where existing [[wikipedia:Schema_(psychology)|schemas]] are modified, or even created to deal with a new event or experience (Jean Piaget, {find specific reference}). ** See also: [[wikipedia:Schema_(psychology)|Schemas]] * ''[Authors Note: Insert table of epistimec emotions such as wonder, elevation, sublime, admiration (Keltner and Haidt), not sure of relevance or format]'' ''- [Authors Note: unsure whether I should spend a sub section on schemas or not, or how much explanation should be done for these sorts of fundamental but still unfamiliar to reader concepts'' - Discussion of what it means to require the need for knowledge structures to be updated. == The role of awe in Maslow's peak experiences == - {| class="wikitable" |+ !Features of peak experience !Example of study |- |disorientation in time and space | |- |ego transcendece & self-forgetfulness |ego death |- |self-forgetfulness |meditation? |- |perception that the world is good, beautiful and desirable |is there a study that supports this resulting notion, |- |feeling passive, receptive and humble |non-attachment?, radical acceptable therapy, meditation/mindfulness |- |sense that polarities and dichotomies have been transcended or resolved |no borders or boundaries, the out of orbit // observer effect |- |feelings of being lucky, fortunate, or graced |gratitude practice |} === Failures to accommodate === * Awe is more specifically defined in (Shiota et al. 2007) as an emotional response to perceptually vast stimuli that overwhelm current mental structures, yet facilitate ''attempts'' at accommodation. ** Segue intro into failures to accommodate, threat-based awe, etc ==== Fear ==== * Keltner and Haidt (2003) argued that awe is situated on the boundary of fear * Threat-based awe (Keltner and Haidt) - flavour of the stimulus * See Nihilism ==== Trauma ==== [x] * failure to accommodate threat-based awe experienes can result in trauma ====== <s>Self</s> ====== * <s>Self in philosophy = individual's own being, knowledge and values (and relationship between these attributes ##CUT##</s> ** <s>self is distingushable from others</s> *** <s>[Authors Note: have a section later in connectedness how when the self is diminished, the concept of otherness is also diminished -> oneness]</s> === Self-perception === [[wikipedia:Self-concept|Self-concept]] = made up of one's self-schemas *possible selves ===== Ego-death ===== * small self -> construct that in the extreme is analogous to those of unitive experience and ego dissolution (diminishing of self) (Hendricks, 2018) * See also [[wikipedia:Ego_death|Ego death]] = complete loss of subjective self concept * ''%% MOVE TO PSYCHEDLICS SECTION OR MOVE PSYCHEDELICS HERE AND REFERENCE HEADING IN THAT PARAGRAPH %%'' ==== Psychedelics ==== ''[Author's Note: I feel psychedelics needs its own subsection here as it is particuarly relevant to my applications, and provides a good segue to need for accommodations.]'' * psychedielic-occassioned mystical experience is characterised by profound awe ** small self -> construct that in the extreme is analogous to those of unitive experience and ego dissolution (diminishing of self) (Hendricks, 2018) * psychedelics have been shown to produce enduring positive change across a range of psychological and behavioural outcomes (Hendricks, 2018) via ''(Griffiths et al., 2018; Griffiths, Richards, Johnson, McCann, & Jesse, 2008, Griffiths et al., 2011; Griffiths, Richards, McCann, & Jesse, 2006; Lebedev et al., 2016; MacLean, Johnson, & Griffiths, 2011)'' as well as treatment of depression. === Oneness / Seperation === See Buddhism, see psychedelics, see Connectedness. [''Author's Note: unsure if this should be it's own section, it feels like it should be in the affect of awe on the self section, after speaking on the feeling of connectedness].'' == Effect of awe on the self == ''[Authors Note: This is where we will have a short section on the self (what it means, collection of experiences and stories we tell ourselves, superceding that, buddhist definitions of the self and awareness and appreciating life as tenets, lojong etc.)]'' === Key feelings and effects === * Awe diminishes the sense of self and shifts attention away from individual interests and concerns (Bai et al., 2017) * Differences among individuas for experiencing awe: ** higher in openness tend to experience greater levels of awe (SK Chen, 2021) -> Shiota et al., 2006; Silvia et al., 2015 ** study found that individuals from higher classes reported less frequently experiencing awe compared to individuals from lower social classes (Piff & Moskowitz, 2017). [''Author's Note: I really like this fact and want to include it somewhere but not sure where to include].'' ** not only religious but also non-religious individuals report experiencing awe; *** however, religious onementions explicitly religious or life-or-death events while non-religious one talks about the wonder of nature (Preston & Shin, 2017). **** simulation of viewing Earth from the space, individuals who experienced awe were non-religious (Reinerman-Jones, Sollins, Gallagher, & Janz, 2013). {| class="wikitable" |+ !Key feelings and effects ! |- |Smallness | |- |Profoundness | |- |Connectedness | |- |Perception of the self | |- |Pro-social behaviour (b) | |} ==== Smallness ==== * study found that smallness was experienced but not insignificance (Edwards, et al. 2025) ** awe shrinks the self without making the individual feel insignificant ==== Profoundness ==== * [x] discuss significance of emotion * the present moment ** forced presence *time dilation (what article was that??) ==== Connectedness ==== * seeking connection after feelings of smallness * higher in openness tend to experience greater levels of awe (SK Chen, 2021) -> Shiota et al., 2006; Silvia et al., 2015 ===== Perception of the self ===== * presenting more balanced view of personal strengths and weaknesses * increased acknowledgment of outside forces in one's accomplishments (Stellar et al., 2018) * epistemic humility = measures of wisdom and moral character * positive self-transcendent experiences can be utilised as driving force for better understanding one's decision and perspectives impact others (Kim et al., 2023) === Behavioural changes === * Link back to need for accommodation and changes [x]. Link back to motivation content [x] ==== Pro-social behaviour ==== [[wikipedia:Prosocial_behavior|Prosocial behaviour]] refers to social behaviours that benefit everyone. * How does this relate back to the need for accommodation? [x] === Failures to accommodate === * Awe is more specifically defined in (Shiota et al. 2007) as an emotional response to perceptually vast stimuli that overwhelm current mental structures, yet facilitate ''attempts'' at accommodation. ** Segue intro into failures to accommodate, threat-based awe, etc ==== Fear ==== * Keltner and Haidt (2003) argued that awe is situated on the boundary of fear * Threat-based awe (Keltner and Haidt) - flavour of the stimulus * See Nihilism ==== Trauma ==== [x] * failure to accommodate threat-based awe experienes can result in trauma == Current applications == * experiencing awe can be regulatory, used for finding balance, and coping with stress (Chen & Mongrain, 2020) * (Rudd et al. 2012) found evidence that experience awe may alletive time scarcity feeling in modern life ''[Authors Note:'' (''link back to appraisal from K&H of time distortion)]'' === Wellbeing === ''[Author's Note: find additional references for five processes and explain in paragraph]'' * experiences in nature, moved by mysic, psychedelics, spiritual contemplation promote mental health? via awe (Monroy and Keltner, 2023) ** awe engages five processes *** shifts in neurophysiology *** diminished focus on self *** increased prosocail realtionality *** greater social integration *** heightened sense of meaning (Monroy and Keltner, 2023) === Therapies === *intro [x] ==== Existential Therapy ==== * death is a main consideration in existential thought (Yalom, 2008) * death awareness reults in anxiety which can be experienced in two different ways: ** a) an overwhelming anxiety (dread or depression) resulting in various dysfunctional behaviors and moodstates associated with the choice of escape from it ** b) an energizing source that leads to live life fully as much as possible. (Furman, 2003) ** awe can be considered as a possible way of dealing with death anxiety (Schenider, 2021) ===== Gestalt ===== * here and now * reliving past experiences in the present moment to reexperience and attempt to accommodate * (Corey, 2009) ==== Psychedelic therapies ==== * evidence [x] * how related to awe [x] == Conclusion == * Awe diminishes the self via the small self affect etc. * Awe requires perceived vastness (stimulus from nature, or learning or challenge to mental schema), and a need for accommodation * Diminishing the self results in pro-social behaviour * Awe and its resulting experience of self-transcendent emotions can be utilised for therapies and healing * Awe promotes wellbeing and underscores other positive stimuli such as psychedelics or spiritual contemplation and learning * Focus Question: [What is awe as an emotion and experience?] ... Prelimary answer: [awe is made up of perceived vastness and a need for an accommodation that often leaves individuals with a smaller sense of self, connectedness and purpose/increased meaning] * Focus Question: [How does the experience of awe diminish the self?] ... Prelimary answer: [awe diminishes the self mostly through the small-self construct, as well as diminishing the ego and seperatedness feeling] * Focus Question: [What is the effect of a diminished sense of self?] ... Prelimary answer: [diminished sense of self results in prosocial behaviour (broad) need to specify what behaviours] * Focus Question: [How can awe be integrated into modern life? i.e. How can we trigger or experience everyday moments of awe?] ** Prelimary answer: [experiencing awe can be regulatory, used for finding balance, and coping with stress (Chen & Mongrain, 2020) ** (Rudd et al. 2012) found evidence that experience awe may alletive time scarcity feeling in modern life ''[Authors Note:'' (''link back to appraisal from K&H of time distortion)]]'' * Focus Question: [How can the experience of awe be applied for social and individual good?] ... Prelimary answer: [therapy, wellbeing, psycotherapy, deliberating triggering awe experiences for healing] * ''[Authors Note: summarise topic, significance, how psych science contributes to understanding and addressing topic, practical take home messages, 150-330 words, mirror in [[#Overview|Overview]]]'' ==See also== * [[wikipedia:Awe|Awe]] (Wikipedia) * [[Motivation and emotion/Book/2016/Awe as an emotion|Awe as an emotion]] (Book chapter, 2016) * [[wikipedia:Ego_death|Ego death]] (Wikipedia) * [[wikipedia:Piaget's_theory_of_cognitive_development|Piaget's theory of cognitive development]] (Wikipedia) * [[wikipedia:Prosocial_behavior|Prosocial behaviour]] (Wikipedia) * [[wikipedia:Schema_(psychology)|Schemas]] (Wikipedia) * [[wikipedia:Self|Self]] (Wikipedia) * [[wikipedia:Self|Self-concept]] (Wikipedia) ==References== {{Hanging indent|1= Arvidson, P. S. (2022). Reverent Awe and the Field of Consciousness. Human Studies, 45(3), 397-416. Bai, Y., Maruskin, L. A., Chen, S., Gordon, A. M., Stellar, J. E., McNeil, G. D., Peng, K., & Keltner, D. (2017). Awe, the diminished self, and collective engagement: Universals and cultural variations in the small self. Journal of personality and social psychology, 113(2), 185. Cavallaro, R. M., Church, M. H., Giddens, B. J., & Rivera, G. N. (2025). Complex effects of awe on meaning in life and true self-knowledge. Self and Identity, 24(3), 189-216. Chen, S. K., & Mongrain, M. (2021). Awe and the interconnected self. The Journal of Positive Psychology, 16(6), 770-778. Edwards, M. E., Mendenhall, K., Sanders, C., & King, L. A. (2026). Small but still significant: Awe and the self. Personality and Social Psychology Bulletin, 52(9), 3032-3045. Furman, R. (2003). Poetry therapy and existential practice. The Arts in Psychotherapy, 30(4), 195-200. Gao, R. (2026). From awe to pro-environmental behavior-a meta-analysis of the relationship between awe and pro-environmental behavior. The Journal of Positive Psychology, 21(5), 958-970. Goldy, S. P., Jones, N. M., & Piff, P. K. (2022). The social effects of an awesome solar eclipse. Psychological Science, 33(9), 1452-1462. Hendricks, P. S. (2018). Awe: a putative mechanism underlying the effects of classic psychedelic-assisted psychotherapy. International Review of Psychiatry, 30(4), 331-342. Jiang, T., Hicks, J. A., Yuan, W., Yin, Y., Needy, L., & Vess, M. (2024). The unique nature and psychosocial implications of awe. Nature Reviews Psychology, 3(7), 475-488. Jiang, T., & Sedikides, C. (2022). Awe motivates authentic-self pursuit via self-transcendence: Implications for prosociality. Journal of personality and social psychology, 123(3), 576. Keltner, D., & Haidt, J. (2003). Approaching awe, a moral, spiritual, and aesthetic emotion. Cognition and emotion, 17(2), 297-314. Kim, Y., Nusbaum, H. C., & Yang, F. (2023). Going beyond ourselves: The role of self-transcendent experiences in wisdom. Cognition and Emotion, 37(1), 98-116. Li, R., Hou, Z., Zhang, C., Xu, Q., & Nie, A. (2024). A meta-analysis examining the relationship between awe and prosocial behavior. Current Psychology, 43(29), 24702-24711. Lucht, A., & van Schie, H. T. (2024). The evolutionary function of awe: A review and integrated model of seven theoretical perspectives. Emotion Review, 16(1), 46-63. Maslow, A. H. (1964). Religions, values, and peak-experiences. Columbus, OH: Ohio State University Press. McMann, G. B. (2025). Role of Vastness and Accommodation of Awe in Personal Change and Growth Grand Canyon University]. Perez, K. A., Lench, H. C., Thompson, C. G., & North, S. (2023). Experimental elicitations of awe: A meta-analysis. Cognition and Emotion, 37(1), 18-33. Piff, P. K., Dietze, P., Feinberg, M., Stancato, D. M., & Keltner, D. (2015). Awe, the small self, and prosocial behavior. Journal of personality and social psychology, 108(6), 883. Piff, P. K., Singhal, I., & Bai, Y. (2025). Bridging me to we: Awe is a conduit to cohesive collectives. Current Opinion in Psychology, 62, 101979. Rivera, G. N., Vess, M., Hicks, J. A., & Routledge, C. (2020). Awe and meaning: Elucidating complex effects of awe experiences on meaning in life. European Journal of Social Psychology, 50(2), 392-405. Shiota, M. N., Keltner, D., & Mossman, A. (2007). The nature of awe: Elicitors, appraisals, and effects on self-concept. Cognition and Emotion, 21(5), 944-963. Shukla, S. (2021). Indian classical music, awe, and healing. The Humanistic Psychologist, 49(3), 355. Stellar, J. E., Gordon, A., Anderson, C. L., Piff, P. K., McNeil, G. D., & Keltner, D. (2018). Awe and humility. Journal of personality and social psychology, 114(2), 258. Sturm, V. E., Datta, S., Roy, A. R., Sible, I. J., Kosik, E. L., Veziris, C. R., Chow, T. E., Morris, N. A., Neuhaus, J., & Kramer, J. H. (2022). Big smile, small self: Awe walks promote prosocial positive emotions in older adults. Emotion, 22(5), 1044. Tanhan, F., DENİZ, M., & Akgün, G. (2022). Awe Therapy: Fundamentals, Formulation and Usage Area Awe Terapi: Temelleri, Formülasyonu ve Kullanım Alanları. Turkish Psychological Counseling and Guidance Journal, 12(65). Tanhan, F., Deniz, M. E., & Akgün, G. E. (2022). Awe Therapy: Fundamentals, formulation and usage area. Turkish Psychological Counseling and Guidance Journal, 12(65), 300-319. Yaden, D. B., Iwry, J., Slack, K. J., Eichstaedt, J. C., Zhao, Y., Vaillant, G. E., & Newberg, A. B. (2016). The overview effect: Awe and self-transcendent experience in space flight. Psychology of Consciousness: Theory, Research, and Practice, 3(1), 1. Yalom, I. D. (2008). Staring at the sun: Overcoming the terror of death. The Humanistic Psychologist, 36(3-4), 283. }} ==External links== * [https://self-transcendence.org/theory Self-transendence] (Self-transcendence.org) * [https://fitmind.org/ Ancient wisdom + modern science] (Fitmind.org) * [https://therapyintampa.com/the-neuroscience-of-awe-how-shared-wonder-supports-mental-health/ The Neuroscience of Awe: How Shared Wonder Supports Mental Health] (North Star Therapy) * Ted Talk Neuroscience video [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Awe]] [[Category:Motivation and emotion/Book/Self]] 8s710h6rkla3bbzwnqacczliahfnbyw 2842522 2842521 2026-10-25T03:41:17Z Amirrorslens 3106278 inserting adapted table from Keltner and Haidt paper 2842522 wikitext text/x-wiki {{title|Awe and the diminished self:<br>How does awe diminish the self and how can this be applied?}} __TOC__ ==Overview== {{RoundBoxTop|theme=2}} [[File:NASA-HS201427a-HubbleUltraDeepField2014-20140603.jpg|right|thumb|200px|'''Figure 1'''. NASA Hubble Ultra Deep Field]] ; Scenario Leo walks into the desert late at night and is taken aback at the night sky above him. Being alone in the vast and flat plane lets him really perceive the vastness of the stars and leaves him feeling very present and aware of his smallness in comparison. This overwhelming feeling leaves Leo seeking connection with others, and a new found sense of purpose. * [Author Note: REPLACE WITH A REAL CASE STUDY - such as near death experience (The Good Place coded) (Might modify scenario to incude this aspect similar to the purple feature box on [[Motivation and emotion/Book/2025/Motivation for using AI companions|this page]])] {{RoundBoxBottom}} * * ''KEY PIECE OF LITERATURE: Keltner and Haidt: Awe 2003'' *''Jean Piaget - Cognitive Development'' * Awe is considered a [[wikipedia:Peak_experience|peak experience]] according to Maslow [x] {{ic|Also consider awe as an emotion}}, (see [[Motivation and emotion/Book/2017/Awe and well-being|Awe and well-being]]) ** Experiencing awe = indicator of reaching highest level of self-actualisation and self-transcendence) * Awe can have positive individual and community effects (pro-social behaviour) {{ic|The topic is not just about awe - it is about the effect of awe on self-perception}} {{RoundBoxTop|theme=3}} '''Focus questions''' * What is awe as an emotion? * What is humanistic psychology and theory of needs? * What is the role of awe in Maslow's peak experiences? * How can we apply awe as peak experiences for individual good? * How does the experience of awe diminish the self? * What is the effect of a diminished sense of self? * How can awe be integrated into modern life? i.e. How can we produce the conditions for everyday and intentional moments of awe?{{RoundBoxBottom}} == Awe as an emotion == [[File:Plutchik-wheel.svg|thumb|Figure 2: Plutchik's wheel of emotions, where awe is modelled as a combination of suprise and fear.]] [[wikipedia:Awe|Awe]] is an emotion generally described as [insert synonyms]. In Robert Plutchik's wheel of emotion as seen in Figure 2, awe is modelled as a combination of suprise and fear (see [[Motivation and emotion/Book/2016/Awe as an emotion|Awe as an emotion]]). ''- KEY REFERENCE: Keltner and Haidt, 2003:'' * ''Additional appraisals account for variation in the hedoic (pleasure) tone of awe experiences'' ** ''threat, beauty, exceptional ability, virtue, and the supernatural'' ** ''%% turn this into a table %%'' ** ''%% should i do a table of definitions %% what other learnings features are there'' - %% we use the experience of awe to define the eliciting emotion . perception of something vast -> triggers awe -> requires need for accomodation In the prototypical perspectice of awe offered by Keltner and Haidt, (2003), two features are central to the experience of awe as an emotion; perceived vastness, and a following need for accommodation. ==== Vastness ==== Vastness refers to experiences that are perceived as being much larger than the self or their ordinary frame of reference. (Keltner and Haidt, 2003). - power Obscurity is another propertry that produces the 'sublime experience' (Burke, 1990, as cited in Keltner and Hiadt., 2014). * "Although the term “vastness” implies great physical size, in this usage “vast” describes any stimulus that challenges one's accustomed frame of reference in some dimension ** A stimulus may convey vastness in physical space, in time, in number, in complexity of detail, in ability, even in volume of human experience. *** For example, one may experience a sense of vastness in a mathematical equation, not because the equation is literally long, but because of the vast number of observed physical processes it is able to explain and predict. ** What is critical is that the stimulus dramatically expands the observer's usual frame of reference in some dimension or domain." (Shiota et al., 2007) ===== Physical vastness ===== ''[Author's Note: Nature, Space, might be wise to mention depth perception studies here. Why particuarly these things. Work in thesis on space video here.]'' * The overview effect = witnessing Earth from the atmosphere = life-altering degree of awe -> transcendent -> complete breakdown of perception of borders and countries = feeling of connectedness (Yaden et al., 2016) * Eclipses: study that used twitter data (over 2 million users) -> individuals exhibited more awe and expressed less self focused and more pro social, affiliative, humble and collective language * Witnessing natural disasters may induce a state of negatively valenced awe (Chaudhury et al., 2022; Piff et al., 2015) ** Link to heading section 'Effect of awe on the self' -> threat-based awe (Keltner and Haidt, 2003). ===== Conceptual vastness ===== Witnessing other people or learning things, coming of age moments, witnessing things for the first time theories here. Music and virtues act of skills. * Acts of Virtue * Acts of skills * Learning things ** A stimulus may convey vastness in physical space, in time, in number, in complexity of detail, in ability, even in volume of human experience. *** For example, one may experience a sense of vastness in a mathematical equation, not because the equation is literally long, but because of the vast number of observed physical processes it is able to explain and predict. ** What is critical is that the stimulus dramatically expands the observer's usual frame of reference in some dimension or domain." (Shiota et al., 2007) == Need for accommodation == * Awe is defined by Keltner and Haidt as an inability to assimilate an experience into current mental structures. (Keltner and Haidt, 2003) * Accommodation is one of the two ongoing processes (assimilation being the other) in [[wikipedia:Piaget's_theory_of_cognitive_development|Piaget's theory of cognitive development]], where existing [[wikipedia:Schema_(psychology)|schemas]] are modified, or even created to deal with a new event or experience (Jean Piaget, {find specific reference}). ** See also: [[wikipedia:Schema_(psychology)|Schemas]] * ''[Authors Note: Insert table of epistimec emotions such as wonder, elevation, sublime, admiration (Keltner and Haidt), not sure of relevance or format]'' ''- [Authors Note: unsure whether I should spend a sub section on schemas or not, or how much explanation should be done for these sorts of fundamental but still unfamiliar to reader concepts'' - Discussion of what it means to require the need for knowledge structures to be updated. {| class="wikitable" |+ ! colspan="8" |Keltner and Haidt's prototype approach to awe and related states |- ! ! colspan="2" |Central features ! colspan="5" |Peripheral or "flavouring" features |- !Eliciting Situations !Vastness !Accommodation !Threat !Beauty !Ability !Virtue !Supernatural |- |Prototype: Powerful leader |X |X |? | | | | |- |Encounter with God |X |X |? |? | |X |X |- |Great skill (Admiration^a) | |X | | |X | | |- |Great virtue (Elevation^a) | |X | | | |X | |- |Tornado |X |X |X |? | | |? |- |Grand vista |X |X | |X | | | |- |Cathedral |X |X | |X |X | |? |- |Awe-inspiring music |X |X | |X |X | |? |- |A beautiful painting (Aesthetic pleasure^a) | | | |X |? | | |- |Grand theory |X |X | |? | | | |- |Seeing an object levitate (Uncanny^a) | |X |? | | | |X |} == The role of awe in Maslow's peak experiences == - {| class="wikitable" |+ !Features of peak experience !Example of study |- |disorientation in time and space | |- |ego transcendece & self-forgetfulness |ego death |- |self-forgetfulness |meditation? |- |perception that the world is good, beautiful and desirable |is there a study that supports this resulting notion, |- |feeling passive, receptive and humble |non-attachment?, radical acceptable therapy, meditation/mindfulness |- |sense that polarities and dichotomies have been transcended or resolved |no borders or boundaries, the out of orbit // observer effect |- |feelings of being lucky, fortunate, or graced |gratitude practice |} === Failures to accommodate === * Awe is more specifically defined in (Shiota et al. 2007) as an emotional response to perceptually vast stimuli that overwhelm current mental structures, yet facilitate ''attempts'' at accommodation. ** Segue intro into failures to accommodate, threat-based awe, etc ==== Fear ==== * Keltner and Haidt (2003) argued that awe is situated on the boundary of fear * Threat-based awe (Keltner and Haidt) - flavour of the stimulus * See Nihilism ==== Trauma ==== [x] * failure to accommodate threat-based awe experienes can result in trauma ====== <s>Self</s> ====== * <s>Self in philosophy = individual's own being, knowledge and values (and relationship between these attributes ##CUT##</s> ** <s>self is distingushable from others</s> *** <s>[Authors Note: have a section later in connectedness how when the self is diminished, the concept of otherness is also diminished -> oneness]</s> === Self-perception === [[wikipedia:Self-concept|Self-concept]] = made up of one's self-schemas *possible selves ===== Ego-death ===== * small self -> construct that in the extreme is analogous to those of unitive experience and ego dissolution (diminishing of self) (Hendricks, 2018) * See also [[wikipedia:Ego_death|Ego death]] = complete loss of subjective self concept * ''%% MOVE TO PSYCHEDLICS SECTION OR MOVE PSYCHEDELICS HERE AND REFERENCE HEADING IN THAT PARAGRAPH %%'' ==== Psychedelics ==== ''[Author's Note: I feel psychedelics needs its own subsection here as it is particuarly relevant to my applications, and provides a good segue to need for accommodations.]'' * psychedielic-occassioned mystical experience is characterised by profound awe ** small self -> construct that in the extreme is analogous to those of unitive experience and ego dissolution (diminishing of self) (Hendricks, 2018) * psychedelics have been shown to produce enduring positive change across a range of psychological and behavioural outcomes (Hendricks, 2018) via ''(Griffiths et al., 2018; Griffiths, Richards, Johnson, McCann, & Jesse, 2008, Griffiths et al., 2011; Griffiths, Richards, McCann, & Jesse, 2006; Lebedev et al., 2016; MacLean, Johnson, & Griffiths, 2011)'' as well as treatment of depression. === Oneness / Seperation === See Buddhism, see psychedelics, see Connectedness. [''Author's Note: unsure if this should be it's own section, it feels like it should be in the affect of awe on the self section, after speaking on the feeling of connectedness].'' == Effect of awe on the self == ''[Authors Note: This is where we will have a short section on the self (what it means, collection of experiences and stories we tell ourselves, superceding that, buddhist definitions of the self and awareness and appreciating life as tenets, lojong etc.)]'' === Key feelings and effects === * Awe diminishes the sense of self and shifts attention away from individual interests and concerns (Bai et al., 2017) * Differences among individuas for experiencing awe: ** higher in openness tend to experience greater levels of awe (SK Chen, 2021) -> Shiota et al., 2006; Silvia et al., 2015 ** study found that individuals from higher classes reported less frequently experiencing awe compared to individuals from lower social classes (Piff & Moskowitz, 2017). [''Author's Note: I really like this fact and want to include it somewhere but not sure where to include].'' ** not only religious but also non-religious individuals report experiencing awe; *** however, religious onementions explicitly religious or life-or-death events while non-religious one talks about the wonder of nature (Preston & Shin, 2017). **** simulation of viewing Earth from the space, individuals who experienced awe were non-religious (Reinerman-Jones, Sollins, Gallagher, & Janz, 2013). {| class="wikitable" |+ !Key feelings and effects ! |- |Smallness | |- |Profoundness | |- |Connectedness | |- |Perception of the self | |- |Pro-social behaviour (b) | |} ==== Smallness ==== * study found that smallness was experienced but not insignificance (Edwards, et al. 2025) ** awe shrinks the self without making the individual feel insignificant ==== Profoundness ==== * [x] discuss significance of emotion * the present moment ** forced presence *time dilation (what article was that??) ==== Connectedness ==== * seeking connection after feelings of smallness * higher in openness tend to experience greater levels of awe (SK Chen, 2021) -> Shiota et al., 2006; Silvia et al., 2015 ===== Perception of the self ===== * presenting more balanced view of personal strengths and weaknesses * increased acknowledgment of outside forces in one's accomplishments (Stellar et al., 2018) * epistemic humility = measures of wisdom and moral character * positive self-transcendent experiences can be utilised as driving force for better understanding one's decision and perspectives impact others (Kim et al., 2023) === Behavioural changes === * Link back to need for accommodation and changes [x]. Link back to motivation content [x] ==== Pro-social behaviour ==== [[wikipedia:Prosocial_behavior|Prosocial behaviour]] refers to social behaviours that benefit everyone. * How does this relate back to the need for accommodation? [x] === Failures to accommodate === * Awe is more specifically defined in (Shiota et al. 2007) as an emotional response to perceptually vast stimuli that overwhelm current mental structures, yet facilitate ''attempts'' at accommodation. ** Segue intro into failures to accommodate, threat-based awe, etc ==== Fear ==== * Keltner and Haidt (2003) argued that awe is situated on the boundary of fear * Threat-based awe (Keltner and Haidt) - flavour of the stimulus * See Nihilism ==== Trauma ==== [x] * failure to accommodate threat-based awe experienes can result in trauma == Current applications == * experiencing awe can be regulatory, used for finding balance, and coping with stress (Chen & Mongrain, 2020) * (Rudd et al. 2012) found evidence that experience awe may alletive time scarcity feeling in modern life ''[Authors Note:'' (''link back to appraisal from K&H of time distortion)]'' === Wellbeing === ''[Author's Note: find additional references for five processes and explain in paragraph]'' * experiences in nature, moved by mysic, psychedelics, spiritual contemplation promote mental health? via awe (Monroy and Keltner, 2023) ** awe engages five processes *** shifts in neurophysiology *** diminished focus on self *** increased prosocail realtionality *** greater social integration *** heightened sense of meaning (Monroy and Keltner, 2023) === Therapies === *intro [x] ==== Existential Therapy ==== * death is a main consideration in existential thought (Yalom, 2008) * death awareness reults in anxiety which can be experienced in two different ways: ** a) an overwhelming anxiety (dread or depression) resulting in various dysfunctional behaviors and moodstates associated with the choice of escape from it ** b) an energizing source that leads to live life fully as much as possible. (Furman, 2003) ** awe can be considered as a possible way of dealing with death anxiety (Schenider, 2021) ===== Gestalt ===== * here and now * reliving past experiences in the present moment to reexperience and attempt to accommodate * (Corey, 2009) ==== Psychedelic therapies ==== * evidence [x] * how related to awe [x] == Conclusion == * Awe diminishes the self via the small self affect etc. * Awe requires perceived vastness (stimulus from nature, or learning or challenge to mental schema), and a need for accommodation * Diminishing the self results in pro-social behaviour * Awe and its resulting experience of self-transcendent emotions can be utilised for therapies and healing * Awe promotes wellbeing and underscores other positive stimuli such as psychedelics or spiritual contemplation and learning * Focus Question: [What is awe as an emotion and experience?] ... Prelimary answer: [awe is made up of perceived vastness and a need for an accommodation that often leaves individuals with a smaller sense of self, connectedness and purpose/increased meaning] * Focus Question: [How does the experience of awe diminish the self?] ... Prelimary answer: [awe diminishes the self mostly through the small-self construct, as well as diminishing the ego and seperatedness feeling] * Focus Question: [What is the effect of a diminished sense of self?] ... Prelimary answer: [diminished sense of self results in prosocial behaviour (broad) need to specify what behaviours] * Focus Question: [How can awe be integrated into modern life? i.e. How can we trigger or experience everyday moments of awe?] ** Prelimary answer: [experiencing awe can be regulatory, used for finding balance, and coping with stress (Chen & Mongrain, 2020) ** (Rudd et al. 2012) found evidence that experience awe may alletive time scarcity feeling in modern life ''[Authors Note:'' (''link back to appraisal from K&H of time distortion)]]'' * Focus Question: [How can the experience of awe be applied for social and individual good?] ... Prelimary answer: [therapy, wellbeing, psycotherapy, deliberating triggering awe experiences for healing] * ''[Authors Note: summarise topic, significance, how psych science contributes to understanding and addressing topic, practical take home messages, 150-330 words, mirror in [[#Overview|Overview]]]'' ==See also== * [[wikipedia:Awe|Awe]] (Wikipedia) * [[Motivation and emotion/Book/2016/Awe as an emotion|Awe as an emotion]] (Book chapter, 2016) * [[wikipedia:Ego_death|Ego death]] (Wikipedia) * [[wikipedia:Piaget's_theory_of_cognitive_development|Piaget's theory of cognitive development]] (Wikipedia) * [[wikipedia:Prosocial_behavior|Prosocial behaviour]] (Wikipedia) * [[wikipedia:Schema_(psychology)|Schemas]] (Wikipedia) * [[wikipedia:Self|Self]] (Wikipedia) * [[wikipedia:Self|Self-concept]] (Wikipedia) ==References== {{Hanging indent|1= Arvidson, P. S. (2022). Reverent Awe and the Field of Consciousness. Human Studies, 45(3), 397-416. Bai, Y., Maruskin, L. A., Chen, S., Gordon, A. M., Stellar, J. E., McNeil, G. D., Peng, K., & Keltner, D. (2017). Awe, the diminished self, and collective engagement: Universals and cultural variations in the small self. Journal of personality and social psychology, 113(2), 185. Cavallaro, R. M., Church, M. H., Giddens, B. J., & Rivera, G. N. (2025). Complex effects of awe on meaning in life and true self-knowledge. Self and Identity, 24(3), 189-216. Chen, S. K., & Mongrain, M. (2021). Awe and the interconnected self. The Journal of Positive Psychology, 16(6), 770-778. Edwards, M. E., Mendenhall, K., Sanders, C., & King, L. A. (2026). Small but still significant: Awe and the self. Personality and Social Psychology Bulletin, 52(9), 3032-3045. Furman, R. (2003). Poetry therapy and existential practice. The Arts in Psychotherapy, 30(4), 195-200. Gao, R. (2026). From awe to pro-environmental behavior-a meta-analysis of the relationship between awe and pro-environmental behavior. The Journal of Positive Psychology, 21(5), 958-970. Goldy, S. P., Jones, N. M., & Piff, P. K. (2022). The social effects of an awesome solar eclipse. Psychological Science, 33(9), 1452-1462. Hendricks, P. S. (2018). Awe: a putative mechanism underlying the effects of classic psychedelic-assisted psychotherapy. International Review of Psychiatry, 30(4), 331-342. Jiang, T., Hicks, J. A., Yuan, W., Yin, Y., Needy, L., & Vess, M. (2024). The unique nature and psychosocial implications of awe. Nature Reviews Psychology, 3(7), 475-488. Jiang, T., & Sedikides, C. (2022). Awe motivates authentic-self pursuit via self-transcendence: Implications for prosociality. Journal of personality and social psychology, 123(3), 576. Keltner, D., & Haidt, J. (2003). Approaching awe, a moral, spiritual, and aesthetic emotion. Cognition and emotion, 17(2), 297-314. Kim, Y., Nusbaum, H. C., & Yang, F. (2023). Going beyond ourselves: The role of self-transcendent experiences in wisdom. Cognition and Emotion, 37(1), 98-116. Li, R., Hou, Z., Zhang, C., Xu, Q., & Nie, A. (2024). A meta-analysis examining the relationship between awe and prosocial behavior. Current Psychology, 43(29), 24702-24711. Lucht, A., & van Schie, H. T. (2024). The evolutionary function of awe: A review and integrated model of seven theoretical perspectives. Emotion Review, 16(1), 46-63. Maslow, A. H. (1964). Religions, values, and peak-experiences. Columbus, OH: Ohio State University Press. McMann, G. B. (2025). Role of Vastness and Accommodation of Awe in Personal Change and Growth Grand Canyon University]. Perez, K. A., Lench, H. C., Thompson, C. G., & North, S. (2023). Experimental elicitations of awe: A meta-analysis. Cognition and Emotion, 37(1), 18-33. Piff, P. K., Dietze, P., Feinberg, M., Stancato, D. M., & Keltner, D. (2015). Awe, the small self, and prosocial behavior. Journal of personality and social psychology, 108(6), 883. Piff, P. K., Singhal, I., & Bai, Y. (2025). Bridging me to we: Awe is a conduit to cohesive collectives. Current Opinion in Psychology, 62, 101979. Rivera, G. N., Vess, M., Hicks, J. A., & Routledge, C. (2020). Awe and meaning: Elucidating complex effects of awe experiences on meaning in life. European Journal of Social Psychology, 50(2), 392-405. Shiota, M. N., Keltner, D., & Mossman, A. (2007). The nature of awe: Elicitors, appraisals, and effects on self-concept. Cognition and Emotion, 21(5), 944-963. Shukla, S. (2021). Indian classical music, awe, and healing. The Humanistic Psychologist, 49(3), 355. Stellar, J. E., Gordon, A., Anderson, C. L., Piff, P. K., McNeil, G. D., & Keltner, D. (2018). Awe and humility. Journal of personality and social psychology, 114(2), 258. Sturm, V. E., Datta, S., Roy, A. R., Sible, I. J., Kosik, E. L., Veziris, C. R., Chow, T. E., Morris, N. A., Neuhaus, J., & Kramer, J. H. (2022). Big smile, small self: Awe walks promote prosocial positive emotions in older adults. Emotion, 22(5), 1044. Tanhan, F., DENİZ, M., & Akgün, G. (2022). Awe Therapy: Fundamentals, Formulation and Usage Area Awe Terapi: Temelleri, Formülasyonu ve Kullanım Alanları. Turkish Psychological Counseling and Guidance Journal, 12(65). Tanhan, F., Deniz, M. E., & Akgün, G. E. (2022). Awe Therapy: Fundamentals, formulation and usage area. Turkish Psychological Counseling and Guidance Journal, 12(65), 300-319. Yaden, D. B., Iwry, J., Slack, K. J., Eichstaedt, J. C., Zhao, Y., Vaillant, G. E., & Newberg, A. B. (2016). The overview effect: Awe and self-transcendent experience in space flight. Psychology of Consciousness: Theory, Research, and Practice, 3(1), 1. Yalom, I. D. (2008). Staring at the sun: Overcoming the terror of death. The Humanistic Psychologist, 36(3-4), 283. }} ==External links== * [https://self-transcendence.org/theory Self-transendence] (Self-transcendence.org) * [https://fitmind.org/ Ancient wisdom + modern science] (Fitmind.org) * [https://therapyintampa.com/the-neuroscience-of-awe-how-shared-wonder-supports-mental-health/ The Neuroscience of Awe: How Shared Wonder Supports Mental Health] (North Star Therapy) * Ted Talk Neuroscience video [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Awe]] [[Category:Motivation and emotion/Book/Self]] tnq6q7o6073hy3utt7amqkla87ked4z 2842526 2842522 2026-10-25T05:26:53Z Amirrorslens 3106278 2842526 wikitext text/x-wiki {{title|Awe and the diminished self:<br>How does awe diminish the self and how can this be applied?}} __TOC__ ==Overview== {{RoundBoxTop|theme=2}} [[File:NASA-HS201427a-HubbleUltraDeepField2014-20140603.jpg|right|thumb|200px|'''Figure 1'''. NASA Hubble Ultra Deep Field]] ; Scenario Leo walks into the desert late at night and is taken aback at the night sky above him. Being alone in the vast and flat plane lets him really perceive the vastness of the stars and leaves him feeling very present and aware of his smallness in comparison. This overwhelming feeling leaves Leo seeking connection with others, and a new found sense of purpose. * [Author Note: REPLACE WITH A REAL CASE STUDY - such as near death experience (The Good Place coded) (Might modify scenario to incude this aspect similar to the purple feature box on [[Motivation and emotion/Book/2025/Motivation for using AI companions|this page]])] {{RoundBoxBottom}} * * ''KEY PIECE OF LITERATURE: Keltner and Haidt: Awe 2003'' *''Jean Piaget - Cognitive Development'' * Awe is considered a [[wikipedia:Peak_experience|peak experience]] according to Maslow [x] {{ic|Also consider awe as an emotion}}, (see [[Motivation and emotion/Book/2017/Awe and well-being|Awe and well-being]]) ** Experiencing awe = indicator of reaching highest level of self-actualisation and self-transcendence) * Awe can have positive individual and community effects (pro-social behaviour) {{ic|The topic is not just about awe - it is about the effect of awe on self-perception}} {{RoundBoxTop|theme=3}} '''Focus questions''' * What is awe as an emotion? * What is humanistic psychology and theory of needs? * What is the role of awe in Maslow's peak experiences? * How can we apply awe as peak experiences for individual good? * How does the experience of awe diminish the self? * What is the effect of a diminished sense of self? * How can awe be integrated into modern life? i.e. How can we produce the conditions for everyday and intentional moments of awe?{{RoundBoxBottom}} == Awe as an emotion == [[File:Plutchik-wheel.svg|thumb|Figure 2: Plutchik's wheel of emotions, where awe is modelled as a combination of suprise and fear.]] [[wikipedia:Awe|Awe]] is an emotion generally described as [''insert'' ''synonyms'']. In Robert Plutchik's wheel of emotion as seen in Figure 2, awe is modelled as a combination of suprise and fear (see [[Motivation and emotion/Book/2016/Awe as an emotion|Awe as an emotion]]). %%''sentence about distinguishing from other emotions'' ==== ''KEY REFERENCE: Keltner and Haidt, 2003 / A prototype approach to distinguising awe from other emotions %%change heading structure here'' ==== In the prototypical perspectice of awe offered by Keltner and Haidt, (2003), two features are central to the experience of awe as an emotion; perceived vastness, and a following need for accommodation. - %% we use the experience of awe to define the eliciting emotion . perception of something vast -> triggers awe -> requires need for accomodation === Vastness === Vastness refers to experiences that are perceived as being much larger than the self or their ordinary frame of reference. (Keltner and Haidt, 2003). - power Obscurity is another propertry that produces the 'sublime experience' (Burke, 1990, as cited in Keltner and Hiadt., 2014). * "Although the term “vastness” implies great physical size, in this usage “vast” describes any stimulus that challenges one's accustomed frame of reference in some dimension ** A stimulus may convey vastness in physical space, in time, in number, in complexity of detail, in ability, even in volume of human experience. *** For example, one may experience a sense of vastness in a mathematical equation, not because the equation is literally long, but because of the vast number of observed physical processes it is able to explain and predict. ** What is critical is that the stimulus dramatically expands the observer's usual frame of reference in some dimension or domain." (Shiota et al., 2007) ===== Physical vastness ===== ''[Author's Note: Nature, Space, might be wise to mention depth perception studies here. Why particuarly these things. Work in thesis on space video here.]'' * The overview effect = witnessing Earth from the atmosphere = life-altering degree of awe -> transcendent -> complete breakdown of perception of borders and countries = feeling of connectedness (Yaden et al., 2016) * Eclipses: study that used twitter data (over 2 million users) -> individuals exhibited more awe and expressed less self focused and more pro social, affiliative, humble and collective language * Witnessing natural disasters may induce a state of negatively valenced awe (Chaudhury et al., 2022; Piff et al., 2015) ** Link to heading section 'Effect of awe on the self' -> threat-based awe (Keltner and Haidt, 2003). ===== Conceptual vastness ===== Witnessing other people or learning things, coming of age moments, witnessing things for the first time theories here. Music and virtues act of skills. * Acts of Virtue * Acts of skills * Learning things ** A stimulus may convey vastness in physical space, in time, in number, in complexity of detail, in ability, even in volume of human experience. *** For example, one may experience a sense of vastness in a mathematical equation, not because the equation is literally long, but because of the vast number of observed physical processes it is able to explain and predict. ** What is critical is that the stimulus dramatically expands the observer's usual frame of reference in some dimension or domain." (Shiota et al., 2007) === Need for accommodation === * Awe is defined by Keltner and Haidt as an inability to assimilate an experience into current mental structures. (Keltner and Haidt, 2003) * Accommodation is one of the two ongoing processes (assimilation being the other) in [[wikipedia:Piaget's_theory_of_cognitive_development|Piaget's theory of cognitive development]], where existing [[wikipedia:Schema_(psychology)|schemas]] are modified, or even created to deal with a new event or experience (Jean Piaget, {find specific reference}). ** See also: [[wikipedia:Schema_(psychology)|Schemas]] * ''[Authors Note: Insert table of epistimec emotions such as wonder, elevation, sublime, admiration (Keltner and Haidt), not sure of relevance or format]'' ''- [Authors Note: unsure whether I should spend a sub section on schemas or not, or how much explanation should be done for these sorts of fundamental but still unfamiliar to reader concepts'' - Discussion of what it means to require the need for knowledge structures to be updated. '''Table 1.''' ''Adapted table from Keltner and Haidt's 2003 article that distguishes awe as an emotion from other related states and appraisals by identifying two central features of vastness and accommodation, as well as other "flavours". '''X''' denotes that the appraisal is usually made in this case. '''?''' denotes that the appraisal is sometimes made in the case and if it is made, it adds a flavour). ^'''a''' denotes states that are related to awe, but should not be labelled as awe.'' {| class="wikitable" |+ ! colspan="8" |Keltner and Haidt's prototype approach to awe and related states |- ! ! colspan="2" |Central features ! colspan="5" |Peripheral or "flavouring" features |- !Eliciting Situations !Vastness !Accommodation !Threat !Beauty !Ability !Virtue !Supernatural |- |Prototype: Powerful leader |'''X''' |'''X''' |? | | | | |- |Encounter with God |'''X''' |'''X''' |? |? | |'''X''' |'''X''' |- |Great skill (''Admiration^'''a''''') | |'''X''' | | |'''X''' | | |- |Great virtue (''Elevation^'''a''''') | |'''X''' | | | |'''X''' | |- |Tornado |'''X''' |'''X''' |'''X''' |? | | |? |- |Grand vista |'''X''' |'''X''' | |'''X''' | | | |- |Cathedral |'''X''' |'''X''' | |'''X''' |'''X''' | |? |- |Awe-inspiring music |'''X''' |'''X''' | |'''X''' |'''X''' | |? |- |A beautiful painting (''Aesthetic pleasure^'''a''''') | | | |'''X''' |? | | |- |Grand theory |'''X''' |'''X''' | |? | | | |- |Seeing an object levitate (''Uncanny^'''a''''') | |'''X''' |? | | | |'''X''' |} Could put a quiz in here and use the descriptions == The role of awe in Maslow's peak experiences == - {| class="wikitable" |+ !Features of peak experience !Example of study |- |disorientation in time and space | |- |ego transcendece & self-forgetfulness |ego death |- |self-forgetfulness |meditation? |- |perception that the world is good, beautiful and desirable |is there a study that supports this resulting notion, |- |feeling passive, receptive and humble |non-attachment?, radical acceptable therapy, meditation/mindfulness |- |sense that polarities and dichotomies have been transcended or resolved |no borders or boundaries, the out of orbit // observer effect |- |feelings of being lucky, fortunate, or graced |gratitude practice |} === Failures to accommodate === * Awe is more specifically defined in (Shiota et al. 2007) as an emotional response to perceptually vast stimuli that overwhelm current mental structures, yet facilitate ''attempts'' at accommodation. ** Segue intro into failures to accommodate, threat-based awe, etc ==== Fear ==== * Keltner and Haidt (2003) argued that awe is situated on the boundary of fear * Threat-based awe (Keltner and Haidt) - flavour of the stimulus * See Nihilism ==== Trauma ==== [x] * failure to accommodate threat-based awe experienes can result in trauma ====== <s>Self</s> ====== * <s>Self in philosophy = individual's own being, knowledge and values (and relationship between these attributes ##CUT##</s> ** <s>self is distingushable from others</s> *** <s>[Authors Note: have a section later in connectedness how when the self is diminished, the concept of otherness is also diminished -> oneness]</s> === Self-perception === [[wikipedia:Self-concept|Self-concept]] = made up of one's self-schemas *possible selves ===== Ego-death ===== * small self -> construct that in the extreme is analogous to those of unitive experience and ego dissolution (diminishing of self) (Hendricks, 2018) * See also [[wikipedia:Ego_death|Ego death]] = complete loss of subjective self concept * ''%% MOVE TO PSYCHEDLICS SECTION OR MOVE PSYCHEDELICS HERE AND REFERENCE HEADING IN THAT PARAGRAPH %%'' ==== Psychedelics ==== ''[Author's Note: I feel psychedelics needs its own subsection here as it is particuarly relevant to my applications, and provides a good segue to need for accommodations.]'' * psychedielic-occassioned mystical experience is characterised by profound awe ** small self -> construct that in the extreme is analogous to those of unitive experience and ego dissolution (diminishing of self) (Hendricks, 2018) * psychedelics have been shown to produce enduring positive change across a range of psychological and behavioural outcomes (Hendricks, 2018) via ''(Griffiths et al., 2018; Griffiths, Richards, Johnson, McCann, & Jesse, 2008, Griffiths et al., 2011; Griffiths, Richards, McCann, & Jesse, 2006; Lebedev et al., 2016; MacLean, Johnson, & Griffiths, 2011)'' as well as treatment of depression. === Oneness / Seperation === See Buddhism, see psychedelics, see Connectedness. [''Author's Note: unsure if this should be it's own section, it feels like it should be in the affect of awe on the self section, after speaking on the feeling of connectedness].'' == Effect of awe on the self == ''[Authors Note: This is where we will have a short section on the self (what it means, collection of experiences and stories we tell ourselves, superceding that, buddhist definitions of the self and awareness and appreciating life as tenets, lojong etc.)]'' === Key feelings and effects === * Awe diminishes the sense of self and shifts attention away from individual interests and concerns (Bai et al., 2017) * Differences among individuas for experiencing awe: ** higher in openness tend to experience greater levels of awe (SK Chen, 2021) -> Shiota et al., 2006; Silvia et al., 2015 '''CRITICISMS / DISUCSSIONS ON AWE Blockages to / some people more likely to experience than others (class, personality, more likely to cultivate interntionally)''' In terms of the state of the research, Keltner and Haidt (2003) suggest in their analysis that one would expect people in upper echelons of societies to experience less awe than those in lower stratas. This was supported by Piff & Moskowitz, (2017) that found that individuals from higher classes reported less frequently experiencing awe compared to individuals from lower social classes. ** not only religious but also non-religious individuals report experiencing awe; *** however, religious onementions explicitly religious or life-or-death events while non-religious one talks about the wonder of nature (Preston & Shin, 2017). **** simulation of viewing Earth from the space, individuals who experienced awe were non-religious (Reinerman-Jones, Sollins, Gallagher, & Janz, 2013). {| class="wikitable" |+ !Key feelings and effects ! |- |Smallness | |- |Profoundness | |- |Connectedness | |- |Perception of the self | |- |Pro-social behaviour (b) | |} ==== Smallness ==== * study found that smallness was experienced but not insignificance (Edwards, et al. 2025) ** awe shrinks the self without making the individual feel insignificant ==== Profoundness ==== * [x] discuss significance of emotion * the present moment ** forced presence *time dilation (what article was that??) ==== Connectedness ==== * seeking connection after feelings of smallness * higher in openness tend to experience greater levels of awe (SK Chen, 2021) -> Shiota et al., 2006; Silvia et al., 2015 ===== Perception of the self ===== * presenting more balanced view of personal strengths and weaknesses * increased acknowledgment of outside forces in one's accomplishments (Stellar et al., 2018) * epistemic humility = measures of wisdom and moral character * positive self-transcendent experiences can be utilised as driving force for better understanding one's decision and perspectives impact others (Kim et al., 2023) === Behavioural changes === * Link back to need for accommodation and changes [x]. Link back to motivation content [x] ==== Pro-social behaviour ==== [[wikipedia:Prosocial_behavior|Prosocial behaviour]] refers to social behaviours that benefit everyone. * How does this relate back to the need for accommodation? [x] === Failures to accommodate === * Awe is more specifically defined in (Shiota et al. 2007) as an emotional response to perceptually vast stimuli that overwhelm current mental structures, yet facilitate ''attempts'' at accommodation. ** Segue intro into failures to accommodate, threat-based awe, etc ==== Fear ==== * Keltner and Haidt (2003) argued that awe is situated on the boundary of fear * Threat-based awe (Keltner and Haidt) - flavour of the stimulus * See Nihilism ==== Trauma ==== [x] * failure to accommodate threat-based awe experienes can result in trauma == Current applications == * experiencing awe can be regulatory, used for finding balance, and coping with stress (Chen & Mongrain, 2020) * (Rudd et al. 2012) found evidence that experience awe may alletive time scarcity feeling in modern life ''[Authors Note:'' (''link back to appraisal from K&H of time distortion)]'' === Wellbeing === ''[Author's Note: find additional references for five processes and explain in paragraph]'' * experiences in nature, moved by mysic, psychedelics, spiritual contemplation promote mental health? via awe (Monroy and Keltner, 2023) ** awe engages five processes *** shifts in neurophysiology *** diminished focus on self *** increased prosocail realtionality *** greater social integration *** heightened sense of meaning (Monroy and Keltner, 2023) === Therapies === *intro [x] ==== Existential Therapy ==== * death is a main consideration in existential thought (Yalom, 2008) * death awareness reults in anxiety which can be experienced in two different ways: ** a) an overwhelming anxiety (dread or depression) resulting in various dysfunctional behaviors and moodstates associated with the choice of escape from it ** b) an energizing source that leads to live life fully as much as possible. (Furman, 2003) ** awe can be considered as a possible way of dealing with death anxiety (Schenider, 2021) ===== Gestalt ===== * here and now * reliving past experiences in the present moment to reexperience and attempt to accommodate * (Corey, 2009) ==== Psychedelic therapies ==== * evidence [x] * how related to awe [x] == Conclusion == * Awe diminishes the self via the small self affect etc. * Awe requires perceived vastness (stimulus from nature, or learning or challenge to mental schema), and a need for accommodation * Diminishing the self results in pro-social behaviour * Awe and its resulting experience of self-transcendent emotions can be utilised for therapies and healing * Awe promotes wellbeing and underscores other positive stimuli such as psychedelics or spiritual contemplation and learning * Focus Question: [What is awe as an emotion and experience?] ... Prelimary answer: [awe is made up of perceived vastness and a need for an accommodation that often leaves individuals with a smaller sense of self, connectedness and purpose/increased meaning] * Focus Question: [How does the experience of awe diminish the self?] ... Prelimary answer: [awe diminishes the self mostly through the small-self construct, as well as diminishing the ego and seperatedness feeling] * Focus Question: [What is the effect of a diminished sense of self?] ... Prelimary answer: [diminished sense of self results in prosocial behaviour (broad) need to specify what behaviours] * Focus Question: [How can awe be integrated into modern life? i.e. How can we trigger or experience everyday moments of awe?] ** Prelimary answer: [experiencing awe can be regulatory, used for finding balance, and coping with stress (Chen & Mongrain, 2020) ** (Rudd et al. 2012) found evidence that experience awe may alletive time scarcity feeling in modern life ''[Authors Note:'' (''link back to appraisal from K&H of time distortion)]]'' * Focus Question: [How can the experience of awe be applied for social and individual good?] ... Prelimary answer: [therapy, wellbeing, psycotherapy, deliberating triggering awe experiences for healing] * ''[Authors Note: summarise topic, significance, how psych science contributes to understanding and addressing topic, practical take home messages, 150-330 words, mirror in [[#Overview|Overview]]]'' ==See also== * [[wikipedia:Awe|Awe]] (Wikipedia) * [[Motivation and emotion/Book/2016/Awe as an emotion|Awe as an emotion]] (Book chapter, 2016) * [[wikipedia:Ego_death|Ego death]] (Wikipedia) * [[wikipedia:Piaget's_theory_of_cognitive_development|Piaget's theory of cognitive development]] (Wikipedia) * [[wikipedia:Prosocial_behavior|Prosocial behaviour]] (Wikipedia) * [[wikipedia:Schema_(psychology)|Schemas]] (Wikipedia) * [[wikipedia:Self|Self]] (Wikipedia) * [[wikipedia:Self|Self-concept]] (Wikipedia) ==References== {{Hanging indent|1= Arvidson, P. S. (2022). Reverent Awe and the Field of Consciousness. Human Studies, 45(3), 397-416. Bai, Y., Maruskin, L. A., Chen, S., Gordon, A. M., Stellar, J. E., McNeil, G. D., Peng, K., & Keltner, D. (2017). Awe, the diminished self, and collective engagement: Universals and cultural variations in the small self. Journal of personality and social psychology, 113(2), 185. Cavallaro, R. M., Church, M. H., Giddens, B. J., & Rivera, G. N. (2025). Complex effects of awe on meaning in life and true self-knowledge. Self and Identity, 24(3), 189-216. Chen, S. K., & Mongrain, M. (2021). Awe and the interconnected self. The Journal of Positive Psychology, 16(6), 770-778. Edwards, M. E., Mendenhall, K., Sanders, C., & King, L. A. (2026). Small but still significant: Awe and the self. Personality and Social Psychology Bulletin, 52(9), 3032-3045. Furman, R. (2003). Poetry therapy and existential practice. The Arts in Psychotherapy, 30(4), 195-200. Gao, R. (2026). From awe to pro-environmental behavior-a meta-analysis of the relationship between awe and pro-environmental behavior. The Journal of Positive Psychology, 21(5), 958-970. Goldy, S. P., Jones, N. M., & Piff, P. K. (2022). The social effects of an awesome solar eclipse. Psychological Science, 33(9), 1452-1462. Hendricks, P. S. (2018). Awe: a putative mechanism underlying the effects of classic psychedelic-assisted psychotherapy. International Review of Psychiatry, 30(4), 331-342. Jiang, T., Hicks, J. A., Yuan, W., Yin, Y., Needy, L., & Vess, M. (2024). The unique nature and psychosocial implications of awe. Nature Reviews Psychology, 3(7), 475-488. Jiang, T., & Sedikides, C. (2022). Awe motivates authentic-self pursuit via self-transcendence: Implications for prosociality. Journal of personality and social psychology, 123(3), 576. Keltner, D., & Haidt, J. (2003). Approaching awe, a moral, spiritual, and aesthetic emotion. Cognition and emotion, 17(2), 297-314. Kim, Y., Nusbaum, H. C., & Yang, F. (2023). Going beyond ourselves: The role of self-transcendent experiences in wisdom. Cognition and Emotion, 37(1), 98-116. Li, R., Hou, Z., Zhang, C., Xu, Q., & Nie, A. (2024). A meta-analysis examining the relationship between awe and prosocial behavior. Current Psychology, 43(29), 24702-24711. Lucht, A., & van Schie, H. T. (2024). The evolutionary function of awe: A review and integrated model of seven theoretical perspectives. Emotion Review, 16(1), 46-63. Maslow, A. H. (1964). Religions, values, and peak-experiences. Columbus, OH: Ohio State University Press. McMann, G. B. (2025). Role of Vastness and Accommodation of Awe in Personal Change and Growth Grand Canyon University]. Perez, K. A., Lench, H. C., Thompson, C. G., & North, S. (2023). Experimental elicitations of awe: A meta-analysis. Cognition and Emotion, 37(1), 18-33. Piff, P. K., Dietze, P., Feinberg, M., Stancato, D. M., & Keltner, D. (2015). Awe, the small self, and prosocial behavior. Journal of personality and social psychology, 108(6), 883. Piff, P. K., Singhal, I., & Bai, Y. (2025). Bridging me to we: Awe is a conduit to cohesive collectives. Current Opinion in Psychology, 62, 101979. Rivera, G. N., Vess, M., Hicks, J. A., & Routledge, C. (2020). Awe and meaning: Elucidating complex effects of awe experiences on meaning in life. European Journal of Social Psychology, 50(2), 392-405. Shiota, M. N., Keltner, D., & Mossman, A. (2007). The nature of awe: Elicitors, appraisals, and effects on self-concept. Cognition and Emotion, 21(5), 944-963. Shukla, S. (2021). Indian classical music, awe, and healing. The Humanistic Psychologist, 49(3), 355. Stellar, J. E., Gordon, A., Anderson, C. L., Piff, P. K., McNeil, G. D., & Keltner, D. (2018). Awe and humility. Journal of personality and social psychology, 114(2), 258. Sturm, V. E., Datta, S., Roy, A. R., Sible, I. J., Kosik, E. L., Veziris, C. R., Chow, T. E., Morris, N. A., Neuhaus, J., & Kramer, J. H. (2022). Big smile, small self: Awe walks promote prosocial positive emotions in older adults. Emotion, 22(5), 1044. Tanhan, F., DENİZ, M., & Akgün, G. (2022). Awe Therapy: Fundamentals, Formulation and Usage Area Awe Terapi: Temelleri, Formülasyonu ve Kullanım Alanları. Turkish Psychological Counseling and Guidance Journal, 12(65). Tanhan, F., Deniz, M. E., & Akgün, G. E. (2022). Awe Therapy: Fundamentals, formulation and usage area. Turkish Psychological Counseling and Guidance Journal, 12(65), 300-319. Yaden, D. B., Iwry, J., Slack, K. J., Eichstaedt, J. C., Zhao, Y., Vaillant, G. E., & Newberg, A. B. (2016). The overview effect: Awe and self-transcendent experience in space flight. Psychology of Consciousness: Theory, Research, and Practice, 3(1), 1. Yalom, I. D. (2008). Staring at the sun: Overcoming the terror of death. The Humanistic Psychologist, 36(3-4), 283. }} ==External links== * [https://self-transcendence.org/theory Self-transendence] (Self-transcendence.org) * [https://fitmind.org/ Ancient wisdom + modern science] (Fitmind.org) * [https://therapyintampa.com/the-neuroscience-of-awe-how-shared-wonder-supports-mental-health/ The Neuroscience of Awe: How Shared Wonder Supports Mental Health] (North Star Therapy) * Ted Talk Neuroscience video [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Awe]] [[Category:Motivation and emotion/Book/Self]] 7tbuiwcw8a1ctl3cg52e9h0l2cqx9v8 1 331605 2842464 2839795 2026-10-24T22:45:34Z Jtneill 10242 Book chapter feedback 2842464 wikitext text/x-wiki == feedback/social contribution == Hi, [[User:U3275984|U3275984]], you are doing great. this looks amazing and you have done so well. [[User:Anne-Lyse Iran|Anne-Lyse Iran]] ([[User talk:Anne-Lyse Iran|discuss]] • [[Special:Contributions/Anne-Lyse Iran|contribs]]) 13:19, 27 August 2026 (UTC) <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]] |2= <!-- Headings --> # See earlier comment about [[#heading casing|heading casing]] <!-- Heading structure --> <!-- 1-level --> # Basic, [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development, perhaps using a 2-level structure (i.e., use subheadings) <!-- 3-level --> # Overly complicated[[Motivation and emotion/Assessment/Major project/Structure|3-level heading structure]] – consider simplifying # The headings focus on FL but not follower motivation <!-- Alignment with focus questions --> # Very good alignment between sub-title, focus questions, and heading structure, but there may be room for improvement (e.g., to better address follower motivation) |3= <!-- Overview--> # Scenario, image, evocative description of the problem/topic, and focus questions <!-- Scenario --> # A scenario or case study is presented in a feature box with an image at the start of this section <!-- Description --> # A clear description of the problem/topic is planned or presented <!-- Focus questions --> # Focus questions are aligned with sub-title and top-level headings # Focus question 2 is awkwardly worded and doesn't directly address the topic; consider rewriting this question |4= <!-- Key points--> <!-- Overall --> # Excellent – key points are well developed for each section # Excellent edit summaries <!-- Scope --> # The scope is excellent (i.e., not too little/narrow or too big/broad) # This topic lends itself to providing practical examples of the pros and cons of CL for employee motivation, as well as recommendations for leadership training and practices #<!-- Theory and research --> # Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples. <!-- Conclusion --> # Conclusion is well developed # What are the practical, take-home messages? (address the focus questions) |5= <!-- Figure --> # Relevant figure(s) presented, captioned, and cited <!-- Caption --> # Figure caption(s) provide(s) a clear, appropriately detailed description that is meaningfully connected with the main text <!-- Cite --> # Figure(s) are cited at least once in the main text <!-- Size --> # Consider increasing image size(s) to make them easier to view |6= <!-- Learning feature --> <!-- Interwiki links ---> # Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Scenarios/examples/case studies --> # Consider use of more scenarios/examples/case studies <!-- Quiz --> # Promising use of quiz question(s) # Place each quiz question in the most relevant section <!-- Tables --> # Also consider using tables to summarise key information |7= <!-- References --> <!-- Overall --> # Excellent <!-- Systematic reviews --> # What are the most relevant systematic reviews/meta-analyses about this topic? <!-- APA style --> # Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]: ## [[Help:Wikitext quick reference|italicisation]] |8= <!-- Resources --> <!-- See also --> # See also ## Excellent <!-- External links --> # External links ## Excellent |9= <!-- User page --> # Very good <!-- Description about self --> # Description about self provided <!-- Links to profile(s) --> # Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks. <!-- Link to book chapter --> # A link to the book chapter is provided |10= <!-- Social contribution --> # Excellent – at least three different types of contributions with direct link(s) to evidence }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 04:28, 5 September 2026 (UTC) == Interesting real-life case study to consider incorporating == Hi there, This chapter is starting to look really interesting! An fascinating (and horrific) real case study of charismatic leadership and follower devotion is the story of the Jonestown cult. The cult leader, Jones, was a charismatic leader who exploited the vulnerabilities of hundreds of followers, which ultimately resulted in a mass suicide of Jones and his followers. I wonder what insight this case could provide into follower motivations? My understanding is that Jones motivated followers to follow him by appealing to their status as minorities and claiming he had the answers to their problems; this was then pushed to the extreme when he moved his followers to an isolated social utopia, where he was in total control of fulfilling his followers needs and remaining loyal became motivating as it was required to survive. Here are some interesting resources you may like to review to explore this topic. Rex, S. (2025). One woman’s path to Jonestown. ''JSTOR.'' https://daily.jstor.org/one-womans-path-to-jonestown/ Tourish, D. (2025). How a “socialist utopia” became the death cult of Jonestown. ''Case studies in political leadership''. Edward Elgar Publishing. https://doi.org/10.4337/9781035328918.00031 [[User:U3253363|U3253363]] ([[User talk:U3253363|discuss]] • [[Special:Contributions/U3253363|contribs]]) 23:25, 5 September 2026 (UTC) == Feedback on Chapter == Hi, I thought this chapter was interesting, especially how it explains the influence leaders can have on their followers. One thing I think could be improved is the discussion of the political examples. It might be helpful to explain more clearly how the research on leadership relates to these examples, since research on leadership in organisations may not apply in exactly the same way to politics. I also think it would be worth mentioning that being a charismatic leader does not necessarily mean someone is a good or ethical leader. Charisma can influence people in different ways, so including this could make the discussion more balanced. Overall, I think explaining this a bit more would help readers understand the theory and its limitations, however I think this is a well written piece. [[Special:Contributions/~2026-51041-43|~2026-51041-43]] ([[User talk:~2026-51041-43|talk]]) 12:22, 12 October 2026 (UTC) == Heading casing == {| style="float: center; background:transparent;color:inherit;" |- | [[File:Crystal Clear app ktip.svg|48px|left]] | {{#if:|Hi [[User:{{{1}}}|{{{1}}}]].|}} FYI, the recommended [[Wikiversity]] heading style uses [[w:Letter case#Sentence_case|sentence casing]]. For example:<br> <big><big>Self-determination theory</big></big> rather than <big><big>Self-Determination Theory</big></big> Here's an example chapter with correct heading casing: [[Motivation and emotion/Book/2019/Growth mindset development|Growth mindset development]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 22:45, 24 October 2026 (UTC) |} <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is an insufficient chapter # The main area(s) for potential improvement: #* define key concepts rather than assuming the reader understands what they are #* use the best psychological theory about the topic #* more detailed review of the best psychological research about the topic #* quality of written expression #* integrate more strongly with other chapters by using embedded links to related topics # Unless fictional characters have been used as part of psychological theory or research, I don't think they have much value in a summary of psychological science about the topic, although they could be perhaps be used as examples/case studies #* I suspect that the [[Motivation and emotion/Assessment#Assessment items|recommended 75 hours]] were not invested in preparing this chapter <!-- Overall - Edit summaries --> #* Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits <!-- Overall – Citations --> # Basic use of academic, peer-reviewed citations to support claims <!-- Overall – Word count --> # Under the [[Motivation and emotion/Assessment/Chapter#Wordcount|maximum word count]], so there is room to expand <!-- Overall – Copyedits --> # For additional feedback, see the following comments and [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FCharismatic_leadership_and_follower_motivation&diff=2842463&oldid=2841180 these copyedits] |2= <!-- Overview – Comments... --> # Reasonably good <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Reasonably engaging scenario or case study <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario # I've increased the size of the Figure 1 image to make it easier to see <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box <!-- Overview – Explains problem --> # Explains the psychological problem or phenomenon reasonably well # What psychological science can help to understand this phenomenon? <!-- Overview – Focus questions --> # The focus questions are reasonably # The focus questions could be improved by: ## Consider reducing the overall number of questions, as some of them are overlapping |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # The chapter lacks sufficiently clear coverage of psychological theory about charismatic leadership # The chapter could be improved by: ## briefly defining core concepts such as: ### leadership ### charisma ### charismatic leadership and it key theoretical principles ### followership motivation ### prosocial motivation ### supervision ## more clearly summarising the key theoretical perspectives and principles about charismatic leadership ## drawing on personality and social psychological theories <!-- Theory – Builds on --> # Builds in a basic way on [[w:|Wikipedia]] articles but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms # The embedded interwiki links do not go to psychological concepts but historical etc. topics <!-- Theory – Depth --> # Insufficient depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Insufficient use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # In some places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic. <!-- Theory – Examples --> # Basic use of examples to illustrate theoretical concepts # The use of two historical examples (Hitler and Trump) is somewhat useful, but consider grounding the chapter is more everyday examples such as promised in the opening scenario or a leader in a contemporary workplace in order to make the chapter more relatable and applicable # Consider using more examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Insufficient review of relevant research # Insufficient use of relevant systematic reviews and/or meta-analyses # More detail about key studies would be ideal <!-- Research – Critical thinking --> # Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out specific critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Basic integration between the most relevant theory(ies) and the best research # The chapter places more emphasis on theory than on research; strive for an integrated balance # Rather than describing individual studies separately, synthesise findings across related studies to identify common themes, patterns, and differences in the evidence # The chapter could be strengthened through deeper integration with related [[Motivation and emotion/Book|chapters]] by including more embedded links |6= <!-- Conclusion comments... --> # Insufficient as a cohesive summary of the best psychological theory and research about the topic # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are summarised in a basic way <!-- Conclusion – Focus questions --> # Address the focus questions <!-- Conclusion – Take-home messages --> # Add practical, take-home message(s) |7= <!-- Written expression – Style comments... --> <!-- Written expression – Written expression --> # Written expression ## The quality of written expression is below professional standaard ## Avoid continual cross-referencing (e.g., As touched on towards the end of the previous section...) because its messy. Just cut to the chase. ## Use past tense when describing research studies, although implications of findings could be in the present tense <!-- Written expression – Sentences --> ## Some sentences could be explained more clearly (e.g., see the {{explain}} and {{rewrite}} tags); consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression ## Avoid starting sentences with a citation unless the author is particularly pertinent. Instead, it is more interesting for the the content/key point to be communicated, with the citation included along the way or, more typically, in [[w:Bracket#Parentheses|parentheses]] at the end of the sentence. <!-- Written expression – Paragraphs --> ## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences. ## Avoid one sentence paragraphs. Communicate one idea per paragraph using three to five sentences. <!-- Written expression – Layout --> # Layout ## The chapter structure is underdeveloped; expand by using more subheadings ## See earlier comment about [[#Heading casing|heading casing]] <!-- Written expression – Grammar --> # Grammar ## The grammar for many sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback <!-- Written expression – Proofreading --> # Proofreading ## More proofreading is needed (e.g., fix punctuation and typographical errors) to bring the quality of written expression closer to a professional standard <!-- Written expression – Figures --> ## Figures ### Well captioned <!-- Written expression – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style <!-- Written expression – Citations --> # Citations use very good [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] ## Use ampersand (&) inside [[w:Bracket#Parentheses|parentheses]] and "and" outside parentheses <!-- Written expression – References --> ## The list of references is brief; deeper reading of the best psychological science about this topic is needed ## References use very good APA style: ### Check and correct use of italicisation <!-- Feedback responsiveness ---> ## No evidence of responsiveness to feedback on talk page |8= <!-- Learning features comments... --> # Basic use of learning features <!-- Learning features – Wikipedia embedded links --> # Reasonably good use of embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding more interwiki links for the first mention of key words and technical concepts would make the text even more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. <!-- Learning features – Wikiversity embedded links --> # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Insufficient use of figure(s) # No use of table(s) <!-- Learning features – Quizzes --> # Reasonably good use of quiz(zes) and/or reflection question(s) <!-- Learning features – See also --> # Insufficient use of the [[{{PAGENAME}}#See also|See also]] section ## Add more links <!-- Learning features – External links --> # Insufficient/No use of the [[{{PAGENAME}}#External links|External links]] section ## Add more links |9= <!-- Social contribution comments... --> # ~10 logged, useful, mostly minor, last-minute unsigned contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] # ~ logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 22:45, 24 October 2026 (UTC) bri5favpqrkzbi7i4gtvqlhspcq353c 1 331614 2842554 2835140 2026-10-25T08:30:05Z Jtneill 10242 Book chapter feedback 2842554 wikitext text/x-wiki == An Australian source on gender and race interacting == Hi U3242837, Your sub-title asks under what conditions achievement motivation is expressed, so you may already be thinking about context. If you want a recent Australian study where gender and race interact rather than operate separately, this one is useful: * Guerzoni, M. A., Prehn, J., & Peacock, H. (2024). Mothers and sportsmen: The gendered and racialised nature of role model selection for Aboriginal and/or Torres Strait Islander youths. ''Australian Journal of Social Issues, 59''(2), 281–296. https://doi.org/10.1002/ajs4.311 They found role model selection among Aboriginal and Torres Strait Islander young people is strongly gendered - family members, particularly mothers, and male sportspeople dominate nominations - which has consequences for whose achievement gets modelled as attainable. Stronach et al. (2023) pairs well with it if you want a media angle: * Stronach, M., O'Shea, M., & Maxwell, H. (2023). 'You can't be what you can't see': Indigenous Australian sportswomen as powerful role models. ''Sport in Society, 26''(6), 970–984. https://doi.org/10.1080/17430437.2022.2088356 I am writing [[Motivation and emotion/Book/2026/Indigenous Australian role models and motivation|Indigenous Australian role models and motivation]] if any of that is useful to you. [[User:Jshottt|Jshottt]] ([[User talk:Jshottt|discuss]] • [[Special:Contributions/Jshottt|contribs]]) 14:23, 27 August 2026 (UTC) == Heading casing == {| style="float: center; background:transparent;color:inherit;" |- | [[File:Crystal Clear app ktip.svg|48px|left]] | {{#if:U3242837|Hi [[User:U3242837|U3242837]].|}} FYI, the recommended [[Wikiversity]] heading style uses [[w:Letter case#Sentence_case|sentence casing]]. For example:<br> <big><big>Self-determination theory</big></big> rather than <big><big>Self-Determination Theory</big></big> Here's an example chapter with correct heading casing: [[Motivation and emotion/Book/2019/Growth mindset development|Growth mindset development]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 12:30, 7 September 2026 (UTC) |} <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]] |2= <!-- Headings --> # See earlier comment about [[#heading casing|heading casing]] <!-- Heading structure --> <!-- 2-level --> # Promising [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] – could benefit from further development and/or refinement <!-- Alignment with focus questions --> # Insufficient alignment between sub-title, focus questions, and top-level headings <!-- GenAI ---> # Are the headings based on [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity. |3= <!-- Overview--> # Basic <!-- Scenario --> # A scenario or case study is presented in a feature box at the start of this section # Add an image to the scenario to help attract reader interest <!-- Description --> # A basic description of the problem/topic is planned or presented <!-- Focus questions --> # Develop closer alignment between the sub-title, focus questions, and top-level headings |4= <!-- Key points--> <!-- Overall --> # Basic development # What are the differences in achievement motivation between genders? <!-- Scope --> # The scope is limited; I think a clearer understanding of what the gender differences are and how theory and research accounts for those differences is needed # Some cultural context is needed; do the gender d9fferences vary between cultures # Consider the role of self-concept <!-- Theory and research --> # Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples. <!-- Other --> # For sections with sub-sections, provide key points for an overview paragraph prior to branching into the sub-headings # Use correct capitalisation ([https://apastyle.apa.org/style-grammar-guidelines/capitalization APA style is a "down" style]) – [https://polishedpaper.com/blog/capitalization-apa-style more info] <!-- Conclusion --> # Conclusion hasn't been developed # What are the practical, take-home messages? (address the focus questions) |5= <!-- Figure --> <!-- Caption --> # Figure caption(s) could better explain how the image connects to key points being made in the main text # Figure caption(s) should include '''Figure X'''. [bold] ... <!-- Cite --> # Cite each figure at least once in the main text using APA style (e.g., see Figure 1) |6= <!-- Learning feature --> <!-- Interwiki links ---> # One in-text [[m:Help:Interwiki linking|interwiki link]] for first mention of key term to [[w:|Wikipedia]]. Also embed links to [[Motivation and emotion/Book|book chapters]]. <!-- Scenarios/examples/case studies --> # Consider incorporating additional scenarios, examples, or case studies to illustrate key concepts. These could build on the Overview scenario or introduce new real-world situations in the main body of the chapter to demonstrate how the concepts apply in practice. <!-- Quiz --> # Consider including quiz question(s) about the take-home messages <!-- Tables --> # Also consider using tables to summarise key information |7= <!-- References --> <!-- Overall --> # Basic # Only cite sources that you have consulted # To be developed <!-- Systematic reviews --> # What are the most relevant systematic reviews/meta-analyses about this topic? <!-- APA style --> # Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]: ## capitalisation ## [[Help:Wikitext quick reference|italicisation]] ## make doi hyperlinks active (i.e., clickable) ## use dois where available instead of other links # A more thorough literature search is recommended. The goal is to identify and use the best academic theory and research about this topic. |8= <!-- Resources --> <!-- See also --> # See also ## Good ## Use bullet-points (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Use [[w:Letter case#Sentence casing|sentence casing]] ## Rename links so that they are more user friendly (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Use alphabetical order <!-- External links --> # External links ## To be developed (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) ## Link to the most relevant external resources about this topic |9= <!-- User page --> # Excellent <!-- Description about self --> # Description about self provided <!-- Links to profile(s) --> # Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks. <!-- Link to book chapter --> # A link to the book chapter is provided |10= <!-- Social contribution --> # Good – two out of three types of contributions made with direct link(s) to evidence. The other type of contribution is making: #* direct improvements to other [[Motivation and emotion/Book|chapters (past or current)]] # To add direct links to evidence of Wikiversity edits or comments: view the page history, select the version of the page before and after your contributions, click "compare selected revisions", and paste the comparison URL on your user page. For more info, see [[Motivation and emotion/Assessment/Chapter#Making and summarising social contributions|Making and summarising social contributions]]. This was demonstrated in [[Motivation and emotion/Tutorials/Wiki editing#Social contributions|Tutorial 2]]. }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 12:30, 7 September 2026 (UTC) == Feedback == I look forward to reading this chapter when it is complete! If you wanted to broaden the applicability of this chapter you could also look into whether the phenomenon you describe also apply in transgender people or whether it is based on biological sex. I found the below dissertation on this topic - whilst it may only be a dissertation it provides an interesting lens. https://scholarworks.waldenu.edu/cgi/viewcontent.cgi?article=14660&context=dissertations [[User:U3254168|U3254168]] ([[User talk:U3254168|discuss]] • [[Special:Contributions/U3254168|contribs]]) 09:01, 30 September 2026 (UTC) <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is a basic to reasonably good chapter that makes reasonably good use of psychological theory and basic use of research to address a real-world phenomenon or problem # The main area(s) for potential improvement: #* more detailed review of the best psychological research about the topic (for example, based on meta-analyses, what are the effect sizes for the gender differences in achievement motivation?) #* quality of written expression #* integrate more strongly with other chapters by using embedded links to related topics <!-- Overall - Edit summaries --> #* [[Motivation and emotion/Wikiversity/Edit summary|Edit summaries]] are used to communicate the changes for most edits <!-- Overall – Copyedits --> # For additional feedback, see the following comments and [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FGender_and_achievement_motivation&diff=2842553&oldid=2841193 these copyedits] |2= <!-- Overview – Comments... --> # Insufficient <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Promising topic/theme but a somewhat awkwardly expressed scenario # Rewriting of the scenario is recommended <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box <!-- Overview – Explains problem --> # Briefly explains the psychological problem or phenomenon; provide more detail # Lacks incisiveness and clarity; e.g., overally focusing on a female disadvantaged perspective rather than openly pursuing gender differences in either direction with regard to achievement motivation # Clarity of written expression can be improved <!-- Overview – Focus questions --> # The focus questions are insufficient # The focus questions could be improved by: ## being more specific to the topic (i.e., the sub-title) ## matching the top-level headings more closely |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Reasonably good—relevant theories are selected, described, and explained, with some room for improvement <!-- Theory – Builds on --> # Doesn't build on [[w:|Wikipedia]] articles and/or/but not related [[Motivation and emotion/Book|chapters]] by embedding interwiki links for key terms <!-- Theory – Depth --> # Reasonably good depth is provided about key theory(ies) <!-- Theory – Tables/Figures/Lists --> # Reasonably good use of tables, figures, and/or lists to clearly convey key theoretical information <!-- Theory – Citations --> # In many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological theory about the topic. <!-- Theory – Examples --> # Use more examples to illustrate theoretical concepts |4= <!-- Research comments... --> <!-- Research – Key findings --> # Reasonably good review of relevant research # Insufficient use of relevant systematic reviews and/or meta-analyses # More detail about key studies would be ideal # Much of the research drawn on is several decades old and the cultural and educational context is not made clear # In many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). This would help to demonstrate a deeper understanding of relevant psychological science about the topic. <!-- Research – Critical thinking --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out specific critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # Reasonably good integration between the most relevant theory(ies) and the best research # The chapter places more emphasis on theory than on research; strive for an integrated balance # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> # Very good summary and conclusion # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are well summarised <!-- Conclusion – Focus questions --> # The take-away messages for each focus question could be spelt out more clearly # Address the focus questions <!-- Conclusion – Take-home messages --> # Somewhat clear take-home message(s) |7= <!-- Written expression – Style comments... --> <!-- Written expression – Written expression --> # Written expression ## The quality of written expression is basic <!-- Written expression – Sentences --> ## Some sentences are overly long. Strive for the simplest expression. Consider splitting longer sentences into two shorter sentences. Shorter words and sentences are more [[w:Readability|readable]]. Try conducting a readability analysis such as via https://www.webfx.com/tools/read-able/. This chapter gets a score of . Aim for 50+. <!-- Written expression – Language --> ## Use [https://www.grammarly.com/blog/first-second-and-third-person/ 3rd person perspective] (e.g., "it") instead of 1st (e.g., "we") or 2nd person (e.g., "you") in the main text. 1st or 2nd person can work well for case studies or feature boxes. <!-- Written expression – Layout --> # Layout ## See earlier comment about [[#Heading casing|heading casing]] <!-- Written expression – Grammar --> # Grammar ## The grammar for some sentences could be improved (e.g., see {{g}} tags); consider using a grammar checking tool, [https://www.studiosity.com/students/ Studiosity], and/or peer feedback ## [https://www.grammarly.com/blog/punctuation-capitalization/possessive-apostrophe/ Possessive apostrophes] are not used correctly (e.g., cats vs cat's vs cats') <!-- Written expression – Copy editing --> # Copy editing ## Further [[w:Copy editing|copy editing]] is needed to improve written expression and bring the chapter closer to a professional standard (e.g., correcting punctuation, spelling, grammar, and typographical errors). ## Remove unnecessary capitalisation – [https://polishedpaper.com/blog/capitalization-apa-style more info] <!-- Written expression – Figures --> ## Figures ### Very well well captioned <!-- Written expression – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style <!-- Written expression – Citations --> # Citations use very good [https://apastyle.apa.org/products/publication-manual-7th-edition APA Style (7th ed.)] ## List multiple citations in alphabetical order by first author surname (e.g., Giraffe, 2024; Zebra & Aardvark, 2020) <!-- Written expression – References --> ## References use very good APA style: ### Check consistency of formatting ### Separate page numbers using an [[w:Dash#En dash|en dash]] (–) rather than a hyphen (-) ### Include dois where available <!-- Feedback responsiveness ---> ## No evidence of responsiveness to feedback on talk page |8= <!-- Learning features comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Insufficient use of learning features <!-- Learning features – Wikipedia embedded links --> # Add embedded in-text [[m:Help:Interwiki linking|interwiki links]] to Wikipedia articles. Adding interwiki links for the first mention of key words and technical concepts would make the text more interactive. See [[Motivation and emotion/Book/2020/Nutrition and anxiety|example]]. <!-- Learning features – Wikiversity embedded links --> # Add embedded in-text links to related [[Motivation and emotion/Book|book chapters]]. Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Reasonably good use of figure(s) # No use of table(s) # No use of feature boxes other than the opening scenario # No use of scenarios, case studies, or examples, other than opening scenario <!-- Learning features – Quizzes --> # Basic use of quiz(zes) and/or reflection question(s) <!-- Learning features – See also --> # Basic use of the [[{{PAGENAME}}#See also|See also]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#See also|template]] (fixed) # [[{{PAGENAME}}#See also|See also]] section not counted for marking purposes due to being over the maximum word count <!-- Learning features – External links --> # Basic use of the [[{{PAGENAME}}#External links|External links]] section ## Use the format shown in the [[Template:Motivation and emotion/Book chapter structure#External links|template]] ## Use bullet points per [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]] (fixed) ## Include sources in [[w:Bracket#Parentheses|parentheses]] after the link |9= <!-- Social contribution comments... --> # ~2 logged, useful, mostly moderate contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] # ~1 logged contributions without [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]], so unable to easily verify and assess. See [[Motivation and emotion/Tutorials|tutorials]] for guidance about how to get direct links to evidence. }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 08:30, 25 October 2026 (UTC) 7lqj1c3f2xrexzmqmmp5veqm3zw0ihy 1 331792 2842482 2832129 2026-10-25T00:17:46Z Jtneill 10242 Book chapter feedback 2842482 wikitext text/x-wiki == Interest In subject == I found this topic really interesting and I think the chapter is developing really well. I also noticed a slight structuring error in your reference list so I made a small edit to ensure it remained consistent and improved the layout. Good luck with the rest of your chapter !!! [[User:Ckopplemann|Ckopplemann]] ([[User talk:Ckopplemann|discuss]] • [[Special:Contributions/Ckopplemann|contribs]]) 04:34, 3 September 2026 (UTC) == Heading casing == {| style="float: center; background:transparent;color:inherit;" |- | [[File:Crystal Clear app ktip.svg|48px|left]] | {{#if:LMM26|Hi [[User:LMM26|LMM26]].|}} FYI, the recommended [[Wikiversity]] heading style uses [[w:Letter case#Sentence_case|sentence casing]]. For example:<br> <big><big>Self-determination theory</big></big> rather than <big><big>Self-Determination Theory</big></big> Here's an example chapter with correct heading casing: [[Motivation and emotion/Book/2019/Growth mindset development|Growth mindset development]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 23:07, 7 September 2026 (UTC) |} <!-- Official topic development feedback --> {{METF/2026 |1= <!-- Title --> # Title and subtitle are correctly worded and use [[w:Letter case#Sentence casing|sentence casing]] |2= <!-- Headings --> # See earlier comment about [[#heading casing|heading casing]] <!-- Heading structure --> <!-- 2-level --> # Basic [[Motivation and emotion/Assessment/Major project/Structure|2-level heading structure]] – would benefit from further development (refinement) # The headings appear to focus on GM itself, rather than how to cultivate it <!-- Alignment with focus questions --> # Develop closer alignment between sub-title, focus questions, and top-level headings |3= <!-- Overview--> # Insufficient <!-- Scenario --> # A scenario or case study is presented in a feature box with an image at the start of this section # However, the scenario doesn't illustrate the development of a GM <!-- Description --> # The description is general about GM; it doesn't sufficiently focus on the ''development of a GM'' <!-- Style --> <!-- Focus questions --> # Develop closer alignment between the sub-title, focus questions, and top-level headings # More specifically, revise the FQs to focus on the ''development of a GM'' |4= <!-- Key points--> <!-- Overall --> # Insufficient development due to lack of emphasis on theory, research, and examples of ''developing a GM'' <!-- Theory and research --> # Strive for an integrated balance of the best psychological theory and research about this topic, with practical examples. <!-- Other --> # For sections with sub-sections, provide key points for an overview paragraph prior to branching into the sub-headings <!-- Conclusion --> # Conclusion hasn't been developed # What are the practical, take-home messages? (address the focus questions) |5= <!-- Figure --> # A figure is presented and captioned # The relevance of the figure to the topic is unclear <!-- Caption --> # Figure caption(s) could better explain how the image connects to key points being made in the main text <!-- Cite --> # Cite each figure at least once in the main text using APA style (e.g., see Figure 1) |6= <!-- Learning feature --> <!-- Interwiki links ---> # Add in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]] (see [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]) <!-- Scenarios/examples/case studies --> # Basic use of scenario/example/case study, but the scenario does not illustrate the topic <!-- Quiz --> # Consider including quiz question(s) about the take-home messages <!-- Tables --> # Also consider using tables to summarise key information |7= <!-- References --> <!-- Overall --> # Very good <!-- Systematic reviews --> # Well done on identifying relevant systematic reviews and/or meta-analyses <!-- APA style --> # Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]: ## capitalisation ## [[Help:Wikitext quick reference|italicisation]] ## [https://apastyle.apa.org/instructional-aids/reference-guide.pdf doi formatting] |8= <!-- Resources --> <!-- See also --> # See also ## Excellent <!-- External links --> # External links ## Very good ## Use [[w:Letter case#Sentence casing|sentence casing]] ## Include source in brackets after link ## Use alphabetical order |9= <!-- User page --> # Used effectively <!-- Description about self --> # Description about self provided <!-- Links to profile(s) --> # Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks. <!-- Link to book chapter --> # A link to the book chapter is provided |10= <!-- Social contribution --> # Excellent – at least three different types of contributions with direct link(s) to evidence # Good – two out of three types of contributions made with direct link(s) to evidence. The other type of contribution is making: }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 23:07, 7 September 2026 (UTC) <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is a reasonably good chapter that makes good use of psychological theory and reasonably good use research to address a real-world phenomenon or problem # The main area(s) for potential improvement: #* more detailed review of the best psychological research about the topic #* quality of written expression <!-- Overall - Edit summaries --> #* Excellent use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to clearly communicate the changes made in most or all of the main edits <!-- Overall – Citations --> # Very good use of academic, peer-reviewed citations to support claims <!-- Overall – Copyedits --> # For additional feedback, see the following comments and [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FDeveloping_a_growth_mindset&diff=2842481&oldid=2839328 these copyedits] |2= <!-- Overview – Comments... --> # Excellent/Very good/Good/Reasonably good/Basic/Underdeveloped <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario <!-- Overview – Scenario - Feature box --> # The scenario uses an appropriate feature box <!-- Overview – Explains problem --> # Clearly explains the psychological problem or phenomenon <!-- Overview – Focus questions --> # The focus questions are excellent (clear and relevant) |3= <!-- Theory comments... --> <!-- Theory – Breadth --> # Very good—key theories are well explained and applied; 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Embedding in-text links to related book chapters helps to integrate this chapter into the broader book project. <!-- Learning features – Figures, tables, feature boxes, scenarios --> # Very good use of figure(s) # Good use of table(s) # Basic use of feature box(es) # Basic use of scenarios, case studies, or examples <!-- Learning features – Quizzes --> # No use of quiz(zes) and/or reflection question(s) <!-- Learning features – See also --> # Good use of the [[{{PAGENAME}}#See also|See also]] section <!-- Learning features – External links --> # Good use of the [[{{PAGENAME}}#External links|External links]] section ## Use [https://www.masterclass.com/articles/sentence-case-explained sentence casing] |9= <!-- Social contribution comments... --> # ~10 logged, useful, mostly minor to moderate contributions with [[Motivation and emotion/Assessment/Chapter/Summarising social contributions#How to add direct links to contributions|direct links to evidence]] }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 00:17, 25 October 2026 (UTC) ijihkvvau6oxi2np7e313pltvslrqvl 0 332412 2842465 2842364 2026-10-24T22:48:31Z Scogdill 1331941 2842465 wikitext text/x-wiki =Event= International championship boxing match between British Jem Smith and Australian Frank Slavin on 23 December 1889 in Bruges, Belgium. ==Overview== On 23 December 1889 a boxing match in Bruges did not end with integrity: British Jim Smith was losing, so his supporters entered the ring, pinned Australian Frank Slavin's arms and beat him up. The referee called it a draw to prevent Slavin from being killed, and then fled. The Cork Constitution called it "the Smith-Slavin piece of brutality."<ref>"Mr. Abington." ''Cork Constitution'' 14 January 1890, Tuesday: 7 [of 8], Col. 5b [of 6]. ''British Newspaper Archive''https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001648%2F18900114&page=7&article=099&stringtohighlight=mandeville+viscount. Same title and p.</ref> ==Logistics== * 23 December 1889, 8:00 a.m. * Stakes: £1,000 ==Related Events== * [[Social Victorians/Boxing/Pelican Club Meeting Smith-Slavin match|Pelican Club Meetings about the Smith-Slavin match]] * the "[[Social Victorians/Boxing/Fracas at the Trocadero|Fracas at the Trocadero]]" ==Who Was Present== # [[Social Victorians/People/Mandeville|Kim Montagu, Viscount Mandeville]] # Jem Smith, British boxer # Frank Slavin, Australian boxer # Lord Rosslyn # Sir M. Sandys # Mr. G. Abington Baird, Smith's manager? and horseman (see description in Anthology) ==Entertainment== ==Anthology== The People On Sunday, 29 December 1899, the People<blockquote>"The Smith-Slavin Fight." ''The People'' 29 December 1889, Sunday: 16 [of 16], Col. 4a–5a [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000729%2F18891229&page=16&article=150&stringtohighlight=mandeville+viscount. Same print title and p. Disgraceful Scenes. Expressions of indignation. The prize fight between Jem Smith, who poses as the fistic "champion of England," and Frank Slavin, the Australian champion, took place early on Monday, and resulted in perhaps the most disgraceful scene ever witnessed in connection with the history of the prize ring. Various spots, both at home and abroad, were from time to time named as the possible battle-field, but finally it became known to a certain select few that an English gentleman, Mr. Atkinson-Grimshaw, residing near Bruges, Belgium, had placed his grounds at the disposal of the combatants. In London it soon became known that the principal supporters of the Englishman had caused a number of Birmingham ruffians to be hired for the purpose of making a "strong corner." The betting changed directly to short odds on the Australian, and also evens, while  offers of 30 to 20 that "Smith did not lose" proclaimed that some ugly and disreputable manœuvres would be attempted. The ring was pitched on a splendid piece of turf — the lawn tennis ground, in fact — and about forty persons assembled, including Viscount Royston, [[Social Victorians/People/Mandeville|Viscount Mandeville]], Mr. Abington, Mr. Harris, Mr. W. Riley, Mr. F. Kotchie, Mr. Ike White, the Hon. M. Sandys, Mr. E. Bayley, Mr. C. Parsons, Mr. J. Kilby, Mr. C. Morton, and other well-known "sports." Smith was dressed first, and and [sic] had to wait a few minutes for Slavin. The Englishman appeared in white flannel drawers, white socks, and lace-up boots, while the Australian wore black drawers, blue socks, and lace-up boots. Harper and Baldock were seconds, Howes bottle-holder, and Mr. Fleming umpire for Smith; Cosnett and Bob Kirby, Mr. Taylor, and Mr. W. Riley acting in similar capacities for Slavin. Mr. Joe Vezey officiated as referee and Mr. Calder as timekeeper. The Fight. The men were in the ring at 8.25, and ten minutes later the fight began. After the first two rounds everything went in favour of Slavin, who in the third round knocked Smith clean off his legs with a terrific right-hander. He repeated this operation four or five times subsequently, and surprised many of the spectators by the determined way in which he fought. Smith looked very serious on coming up for the fourth round. At this time a suspicious movement was noticeable among the brigade of Birmingham ruffians, and one of them came round to the colonial's corner with no good intentions. Had it not been that Mick Taylor and Frank Hinde kept zealous guard over all the water provided for the use of the combatants, there is no knowing what fiendish trick might not have been perpetrated. Getting to close quarters, Slavin delivered heavily right and left on Smith's body, the latter being very slow in responding to this vigorous attack. He had, however, other assistants, for a vile ruffian from over the ropes deliberately struck Slavin a blow on the head. A shout was at once raised, but when charged with it the rascal denied that he had done anything. A scuffle ensued between the men, and to escape from the difficulty Smith cross-buttocked his opponent, who fell. In the fifth and sixth rounds Slavin again brought Smith to the ground, and made a nasty gash on his forehead. All this time the most horrible language proceeded from Smith's corner. In the seventh round a desperate rally took place, during which Smith fought Slavin on to the ropes, and there held him. Here was a grand opportunity for the miscreants, who did not fail to make use of it. One of them kicked Slavin repeatedly, and others beat him about the head, knuckle-dusters and life preservers being used. Slavin during the progress of the eighth round further received a couple of smart blows from a stick which a cowardly ruffian carried. Taking not the smallest notice of these interruptions Slavin pursued his legitimate opponent, both Smith's seconds following him about and jeering in his face. The referee entered the ring, and cautioned the seconds to move away. This had very little effect, and in the midst of half-arm fighting of a very severe kind, Smith was one more brought to the ground. In the ninth round the Australian again ventured into Smith's corner, and the fighting brought him close to the ropes. The  Birmingham mob here set about Slavin with great brutality, kicking him and punching him on all parts of the body. Getting away from the ropes Slavin let go the right, and, reaching Smith's mouth, caused the blood to spring forth profusely, bringing the Englishman to grass. The tenth and eleventh rounds were not quite so exciting, but in the twelfth Slavin drove his right with terrific force on to Smith's chin, knocking him clean off his legs. In the thirteenth round the Australian doubled Smith up with a fearful blow under the left ribs, and the Englishman's friends, seeing that Smith was almost beaten, raised a cry of " Police!" Smith was hurried from the ring, but Slavin and his friends refused to budge, and after "Time" had been called they claimed the fight. The referee, who appeared afraid to give a just decision, was surrounded by partisans of Smith, who had meanwhile come back, and eventually ordered the fight to proceed. The next round, the fourteenth, was the last. Slavin fought in a most determined fashion, and when he was punishing Smith heavily, the latter caught the Australian round the waist and held him on to the ropes, whilst the ruffians standing round beat and kicked him in a most cowardly fashion. Amid cries of "Foul," the pair fell to the ground, and, on the men taking their corners, the referee said he would act no longer, as it was perfectly clear Slavin was not to have fair play. This created immense excitement, during which the referee declared the fight a draw, and, crossing to Slavin, said, "I do not want to stay here to see you killed." Afterwards the roughs tried to get at Slavin, who, however, managed to hide in a barn. Later on he went with a few of his friends to Bruges, where he caught the train for Ostend, and came over to Dover, thence proceeding to  Margate. Slavin's Opinion of his Treatment. The following conversation took place during an interview with Slavin, a few hours after the fight:— Did you expect any trouble at the ring  side? — Slavin: Yes, I was prepared for that, as I was constantly receiving letters of a threatening nature, and though these things as a rule are of no account, I had reason to believe there was some truth this time. Besides, I had not forgotten the affair at Astley's Theatre. — Do you feel any effects of the fighting? — Slavin: The fighting? I feel nothing of what I got from any one in the ring; it's what those beauties gave me outside. The only time Smith hit me in anything like form was on the jaw. — Why did you persist in going to his corner? — Slavin: I wanted to get the affair over and enjoy my Christmas dinner, and I knew if the fighting was left to Smith I should want it served in the ring. — Of course you heard a mob was got together for Smith? — Slavin: Yes, I knew all about that, but I relied on British fair play, and expected the referee to do his duty. Directly I got in the ring I saw what the game was, and knew my only chance was to knock Smith out. — What do you think of Smith? — Well, I think so much of him that I'll bet him £200 I stop him in six rounds, but we must have no more private shows. I've had enough of that business. — l suppose you have a poor opinion of English fair play? — Slavin: I have been treated like a prince generally, but I can see the fighting men are easily led astray, and I expected trouble directly Smith objected to a referee. If we served an Englishman in the same manner in Australia we should get drummed out of the town, and rightly too. — What are your plans for the future? Well, I shall stay in England some time and challenge Sullivan, and you might as well cable it over to America for me. — With the gloves? — No. I want to fight him with the knuckles. I think I should get fair play out there. Indignation in London. Directly the news became known in London the greatest indignation prevailed in all the West-end, [Col. 4c–5a] clubs at the dastardly way in which Slavin had been treated. Subsequently some forty or fifty gentlemen proceeded to the Pelican to demand the instant and public repudiation on the part of the committee with any official cognisance of the arrangements for the fight, and also for the summary punishment of those who had dragged dowr [sic] a manly pastime to the lowest depth of degradation and blackguardism. In accordance with these demands, and in full sympathy with them, those members present, conjointly with the proprietor, immediately sent out telegrams summoning an extraordinary meeting of the cornmittee. The Pelican Club Meeting. At tje special meeting of the committee of the Pelican Club to was unanimously determined to address a letter of sympathy to Slavin, expressing the appreciation of the committee of "the courage he exhibited in face of the most blackguardly conduct which ever disgraced the ring side.["] The committee added that they fully recognised that he had indisputably established his claim to the championship of England, and they further informed him of their intention to present to him a belt and purse in reccognitin</blockquote> === The ''Sportsman'' === On Monday, 30 December 1889, the ''Sportsman'' printed a list of people and organizations that had contributed to the "testimonial" fund for Slavin as well as some letters to the editor about the Slavin-Smith match:<blockquote> A committee meeting of the above was held on Saturday afternoon, Mr W. J. Innes in the chair.  There were also present Mr W. Riley, Mr Fred H. [Gala?], Mr C. Parsons, Mr A. Butler, Mr M. J. Harrow (Margate), Mr Ike White, Mr E. C. Irish, and Mr G. T. Dunning (hon. sec. and treasurer). The chairman read a letter from Mr John Lees, regretting his inability to attend owing to illness, whilst the remaining committeeman, Mr E. W. Bayly, was absent in Paris. The appointment of the hon. sec. and treasurer was confirmed by the committee, and after some discussion it was decided to close the subscription-list on Monday, January 20, and to present the proceeds to Slavin at the Cambridge Music Hall, Bishopsgate-street, on a day to be named. Other business of minor importance was transacted, and after the next committee meeting had been fixed for Wednesday, January 15, the meeting closed with a vote of thanks to the chairman. Additional subscriptions to the fund, which now reaches 300''l''., will be found appended [names set as unordered list to save space, £ ''s.'' ''d.''] * Lord Mandeville 2 2 0 * Lord Royston 2 2 0 * Lord James Douglas 2 2 0 * Hon. Michael Sandys 2 2 0 * Fred Kotchie 2 2 0 * Mr Ike White 2 2 0 * Mr W. Riley 2 2 0 * Mr C. Parsons 2 2 0 * Mr J. Kilby 2 2 0 * Mr Frank Hinde 2 2 0 * Proprietors of The Sportsman 2 2 0 * Mr. J. Purdey 2 2 0 * Mr. E. W. Bayly 2 2 0 * Messrs H. A. Domby and Son 2 2 0 * "Caption Coe" 2 2 0 * Mr C. W. Bulpett 2 2 0 * Mr P. W. Carr 2 2 0 * Mr W. Morley Pegge 2 2 0 * W. D. S. 2 2 0 * J. E. S. 2 2 0 * Mr W. H. Grenfel 2 2 0 * Mr Quartermaine East 2 2 0 * Mr H. H. Hungerford 2 2 0 * Mr Noel Fenwick 2 2 0 * Sir John Leslie, Bart. 2 2 0 * Mr T. Brinkman 2 0 0 * Mr Calder 1 1 0 * Mr H. W. Hill 1 1 0 * Mr John Young 1 1 0 * Mr D. A. Neilson 1 1 0 * Mr W. Neilson 1 1 0 * Mr Julius Simon 1 1 0 * Mr J. Edge 1 1 0 * Mr H. Marshall 1 1 0 * Mr F. Aynsley 1 1 0 * Col. Hon. Wellesley 1 1 0 * Mr Charles Hawtrey 1 1 0 * Mr W. F. Ashton 1 1 0 * Mr Barratt 1 1 0 * Old Durham Ox 1 0 0 * Sir John Willoughby 1 0 0 * A friend, per C. H. A. 0 10 6 * Mr Ben Giles 0 10 6 * Mr Tom Vale 0 10 6 * [E.?] W. 0 10 6 * Officer 0 10 0 * Per Mr Ike White, Exchange Club: ** S. W. 2 2 0 ** Mr G. B. Knight 1 1 0 ** Mr A. Powles 1 1 0 ** Mr S. Blandford 1 1 0 ** T. F. W. C. 1 1 0 ** Mr Leo Harward 1 1 0 ** Mr J. Harward 1 1 0 ** Mr H. Harward 1 1 0 ** Mr [Monteagle?] [Moutcagie??] 1 1 0 ** Mr C. Orate 0 10 6 ** W. G. P. 0 10 6 ** H. R. T 0 10 6 ** Mr Josiah Smith 0 10 6 ** W. H. R. 0 10 6 ** Mr R. Wilson 0 5 0 * Per Mr M. J. Harlow, Margate: ** Mr A. H. [Cosi? Coni?] 2 2 0 ** Mr J. Thurston 1 1 0 ** Mr J. A. Fagg 1 1 0 ** Mr F. [Miles?] 1 1 0 ** Mr Adolphe [Drineqbeir?] 1 1 0 ** Mr Ernest [Drineqbier?] 1 1 0 ** Mr Lacy Brown 1 1 0 ** Mr Algy Brown 1 1 0 ** Mr M. J. Harlow 1 1 0 ** An Admirer 1 1 0 ** Mr J. W. Robinson 1 1 0 ** Mr W. Searles 1 1 0 ** Mr J. Smith 1 1 0 ** Mr J. Hume 1 1 0 ** Mr J. Painter 1 1 0 ** A Scotchman 1 1 0 ** Mr J. Mundell 0 10 6 ** Mr Bob Redman 0 10 6 ** Mr G. A. Body 0 10 6 ** Mr J. Phipps 0 10 6 ** Mr J. Doughty 0 10 6 ** Old Fisherman 0 10 6 ** Mr Joe Knott 0 10 6 * Per Mr B. J. Angle ** Some mebers of the Stock Exchange 125 0 0 ** Hr Henry J. Turner 2 2 0 * Per Mr W. J. Innes: ** Mr Sidney Fry 2 2 0 ** Mr W. J. I. 2 2 0 ** Mr W. Schlesinger 2 2 0 ** Mr Joseph Thompson 2 2 0 ** Mr Barney Thompson 2 2 0 ** Mr L. Meyers 2 2 0 ** Mr W. Shoe 2 2 0 ** Mr George Cooper 2 2 0 ** Mr J. T. Crossley 2 2 0 ** Mr Andy Anderson 2 2 0 ** Mr G. M. T. 2 2 0 ** Mr F. R. 2 2 0 ** Mr W. Bertram 1 1 0 ** Mr J. W. Hammon 1 1 0 ** Mr Alec Harris 1 1 0 ** Mr John Isaacs 1 1 0 ** Mr Edward [Dance? Dasce?] 1 1 0 ** Mr George Blanchett 1 1 0 ** Mr F. [Leleu? Lolou?] 1 1 0 ** Mr W. J. R. 1 1 0 ** Mr H. C. 1 1 0 ** Mr A. A. M. 1 1 0 ** Mr D. W. Fowler 0 10 6 * Per Mr F. H. Gale: ** Mr R. Moore 2 2 0 ** Mr A. H. Goode 2 2 0 ** Mr [B.? E.? R.?] J. G. Goode 2 2 0 ** Mr F. H. Gale 2 2 0 ** Mr H. Ulph 1 1 0 ** Mr J. Hooper 1 1 0 ** Mr C. Kittall 1 1 0 ** Mr C. Moore 1 1 0 * Per Mr W. Riley: ** Mr Brand (Wiltshire) 2 2 0 ** Mr Charles Godfrey 2 2 0 ** Mrs Lane (Britannia) 2 2 0 ** Mr Sam Adams 2 2 0 ** B. and M. 0 7 6 * Per Mr A. Cockburn: ** Mr Richardson 2 2 0 ** Mr A. Cox 1 1 0 ** Mr Hy. Brackenbury 1 1 0 To the Editor of T<small>HE</small> S<small>PORTSMAN</small>. Sir,— From the evidence of your eye-witness reporter there seems to me the clearest possible case against Smith, Harper, and Baldock — that Smith held Slavin on the ropes while the roughs kicked and "knuckle-dustered" him, that Harper put his knee on Slavin when down, and that Baldock did likewise, and that throughout they advised Smith to stand like a coward under the protection of the savages till the instinct which filled Slavin's manly bosom prompted the honest outburst to Smith, "Come and have a round." The duty for sportsmen to fulfil with the assistance of their representative journal, is, firstly, to see that Slavin is elected Champion and that he is handsomely recompensed, and that these so-called "seconds" should be properly punished and our splendid timber-yard Champion put in his proper place. The names of those who hired the roughs should also be published. I have spoken to at least fifty patrons of boxing shows, and one and all have made the most positive and determined objection to supporting any entertainment where the three men named above are concerned directly or indirectly. They, by their own stupidity and cupidity, have put the last nail in their own coffin, I hope. With regard to the referee, this individual is decidedly unsatisfactory. He allowed Smith to bolt when the cry of "Police" was raised without disqualifying him in any way, and then put an end to the business by declaring it a draw. Now, if the said referee is not a determined or courageous man, then he acted to the best of his judgment, for he was afraid of personal violence, having literally to escape to London; but on his arrival here, I and numberless others naturally expected some explanation or a reversal of his decision. The whole sporting community are on the tiptoe of excitement, and Mr Vesey in the coolest  possible manner washes his hands of the affair, declining absolutely to expose any opinion. This is rather rough on the public considering that be consented to act as referee of his own free will, and he is no chicken in such matters. He does not tell us whether he is aware that he has the power to reverse a decision given under fear; nor does he say whether he still considers the decision made on the spot a happy one. How was it that no steps were taken to protect the referee? There is no legitimate courage in the Brummagem rough. He is an imposter. With assistance of our excellent so-called Champion, and about another ten, armed with knuckle-dusters, he will doubtless attack Slavin, and if a torrent of filthy, cowardly abuse could kill he would beat Slavin single-handed. One of these gallant gentry actually returned to Birmingham without going to Bruges, being positively afraid to chance crossing the tiny stretch of sea. An American reader will bear me out that had our unsatisfactory referee been under the protection of one single genuine Nebraska "tough," as they are called out there, he would, without uttering one word of bad language, have cleared out the lot had they attempted violence to his charge. But it would never have come to that, for the sight of a real, genuine armed rough, with determined set features, would have struck terror into the hearts of the pluckless Brum shoddy bully, with the proverbial dull, stupid, animal-like pudding face. As an Englishman I [Col. 1c–2a] positively feel guilty among my American friends when they ask me what about my vaunted British fairplay, and I quite fail to see or understand how "seconds" — no matter, for argument's sake, how dishonest they may be can such fools as to kill the goose which lays their golden eggs. Prize-fighting and boxing are their bread-and-butter, and some of those who accepted the assistance of the roughs in the recent important international fight must have surely prognosticated that trouble and disgrace would follow, and they of all men should have been the last to wantonly risk breaking up one of our favourite English sports. In conclusion, contrast the courage of the two men. Slavin practically fought about six men, not knowing the moment that one well-directed blow from a "duster" might have cost him his life or lost him the fight. Think of his manly appeals to the cowards for a few minutes' fairplay. I only wish they each had to have four rounds singly with him! — Yours, &c., A Lover of Boxing. To the Editor of T<small>HE</small> S<small>PORTSMAN</small>. Sir, — Previous to the late show of ruffianism I was in sympathy with and respected Smith as the Champion, but I am grieved to say that nothing of late years has so thoroughly disgusted me as the ill-treatment which was meted out to Slavin by those who disgrace not only sport but the very name of man. I sincerely hope that Slavin will still continue to think that one of the dearest wishes of the true sportsman of this country is to see fair play. I enclose my mite in the shape of one guinea, and subscribe myself, yours, &c., Leicester, December 28. J. Barrett. To the Editor of T<small>HE</small> S<small>PORTSMAN</small>. Sir, — Herewith I inclose cheque 1''l''. 1''s''. to the Slavin Fund, and hope that his manly and honourable conduct will be well rewarded. I cannot, however, agree with those who hope "that we have seen the last of prize-fighting," and other sentiments of the kind. It will be an evil day for England when the fighting instinct dies out from among her sons. No, let us rather hope that the Slavin-Smith ''fiasco'' will be the beginning of a new era, and that for the future fights will not be allowed to take place without proper ring-keepers, as in the days of Spring, Langham, and Bendigo, and that the Championship of England will not be long without a native defender.—Yours, &c., J. W. B. C., Junior Carlton Club, December 27. To the Editor of T<small>HE</small> S<small>PORTSMAN</small>. Sir, — l beg to inclose you the sum of one guinea, my subscription to the Slavin Fund. I was indeed sorry to read of the outrageous and murderous treatment Slavin received from Smith's supporters, which support was tacitly sought after by Smith not leaving his corner and meeting his man in the centre of the ring, as any Englishman should have done that presumed to call himself "England's Champion." I feel quite certain that had this outrageous conduct passed away without the strong condemnation it received at all good sporting men's hands in this country, and the hearty and the thoroughly unanimous sympathy expressed for Slavin, the good feeling and perfect bon accord that has existed between the sporting fraternity of the two countries, New and Old England, would soon have ceased to exist. Australians are great stickers to and warm supporters of England and the English, and they will all feel most keenly the slur cast upon "British fair-play and courage." — Yours, a lover of it, Sydney Galvayne. York, December 26. To the Editor of T<small>HE</small> S<small>PORTSMAN</small>. Sir, — Enclosed you will cheque for one guinea as a simple tribute to Slavin, who so squarely held his own, and who should not be looked upon as a stranger in a strange country, but as a cousin Englishman, who deserves support, not only as a true sportsman but also against the fearful ordeal of ruffianism to which he has been subjected. As a lover of fair play believe me, yours faithfully, Birmingham, December 27. Tom B. Sharp. To the Editor of T<small>HE</small> S<small>PORTSMAN</small>. Sir, — Please add 9''l.'' 14''s.'' to the list I have already sent you for Slavin testimonial, and oblige, yours, &c., I. White. Exchange Club, Queen-street, Cheapside, Dec. 28.<ref>"The Ring. The Slavin Testimonial." ''The Sportsman'' 30 December 1889, Monday: 4 [of 4], Col. 1a–2b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001055%2F18891230&page=4&article=020&stringtohighlight=mandeville+viscount. Same print title and p.</ref> </blockquote> On 9 January 1890 in part of a much larger partisan report with critiques of the investigative committee, its chair and its results as well as other newspapers printing rumors:<blockquote>JOHN L. SULLIVAN ON ENGLISH FAIR PLAY.<p> J<small>OHN</small> L S<small>ULLIVAN</small> has been speaking his mind on the Smith-Slavin fight to some purpose. "I can't see how an English fighter can be beaten," he said. This man Smith is little better than a middle-weight, and I haven't the slightest doubt that Slavin beat him fairly. Some people wonder why I did not go to England to fight Smith. Now you understand why I insisted on meeting him in this country. In England fair play is unknown. The man with the greatest number of 'heelers' always wins or saves himself from defeat. Slavin or Jackson will, in all probability, look for a match with me when they arrive in this country, and | am perfectly willing to meet either or both of them for a proper consideration. I want a big purse, and I am not particular where I fight so long as it is in a club-house. No more law-breaking for  me."<ref>"The Late Fight. Meeting of the Pelican Club. Stormy Proceedings. The Committee Resigns." ''The Sportsman'' 9 January 1890, Thursday: 4 [of 4], Col. 3b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001055%2F18900109&page=4&article=049&stringtohighlight=mandeville+viscount. Same print title and p.</ref></blockquote> Also on January 9, the ''Sportsman'' reported on a collection made for Slavin:<blockquote>THE SLAVIN TESTIMONIAL<p> The subscriptions towards the above now amount to over 400''l.'', and it having been decided to close the list on the 20th, the committee will be obliged if those who have promised donations will forward the same to the hon. sec. and treasurer, Mr. G. T. Dunning, T<small>HE</small> S<small>PORTSMAN</small> Office.</blockquote> {| class="wikitable" ! !£ !s. !d. |- |Amount already acknowledged |407 |3 |6 |- |Mr E.Tabernacle |1 |1 |1 |- |H. S. |1 |1 |0 |} <blockquote>The guinea acknowledged per Mr Riley, from C. R., yesterday, should be from C. K.<ref>"The Late Fight. Meeting of the Pelican Club. Stormy Proceedings. The Committee Resigns." ''The Sportsman'' 9 January 1890, Thursday: 4 [of 4], Col. 3b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001055%2F18900109&page=4&article=049&stringtohighlight=mandeville+viscount. Same print title and p.</ref></blockquote> === The ''Huddersfield Daily Chronicle'' === On 10 January 1890:<blockquote> Mitchell and Lord Mandeville. As an addendum to the important meeting of the Pelican Club committee, on Wednesday, came a gathering of "sports" at the Trocadero, Piccadilly Circus. Lord Mandeville and Mr. Abington., who had been in attendance at Denman-street, were joined by "Pony" Moore, his son-in-law, C. Mitchell, J. Carney (the Birmingham light-weight), J. Baldock, B. Habbijam, C. Bates, &c. The party, who occupied seats in the stalls, naturally discussed things pugilistic, the talk being mostly devoted to the recent international fight between Frank P. Slavin and Jem Smith. " [sic] The Squire's opinions on the matter happened not to coincide with those of Mitchell, and high words were exchanged between the pair. Suddenly the Birmingham boxer jumped up and "went for" Mr. Abington, but did not strike him. Lord Mandeville immediately interposed and asked Mitchell to be quiet, whereupon the latter directed his attention to his lordship, and struck him sharply on the forehead. Carney and Baldock sprang to his lordship's assistance, and Mitchell and "Pony" Moore at once left the building, the whole proceedings not lasting longer than a few seconds. It soon became noised abroad that there was "something up," and a crowd gathered, but the police saw Mitchell and Mr. G. W. Moore safely across the road.<ref>"The Smith-Slavin Fight." ''Huddersfield Daily Chronicle'' 10 January 1890, Friday: 4 [of 4], Col. 6b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0002876%2F18900110&page=4&article=071&stringtohighlight=mandeville+viscount. Same print title and p.</ref></blockquote> After the Pelican Club meetings, Slavin comments:<blockquote>On Wednesday at Westminster Aquarium, F.  P. Slavin, the colonial pugilist, who received such bad treatment at the hands of roughs in his recent prize-fight with Jem Smith, the English "champion," made his first public appearance since the fight. Mr. Frank Hinde, in introducing Slavin, said he would give an exhibition with Cosnett of Birmingham, who seconded him in his fight. Slavin was the gamest man that ever  stepped into a ring, he hoped the audience would rive him a good old English welcome. On appearing on the stage Slavin met with a most enthusiastic reception, and the three rounds' sparring with Cosnett was much appreciated. At the conclusion Slavin, in response to loud calls, bowed his acknowledgments.<ref>"Slavin at the Royal Acquarium." ''Liverpool Weekly Courier'' 11 January 1890, Saturday: 3 [of 8], Col. 5a [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0003067%2F18900111&page=3&article=042&stringtohighlight=mandeville+viscount. Same print title and p.</ref></blockquote> === ''Reynolds's Newspaper'' === ''Reynolds's Newspaper'' quotes J. L. Sullivan on the Smith-Slavin match:<blockquote>Mr. John L. Sullivan has been speaking his mind on the Smith-Slavin fight to some purpose. "I can't see how an English fighter can be beaten," he said. "This man Smith is little better than a middle-weight, and I haven't the slightest doubt that Slavin beat him fairly. Some people wonder why I did not go to England to fight Smith. Now you can understand why I insisted on meeting him in this country. In England fair play is unknown. The man with the greatest number of "heelers" always wins or saves himself from defeat. Slavin or Jackson will in all probability look for a match with me when they arrive in this country, and I am perfectly willing to meet either or both of them for a proper consideration. I went a big purse, and I am not particular where I fight so long as it is in a club-house. No more law-breaking for me."<ref>"The Recent Prize Fight." ''Reynolds's Newspaper'' 12 January 1890, Sunday: 6 [of 8], Col. 4a–b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000101%2F18900112&page=6&article=053&stringtohighlight=mandeville+viscount. Same print title and p.</ref></blockquote> === The ''Cork Constitution'' === About Mr G Baird (Mr. Abington) from the ''Cork Constitution'':, referring at the end to the [[Social Victorians/Boxing/Fracas at the Trocadero|Fracas at the Trocadero]] or a restaurant:<blockquote>This well-known sportsman (Mr G Baird) who is occupying an unenviable position owing to his conduct at the Smith-Slavin piece of brutality, is the largest owner of race horses in England. He has no less than 100 horses in training, and though he won £10,000 in stakes last year it is computed that he should have won £25,000 additional to even pay his way. Anent that fight which has caused such indignation in the country, a London music hall was last week the scene of a pretty disturbance. Mr Abington and Lord Mandeville occupied a box during the entertainment, and received, amongst others, Charlie Mitchell, the well-known pugilist; an angry discussion ''re'' the fight ensued, and concluded by the peer measuring himself on the floor from one Mitchell’s "lefts."<ref>"Mr. Abington." ''Cork Constitution'' 14 January 1890, Tuesday: 7 [of 8], Col. 5b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001648%2F18900114&page=7&article=099&stringtohighlight=mandeville+viscount. Same title and p.</ref></blockquote> === The ''Colonial Standard and Jamaica Despatch'' === This report from the ''Colonial Standard and Jamaica Despatch'' mentions fighting in the streets (at a corner in Fleet Street) and tosses in an unrelated sentence about an upcoming match between a West Indian and J. L. Sullivan:<blockquote>The disgraceful affair, called by courtesy and [sic] "International Prize Fight," which desecrated the peaceful dignity of the quaint old city of Bruges on Monday morning ought to convince people that there is nothing left to hope for in the Prize Ring. [2c–3a] At eight o'clock in the morning, Jem Smith, self-styled Champion of England, and Frank Slavin, the Australian Champion, with their seconds and parties, met in the grounds of a private gentleman to try conclusions. The "fight" lasted for twenty-five minutes, and was then brought to an abrupt conclusion by the referee declaring it drawn — he would have been killed had he given it to Slavin, who proved to be incomparably the better man. Fourteen rounds were fought in all, during the whole of which time Smith's friends and Smith's seconds worried, and at last assaulted, the Australian, who behaved quite pluckily. It is a disgrace to the name of Englishmen that this brutal scene should go forth to the world. Smith's backers knew well that he had no chance of winning, so they laid their plans to smash Slavin on their own account. It is a wonder that he was not killed — as was reported in Fleet-street at ten o'clock — for Smith's roughs struck and kicked him whenever he came near their corner. Although Lord Mandeville, Lord Rosslyn, Sir M. Sandys, and a few other members of the upper circle, were present at the fight, the bulk of the spectators were the refuse of the gambling rings of London, blackguards and ruffians of the deepest dye. The stakes were £1,000. So great an amount of indignation has been expressed on all sides at the treatment to which the Australian was exposed, that a subscription has been opened up for him. The West Indian, Jackson, is just leaving London for San Francisco, in order to get ready to meet J. L. Sullivan.<ref>"London." ''Colonial Standard and Jamaica Despatch'' 16 January 1890, Thursday: 2 [of 4], Col. 2c–3a [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0004059%2F18900116&page=2&article=019&stringtohighlight=mandeville+viscount. Same print title, n.p.</ref></blockquote> == Notes and Questions == # ==References== {{reflist}} 6s1am0o1qxxbfh8uxiwiwjbyp4v88tc 0 332549 2842450 2842429 2026-10-24T20:03:57Z DavidMCEddy 218607 wdsmth 2842450 wikitext text/x-wiki {{Research project}} :''This article invites all to contribute references and interpretations to the growth in average annual income, Gross Domestic Product (GDP) per capita at purchasing power parity, as applied to leading countries in the [[w:Maddison Project|Maddison Historical Statistics Project]].'' :''This essay is on Wikiversity to encourage a wide discussion of the issues it raises moderated by the Wikimedia rules that invite contributors to [[w:Wikipedia:Be bold|“be bold but not reckless,”]] contributing revisions written from a [[Wikiversity:Disclosures|neutral point of view]], [[Wikiversity:Cite sources|citing credible sources]] -- while raising other questions and concerns on the associated [[Wikiversity:FAQ|'''{{'}}“Discuss”'''{{'}} page]] -- and treating others with respect.'' ::''This article uses [[w:ISO 8601|ISO 8601]] dates except for References, which are controlled by standard Wikidata formatting, and direct quotes. In the initial author's experience, [[ISO 8601 and computing differences between dates|ISO 8601 dates seem to make it easier to remember dates and to compute differences between them.]]'' == Abstract == This article first identifies and plots GDP per capita for leading countries in the Maddison data then discusses comments by Nobel prize winning economists and others to interpret what appears in said plot. No one really knows how to make economies grow. However, growth at the technology frontier requires creative destruction. Also, inequality is limited by diversity of ownership and control of the major media. == Maddison Historical Statistics == [[File:MaddisonDataLeaders1349-2022.svg|thumb|''Figure 1. World leaders in GDP per capita 1349-2022 (NLD = initially Holland to 1809 and the Netherlands after; GBR = initially England to 1700, Great Britain to 1870, then the UK; USA starting with the 13 colonies that became the US; SGP = Singapore).<ref>See Bolt and van Zanden (2024) for the Maddison Data generally, Broadberry et al. (2015) for the data on England 1252–1700 and on Great Britain until 1870, Sugimoto (2011) for Singapore to 2007, van Zanden and van Leeuwen (2012) for the data on Holland 1349–1807, and Smits et al. (2000) for the data on the Netherlands 1808-1913. The dramatic drop in the orange line between 1807 and 1808 could be due to the change in reference from [[w:Holland|Holland]] to the Netherlands, as Holland at that time was more prosperous than the rest of the Netherlands.</ref>'']] The [[w:Maddison Project|Maddison Historical Statistics Project]] (aka Maddison Project) is widely recognized as the premier data source on historical macroeconomic data. It is named after [[w:Angus Maddison|Angus Maddison]] (1926-2010), who was a pioneer in [[w:National accounts|national accounts]], especially estimating [[w:Gros Domestic Product|gross domestic product]] (GDP) per capita (GDPpc) and population (pop) at various times dating back to the year 1 of the [[w:Common Era|Common Era]] (CE / AD). After Maddison's death, the Maddison Project has continued to update the database to include the best sources available on any country for any historical period since year 1. When accessed 2026-10-10 the version of the data available had been released 2024-09-18 containing data to 2022.<ref>Bolt and van Zanden (2024).</ref> This dataset includes 23,280 observations on at least one of GDPpc and pop for an average of 137 years for 169 countries. Only 18 countries had the highest GDPpc for at least one year. Annual data starts in 1252, and that's only for one country, GBR = England / Great Britain / the United Kingdom. NLD (Holland / the Netherlands) had the highest GDPpc for 97 percent of the years between 1349 and 1807, and the Dutch leadership in Figure 1 may have ended then, not because of the devastation from the [[w:Napoleonic Wars|Napoleonic Wars]] (which negatively impacted them) but because the available data prior to 1807 are for [[w:Holland|Holland]], while more recent data are for the Netherlands, of which Holland is only a part, though probably the most prosperous part at that time. This analysis focuses on data since 1349, in part because data were available for at most 4 countries for most years prior to that date. (The 3-letter country codes used here are [[w:ISO 3166-1 alpha-3|ISO 3166-1 alpha-3]], used in the Maddison data.) We are interested in technology leaders, so we eliminate the [[w:petrostates|petrostates]] of Qatar (QAT), Kuwait (KWT), and United Arab Emirates (ARE), plus Norway (NOR), which owes a substantial portion of their wealth to democratic management of [[w:North Sea oil|North Sea oil]].<ref>Dośpiał-Borysiak (2017).</ref> With these restrictions 12 countries had the highest GDPpc for at least one year. However, FRA (France), ITA (Italy), SWE (Sweden), BEL (Belgium), NZL (New Zealand) and CHE (Switzerland) led for at most 7 of the 675 years between 1349 and 2022; we eliminate those, because including them would seem to add more clutter than clarity. We also eliminate AUS (Australia), which led for 17 of the 39 years between 1853 and 1891, when it was competing for the leadership with GBR and USA, for the same reason. And we eliminate LUX (Luxembourg), which has a population under a million. This leaves NLD, GBR, USA, and SGP (Singapore), plotted in Figure 1. This plot has some interesting features, which we discuss in the remainder of this research report. == Nobel economists and others == Banerjee and Duflo (2019, p. 207) said, "despite the best efforts of generations of economists, the deep mechanisms of persistent economic growth remain elusive." That appeared shortly before they shared the 2019 [[w:List of Nobel Memorial Prize laureates in Economic Sciences|Nobel Memorial Prize in Economics]] with [[w:Michael Kremer|Michael Kremer]] for documenting how [[w:Randomized controlled trial|randomized controlled trial]]s could be used to alleviating poverty in [[w:Development economics|development economics]]. While Banerjee and Duflo insist that no one knows how to make an economy grow, other leading economists recommend partial solutions. For example, [[w:Joel Mokyr|Mokyr]] (2017, p. 53) said, "The willingness of the Netherlands and later Britain and the United States to tolerate Jews and dissenting Christians contributed a great deal to their economies, especially in high-skilled manufacturing and financial services." In other words, immigrants made an outsized contribution to the growth of NLD, GBR, and US, visible in Figure 1. Mokyr's comment appeared over a year before Banerjee and Duflo (2019) and roughy eight years before Mokyr shared the 2025 Nobel Memorial Prize in Economics with [[w:Philippe Aghion|Aghion]] and [[w:Peter Howitt (economist)|Howitt]]. Mokyr (2017, p. 255) also said that, "intellectual innovation will eventually end up being increasingly resisted by an incumbent technology; if that resistance is not overcome, technological progress may be extinguished." That resistance, he suggested, drives {{quote| the empirical regularity that no society remains at the cutting edge of technological creativity for very long<ref>Mokyr (2017, p. 260). [[w:Robert Michels|Michels]] (1911) called this "the [[w:Iron law of oligarchy|iron law of oligarchy]]".</ref>.}} Mokyr (2017, p. 260; 1994) calls this "Cardwell's Law" after Cardwell (1972), which describes {{quote| the interaction between technological creativity and political competition ... based on the observation that technological progress encounters resistance from various groups that believe they stand to lose from innovation. These pressure groups will try to manipulate the political system to suppress successful innovation.}} Further documentation of this appears in Acemoglu and Johnson (2023) and Acemoglu and Robinson (2012). This provides one explanation for the end of growth of the economy of Holland shortly before 1600 and for the slowing of economic growth in recent decades in the US, the UK, and the Netherlands, visible in Figure 1. It can also explain [[w:History of the Roman Empire|the rise and fall of the Roman Empire]], of the [[w:Maya civilization|Maya civilization]], and other societies that became very rich from allowing commoners to become entrepreneurs and then declined after previous winners changed the rules of the political economy to block further [[w:creative destruction|creative destruction]]. This includes how [[w:History of the Republic of Venice|Venice]] became "possibly the richest place in the world, with the most advanced set of inclusive economic institutions", dominating trade in the [[w:Mediterranean Sea|Mediterranean]] based on a system that encouraged entrepreneurial commoners to undertake risky trading ventures starting after 810 CE and ending when leaders made wealthy by such ventures changed the rules to end that system starting in 1297, destroying their dominance of Mediterranean trade "and converting Venice into a museum", in the words of Acemoglu and Robinson (2012, pp. 152-155). Mokyr (2017, p. 60) also mentioned "entrepreneurs of hate", who convince others to dislike some group in a way that may benefit those "entrepreneurs" and major media organizations, who promote such "entrepreneurs" to please those who control much of the money for the media. For-profit social media have been particularly effective in herding their audiences into echo chambers, reinforcing their preconceptions in ways that increase political polarization and violence and threaten democracy.<ref>See, e.g., DiResta (2026), DiResta and Keinfeld (2025), and the Wikiversity article on "[[Managing the good and the evils of social media]]", accessed 2026-10-14.</ref> When successful this protects current elites from competition that would otherwise drive [[w:creative destruction|creative destruction]], benefitting the public at large while threatening existing elites. For example, most research on immigrants suggest that on average immigrants are more entrepreneurial and no more likely to commit crime than native born,<ref>e.g., Martínez et al. (2017); Muchow and Stanishevska (2026); Ajzenman(2023).</ref> and some studies report that [[w:Sanctuary city|sanctuary jurisdiction]]s have on average higher median incomes and less crime.<ref>Martínez et al. (2017); Wong (2017). See also the Wikipedia article on "[[w:Sanctuary city|sanctuary city]]", accessed 2026-10-12.</ref> Where immigrant crime is higher, it typically result from anti-immigrant labor market restrictions or from young less-educated males with irregular status.<ref>Marie and Paolo (2024). See also the Wikipedia article on "[[w:Immigration and crime|Immigration and crime]]", accessed 2026-10-12.</ref> But major media outlets make money from featuring charismatic figures, who attract an audience without offending people who control much of the money for the media. In 2017 President [[w:Immigration policy of the first Trump administration|Trump]]'s Attorney General [[w:Jeff Sessions|Jeff Sessions]] quoted a research study that supposedly showed that sanctuary cities had more crime than cities without sanctuary policies. The authors of that study insisted that Sessions had ''misrepresented'' their work, which concluded "that sanctuary policies have no effect on crime rates, despite narratives to the contrary."<ref>Roll (2017); Gonzalez O’Brien, et al. (2017).</ref> To the extent that these comments provide a balanced summary of the research in this area, it suggests that leaders like Trump and Sessions, to name only two, can be fairly describes as "entrepreneurs of hate". We next discuss competition in media markets before discussing immigrants further. === Diversity of ownership of the major media === Acemoglu and Johnson (2023, esp. ch. 4) describe how religious leaders in hierarchical societies prior to [[w:James VI and I|King James]] of the [[w:King James Version|King James Bible]] convinced commoners to live in poverty while giving increasing shares of what they produced, so religious and secular elites could live in opulence. Prior to King James the major media were religious organizations. In such hierarchical societies some religious leaders were entrepreneurs of hate, exhorting the faithful to fear and hate designated evil others and give increasing shares of what they produce so secular elites would "protect" them in wars against designated evil others. Acemoglu and Robinson (2012, p. 461) said, "Empowerment of society at large is difficult to coordinate and maintain without widespread information about whether there are economic and political abuses by those in power. ... Pamphlets and books informing and galvanizing people played an important role during the [[w:Glorious Revolution|Glorious Revolution]] [1688] in England, the French Revolution [1789-1799]" and in the US. ([[w:Daron Acemoglu|Acemoglu]], [[w:Simon Johnson (economist)|Johnson]] and [[w:James A. Robinson|Robinson]] shared the 2024 [[w:List of Nobel Memorial Prize laureates in Economic Sciences|Nobel Memorial Prize in Economics]].) But pamplets started challenging established power structures much earlier. The publication of [[w:Martin Luther|Martin Luther]]'s ''[[w:Ninety-five Theses|Ninety-five Theses]]'' in Latin 1517-10-31 was quickly condemned by [[w:Pope Leo X|Pope Leo X]] and his hierarchy. [[w:Johann Tetzel|Johann Tetzel]], who had been selling [[w:Indulgence|indulgence]]s, published a pamphlet calling for Luther to be burnt for [[w:heresy|heresy]]. Eight hundred copies were reportedly sent to [[w:Wittenberg|Wittenberg]] to be sold, but students seized and burned them.<ref>Hendrix (2015, p. 64), quoted from the Wikipedia article on the ''[[w:Ninety-five Theses|Ninety-five Theses]]'', accessed 2026-10-14.</ref> This led to a pamphlet war. Luther's brief German-language reply was short and easy for commoners to understand<ref>Leppin and Wengert (2015, p. 389), cited from the Wikipedia article on the ''[[w:Ninety-five Theses|Ninety-five Theses]]'', accessed 2026-10-14.</ref> and was much more popular than Tetzel's.<ref>Pettegree (2015, p. 145), cited from the Wikipedia article on the ''[[w:Ninety-five Theses|Ninety-five Theses]]'', accessed 2026-10-14.</ref> Less than twenty years later, in 1534, King [[w:Henry VIII|Henry VIII]] declared himself head of the [[w:Church of England|Church of England]], denying the authority of the Pope. To secure the support of his subjects, he drove a propaganda war that included print literature<ref>Raymond (2006), cited from the Wikipedia article on "[[w:Pamphlet war|pamphlet war]]", accessed 2026-10-15.</ref> and theatre and minstrel troupes funded by the king. The latter traveled around the land to promote the new religious practices. Henry was portrayed as the glorious king of England, bravely and heroically defending the true faith against foreign devils, especially the Pope and Catholic priests, monks, etc.<ref>Hibbert et al. (2010, p. 928), cited from the Wikipedia article on "[[w:Henry VIII|Henry VIII]]", accessed 2026-10-15.</ref> Henry died in 1547. Less than a dozen years later, the crown passed to [[w:Elizabeth I|Elizabeth I]]. Her limited personal wealth and English tradition required her to negotiate with Parliament for tax money. Acemoglu and Robinson (2012, pp. 105-111) noted that she was forced to limit the granting of monopolies and allow a class of merchants to compete in international trade. Many English merchants became wealthy, helped by the defeat of the [[w:Spanish Armada|Spanish Armada]] in 1588. When Elizabeth died in 1603, she was replaced by her cousin, King [[w:James VI and I|James VI of Scotland]], who then became also James I of England. He claimed [[w:Divine right of kings|divine right of kings]], threatening this class of merchants. To raise money without asking Parliament, he granted patents and monopolies to those who would pay. Parliament responded by passing the [[w:Statute of Monopolies|Statute of Monopolies]] on 1624-05-29. This statute repealed some past and future patents and monopolies except for novel inventions. But that Statute of Monopolies did not end the conflict. It ultimately went public, described as, "A Pamphlet War in England, 1641-1643".<ref>Washington (2018).</ref> Some pamphlets supported [[w:Charles I of England|Charles I]], who became King after James died in 1625. Others supported Parliamentarians and [[w:Levellers|Levellers]]. The latter supported radical ideas like [[w:popular sovereignty|popular sovereignty]], extended suffrage, equality before the law and religious tolerance. In 1642 war broke out between Parliament and Charles I, though many in Parliament sided with Charles -- especially those, who had received monopolies from him.<ref>Acemoglu and Robinson (2012, p. 189).</ref> The war was won by the [[w:New Model Army|New Model Army]], which represented many in Parliament plus Levellers. The latter were excluded from the government that followed the [[w:Execution of Charles I of England|beheading of King Charles I]] in 1649.<ref>Foxley (2013, p. 207), cited from the Wikipedia article on [[w:Levellers|Levellers]], accessed 2026-10-15.</ref> However, that new government stopped granting as many monopolies and allowed so many commoners to become entrepreneurs that the GDP per capita began to grow at rates that are key features of Figure 1 for England and the 13 English colonies that became the US over a century later. One explanation for that economic growth was the diversity of independent publishers of pamphlets, books and newspapers. [[w:Robert W. McChesney|McChesney]] and [[w:John Nichols (journalist)|Nichols]] (2010, 2016) suggested that the US [[w:Postal Service Act|Postal Service Act]] of 1792<ref>Wikipedia "[[w:Postal Service Act|Postal Service Act]]", accessed 2023-07-11.</ref> made a major contribution to making the US what it is today. Under that act, newspapers were delivered up to 100 miles for a penny, when postage for letters was between 6 and 25 cents depending on distance.<ref>Historian, USPS (2010).</ref> McChesney and Nichols estimated that between 1840 and 1844, the US postal subsidy was a little over 0.2% of GDP.<ref name=McC-N2010>McChesney and Nichols (2010, pp. 310-311, note 88).</ref> During the first half of the nineteenth century the US had more independent newspaper publishers (more independent media organizations) per million population than at any other time or place in human history.<ref>John (1995). See also the interview with him at "[[Media concentration per Columbia History Professor Richard John]]" on Wikiversity, accessed 2026-10-12.</ref> In 1932 with official unemployment over 20 percent, the conservative argument that blames the poor for their poverty did not sell newspapers, and the US still had sufficient competition in media markets that presidential candidate Franklin Roosevelt (FDR) got a mandate to spend the money to put humans back to work while taxing the ultra-wealthy like they had never been taxed before or since and during World War II to institute wage and price controls under the [[w:Office of Price Administration|Office of Price Administration]]. The latter prevented price gouging by government contractors that had generated inflation and stifled economic growth in previous major wars (World War I, the Civil War, and the War of 1812). Instead, the US averaged 7 percent per year growth for FDR's 12 years.<ref>The "6 percent per year" growth during FDR's 12 years was computed as the average of log(GDPpc) in the Maddison Data. To be precise, that computation produced 0.0597; converting that from a log scale to percent increase gives 0.0615. Using data from [[w:MeasuringWorth|MeasuringWorth]] instead of Maddison, we get 0.783 and 0.815, respectively. More documentation of this is available in "[[Media Literacy and You/Fox, the Great Depression, the Great Recession, and our future]]" on Wikiversity, accessed 2026-10-13.</ref> One of the most interesting features of Figure 1 is the behavior of the US economy in the administrations of [[w:Herbert Hoover|Herbert Hoover]] and FDR: Under Hoover, the economy declined at 9 percent per year for 4 years, a financial disaster that created the public demand for change that gave FDR the mandate to do what he did. Economic growth under two other presidents are numerically comparable to FDR's numbers but for much shorter periods: (1) [[w: Warren G. Harding|Warren G. Harding]] averaged 7 percent annual growth for two years before he died in 1923 during the [[w:Roaring Twenties|Roaring Twenties]]. (2) [[w:Rutherford B Hayes|Rutherford B Hayes]], who took office in 1877 at the end of the [[w:Panic of 1873|Panic of 1873]], oversaw 5.5 percent growth per year for his 4 years as president. Those impressive sounding average growth figures are so short that they are not noticeably different from other minor features visible in Figure 1. FDR also dramatically reduced inequality. For over a century between about 1820 and 1932 the top 1 percent of the US population owned roughly 45 percent of net personal wealth in the US, while the bottom 90 percent owned roughly 22 percent. Shortly after FDR died, the wealth of the top 1 percent and bottom 90 percent were even at roughly 30 percent. FDR also oversaw dramatic growth in the power of organized labor. Their successes continued to reduce inequality to roughly 22 percent for the top 1 percent vs. 37 percent for the bottom 90 percent when Ronald Reagan became president. Since then, [[w:Reaganomics|Reaganomics]] have been rebuilding the [[w:Robber baron (industrialist)|Robber Baron]]s.<ref>This history of inequality is documented in [[:File:Share of US wealth 90p99.svg|Figure 7. Shares of US wealth - bottom 90 and top 1 percent, 1820-2023]] in the Wikiversity article on "[[w:Media Literacy and You/Fox, the Great Depression, the Great Recession, and our future|Media Literacy and You/Fox, the Great Depression, the Great Recession, and our future]]" in Wikiversity, accessed 2026-10-13.</ref> FDR made enemies among the wealthiest and most powerful. In a [[q:Franklin D. Roosevelt#Address at Madison Square Garden (1936)|speech in Maddison Square Garden in 1936]], FDR mentioned, "the old enemies of peace: business and financial monopoly, speculation, reckless banking, class antagonism, sectionalism, war profiteering. ... They are unanimous in their hate for me -- and I welcome their hatred." They doubtless could see that they could no longer afford as many personal servants -- maids, butlers, cooks, chauffeurs, etc. -- as they had before FDR. Broadcasting was relatively new when FDR took office. The economics of broadcasting naturally favored a very few networks, whose power concerned FDR. In 1941 his [[w:Federal Communications Commission| Federal Communications Commission]] (FCC) adopted the [[w:Mayflower doctrine|Mayflower doctrine]], which prohibited broadcasters from advocacy. When FDR died, people with power were able to get the Mayflower doctrine replaced by the more lenient [[w:Fairness Doctrine|Fairness Doctrine]], which opened the floodgates for entrepreneurs of hate, who drove the [[w:McCarthyism|McCarthy Red Scare]] in the US and many other problems since. Over time this became worse, facilitated by increasing concentration of ownership of the media. That contributed to the end of the Fairness Doctrine starting in 1985. That revocation and especially the more recent rise of for-profit social media have amplified the ability of entrepreneurs of hate to make money both for commercial social media and for themselves by disseminating lies about others, e.g., immigrants.<ref>On how for-profit social media companies promote entrepreneurs of hate, see, e.g., DiResta (2026) and the Wikiversity article on, "[[Managing the good and the evils of social media]]", accessed 2026-10-12. DiResta (2026, p. 1) described the promotion of "influencers" by for-profit social media as essentially unleashed "a war for power, influence, and reality itself—a conflict in which the ends justify any means, and one that has ripped America apart, threatened our very democracy, and consumed my own life for the past decade, leading to death threats, congressional subpoenas, lawsuits, and legal bills running into seven figures." In the interview on Wikiversity, DiResta said her legal bills to that point had exceeded $7 million.</ref> And the rate of economic growth in the US (and the Netherlands and the UK) have been slowing in recent decades, as is visible in Figure 1. Inequality expert [[w:Thomas Piketty|Piketty]] (2022, p. 139) noted, {{quote| In the United States, the national income per inhabitant rose at a rate ... of 2.2% between 1950 to 1990 when the top tax rate reached on average 72%. The top rate was then cut in half, with the announced objective of boosting growth. But in fact, growth fell by half, to 1.1% per annum between 1990 and 2020 ... . Beyond a certain level of inequality, repeatedly increasing differences in income and wealth has clearly had no positive effect on economic dynamism. In sum, all the data at our disposal today suggest that virtually confiscatory tax rates have been an immense historical success. They have made it possible to greatly reduce the divergences of fortunes and incomes, while at the same time improving the situation of the middle and lower classes, developing the welfare state, and stimulating better economic and social performance overall. Historically, it is the battle for equality and education that has made economic development and human progress possible, and not the veneration of property, stability, and inequality.}} Table 1 compares Piketty's growth numbers with computations based on the data from both Maddison and [[w:MeasuringWorth|MeasuringWorth]]: The numbers are close but not exact. {| class="wikitable sortable" style="text-align: center;" |+ Table 1. US growth rate 1950 to 1990 to 2020 |- ! ! colspan=3 | US GDP per capita !! colspan=2 | Growth (% / yr) |- ! source || 1950 !! 1990 !! 2020 !! 1950 - 1990 !! 1990 - 2020 |- | Piketty || - || - || - || 2.2% || 1.1% |- | Maddison || $15,240 || $36,982 || $54,379 || 2.24% || 1.29% |- | MeasuringWorth || $16,208 || $40,191 || $61,127 || 2.30% || 1.41% |} Acemoglu and Robinson (2012, pp. 105-106) provide a historical analysis supporting Piketty's concern about inequality slowing growth. They note that in 1588, England, Spain and France {{quote| were ruled by relatively absolutist monarchs: [[w:Elizabeth I|Elizabeth I]] in England, [[w:Philip II of Spain|Philip II]] in Spain, and [[w:Henry III of France|Henry III]] in France. All were battling with assemblies of citizens -- such as the [[w:Parliament of England#Tudor era (1485–1603)|Parliament]] in England, the [[w:Cortes Generales#Habsburg rule (16th–17th centuries)|Cortes]] in Spain, and the [[w:Estates General (France)#Revival in 1560–1614|Estates-General]] in France -- that were demanding more rights and control over the monarchy. ... [T]he English Parliament and the Spanish Cortes had power over taxation, while the Estates-General did not. In Spain this mattered little, because after 1492 the Spanish Crown had a vast American empire and benefited massively from the gold and silver found there. In England ... Elizabeth I was far less financially independent, so she had to beg Parliament for more taxes. In exchange, Parliament demanded concessions, in particular restrictions of the right of Elizabeth to create monopolies. ... In Spain the Cortes lost a similar conflict.}} Consistent with Acemoglu and Johnson (2023) and the discussion above, inequality tends to increase elite control of the media, which it makes it harder for commoners to get a fair hearing of their concerns. === Media conflicts of interest in IP law === Major media have [[w:Conflict of interest|conflicts of interest]] in honestly reporting on anything involving [[w:Intellectual property|intellectual property]] (IP) law. An example of Cardwell's law, mentioned above, is the [[w:Copyright Term Extension Act|Sonny Bono Copyright Term Extension Act]], also known as the Mickey Mouse Protection Act of 1998. The copyright term on ''[[w:Steamboat Willie|Steamboat Willie]]'', the first [[w:Mickey Mouse|Mickey Mouse]] animated film that Disney distributed, was extended multiple times by the US Congress following extensive lobbying by the film industry led by [[w:The Walt Disney Company|the Walt Disney Company]]. ''Steamboat Willie'', which debuted in 1928, finally entered the [[w:Public domain in the United States|US public domain]] on January 1, 2024. Eric Eldred and others filed suit, claiming that allowing any number of retroactive extensions violated the [[w:Copyright Clause|Copyright Clause]] of the US Constitution, which says, {{quote| [the United States Congress shall have power] To promote the Progress of Science and useful Arts, by securing for limited Times to Authors and Inventors the exclusive Right to their respective Writings and Discoveries.}} In ''[[w:Eldred v. Ashcroft|Eldred v. Ashcroft]]'' (2003), the [[w:Supreme Court of the United States|US Supreme Court]] disagreed with Eldred and allowed the statute to remain in effect. However, that case drew enough public attention that copyright terms have not (yet) been extended further, and ''Steamboat Willie'' is currently in the pubic domain. [[w:Lawrence Lessig|Lessig]] (2004) ''[[w:Free Culture (book)|Free Culture]]'' describes other examples, where US copyright law seems to ''limit'' rather than "promote the Progress of Science and useful Arts". This includes a case where a 2002 freshman, Jesse Jordan, modified a search engine to make it easier for other student to find pictures, research, notes, movie clips, etc., on an [[w:intranet|intranet]] at [[w:Rensselaer Polytechnic Institute|Rensselaer Polytechnic Institute]] (RPI). The [[w:Recording Industry Association of America|Recording Industry Association of America]] (RIAA) sued Jordan and three other students for close to $100 billion for copyright violations -- six times the total profit of the film industry in 2001. Jordan had an uncle, who was an attorney, who explained that the legal bills for fighting that case would be at least $250,000. That would bankrupt his parents; they could not recover that money, even if they won. Jordan had saved $12,000 to cover part of his college expenses. RIAA accepted that and settled the case.<ref>Lessig (2004, p. 42).</ref> This is an example of a [[w:Strategic lawsuit against public participation|strategic lawsuit against public participation]] (SLAPP), which in this case rested on questionable interpretations of copyright law. The major media have conflicts of interest in honestly reporting on anything involving (a) copyright laws, because much of their income is derived from copyright royalties, or (b) anything that might offend their major accounts, including major corporations. And anti-SLAPP laws, which exist in some jurisdictions but not others,<ref>[[w:Robert Corn-Revere|Robert Corn-Revere]], Chief Counsel for the [[w:Foundation for Individual Rights and Expression|Foundation for Individual Rights and Expression]], supports anti-SLAPP laws and the "JAWBONE act", to limit the power of the government to threaten questionable actions, like the Trump administration's threats against the ABC network over a Jimmy Kimmel monolog. See the Wikiversity article on "[[Foundation for Individual Rights and Expression]]", accessed 2026-10-16.</ref> limit the ability of major corporations to abuse their power against potential challengers without such deep pockets. Similarly Graves (2023) asked, "Do copyrights and paywalls on refereed academic journals violate the US constitution?" No one expects to make money from copyright royalties for drafts submitted to refereed academic journals. They do it out of a felt need (a) to contribute to the future of humanity or (b) to build their reputation or (c) because promotions at many universities are based partly on publication records. Graves is not alone in his concern about this: Many European countries do not allow reports of research they fund to be published behind paywalls.<ref>Dimi Zagorski is concerned that copyright exceptions for scientific research are different in different European countries, and that's a problem for the Wikimedia Foundation. See the discussion of this in the Wikiversity article on "[[Wikimedia concerns with European copyright rules including AI and scientific research]]". That's the opposite of US law, which grants patents to universities, research institutions, private companies, and individuals for innovations developed under government contracts, per the Bayh-Dole Act of 1980, described by Baker (2016, p. 80).</ref> Regarding patent law, the major media have conflicts of interest in discussing proposed changes affecting major accounts, but they make little use of patent law themselves. This seems to have contributed to the claim that, "[O]ver the last four decades ... stronger patent protection seems to be associated with slower growth" in the US, according to [[w:Dean Baker|Baker]] (2016, pp. 77-78). A potentially similar phenomenon appears in a comparison of 21 different European countries in an analysis that displays "a positive relationship between anticorruption measures in a country and rate of innovation in that country as measured by patents per capita", discussed by Aghion et al. (2021, Figure 15.1b, pp. 292-293).<ref>Usher and Kim-Leffingwell (2022) documented how each member of the [[w:Institute for Nonprofit News|Institute for Nonprofit News]] (INN), relatively small nonprofits independent of major media organizations and not funded by advertising, in a US federal court district was associated with on average 1.4 additional prosecutions for political corruption per year between 2003 and 2019. They found on average 10 federal prosecutions for political corruption for each federal court district per year. Thus, 1.4 additional prosecutions per year per district represents an increase of 14 percent per member of INN.</ref> === Immigrants === We now return to discussing the initiation of economic growth following the beheading of Charles I in 1649 visible in Figure 1. Charles I was replaced by [[w:English Council of State|Lord Protector Cromwell]], who refused to authorize as many monopolies as his predecessors, as mentioned above. He also encouraged Jews and dissenting Christians to settle in England, as mentioned by Mokyr (2017, p. 53), cited above. But Mokyr is not alone in claiming that immigrants contribute to economic growth. Aghion et al. (2021, p. 266) cite Arkolakis et al. (2020) for explaining how the period 1880–1920 is noteworthy for two parallel evolutions. First, the United States caught up with England and France to become the most technologically advanced country in the world as well as the wealthiest in terms of GDP per capita. Second, the United States welcomed large numbers of immigrants, most notably from Europe. The purpose of the cited study was to demonstrate a causal connection between these two phenomena: in other words, to show that European immigration was responsible for the United States’ taking the lead in technology. ... More precisely, if the United States had completely closed its borders to all foreign immigrants in 1880, its per capita GDP would have been 30 percent lower than the level actually attained in 1920". === Industrial Revolution === There seems to be a broad consensus that the [[w:Industrial Revolution|Industrial Revolution]] began in England. However, there is less agreement on when it began. Mokyr (2017, p. 3) says, "The British Industrial Revolution of the late eighteenth century unleashed a phenomenon never before even remotely experienced by any society." He described "three cultural elements" required for "sustainable economic growth propelled by technological progress": (1, 2) The belief that it was both ''possible'' and ''desirable''. (3) Supportive policy measures had to be implemented. "[B]efore 1750 they did not produce anything like an Industrial Revolution or sustainable economic growth propelled by technological progress.<ref>Mokyr (2017, pp. 19-20).</ref> Acemoglu (2026, p. 53) similarly mentioned the eighteenth century in saying, "The breakthroughs in production technology that started in the middle of the eighteenth century in Britain unleashed the Industrial Revolution." Aghion et al. (2021, esp. pp. 29-30) offer a ''later'' date for the start of the Industrial Revolution: They report that, "European economies prior to 1820 experienced a series of phases of growth and decline. This raises a question: How did the pre-1820 period differ from the takeoff that started with the first Industrial Revolution?" Aghion et al. (2021, p. 20) describe the "first Industrial Revolution" as having been triggered by "the invention of the steam engine by James Watt in the 1770s ... . The invention of electricity set off the second Industrial Revolution, whose golden age was in the first half of the twentieth century. This second revolution originated with the invention of the [[w:Incandescent light bulb|light bulb]] by [[w:Thomas Edison|Thomas Edison]] in 1879 and of the [[w:dynamo|dynamo]] by [[w:Werner von Siemens|Werner von Siemens]] in 1866."<ref>The dynamo was independently invented by [[w:Henry Wilde (engineer)|Henry Wilde]], [[w:Charles Wheatstone|Charles Wheatstone]], [[w:Samuel Alfred Varley|Samuel Alfred Varley]] and Siemens, all about the same time, according to the Wikipedia article on [[w:Dynamo#Practical designs|dynamo]], accessed 2026-10-18.</ref> The image in Figure 1 raises questions about these claims by Mokyr, Acemogu, and Aghion et al.: It shows growth followed by stagnation in Holland but not "phases of growth and decline" with the exception of wars and [[w:Herbert Hoover|Herbert Hoover]]. It also does ''not'' show the slope (i.e., rate of growth) increasing starting around 1750 or the 1770s or 1879 or 1886. Instead it shows sustained growth in GDP per capita for England / Great Britain / the UK starting ''abruptly'' after 1649 with economic disruptions during the reign of [[w:Anne, Queen of Great Britain|Queen Anne]] (1702-1714) and the [[w:War of the Spanish Succession|War of the Spanish Succession]] (1701-1714). This was followed by relatively steady but slower growth as England was ruled by Kings [[w:House of Hanover#Monarchs of Great Britain, Ireland, and Hanover|George I, II, and III]] (1714-1820). Another dramatic increase in the slope started abruptly about 1819. Acemoglu and Johnson (2023, chapter 1) are more consistent with Figure 1. They said that, "the Industrial Revolution ... [was] rooted in institutional changes that started empowering the middling sort of English people from the sixteenth and seventeenth centuries onward." ... However, "the quality of life of a European peasant circa 1700 was not much different from that of an Egyptian peasant two thousand or seven thousand years previously." (chapter 5) They also claim that, "English national production per capita doubled from 1500 to 1700", (chapter 5), and "Growth emerged slowly in the second half of the eighteenth century as the Industrial Revolution, based on major technological breakthroughs and their application in industry, took root." (p. 45, chapter 2) Indeed, Figure 1 shows English GDP per capita doubling between 1500 and 1700, but it appears to have been fairly constant from 1500 to 1649, then roughly doubled in the next half century. [The data on England up to 1700 in Figure 1 came from Broadberry et al. (2015), which was not available when Acemoglu and Robinson were writing ''Why Nations Fail'', which appeared in 2012.] Acemoglu and Robinson (2012, ch. 11) describe the financial instability and slow growth during the reigns of Queen Anne and George I, II, and III. After the Glorious Revolution, France, Spain, the pope, and others refused to recognize [[w:William III of England|William and Mary]] (1689-1702) and then Queen Anne as heads of state of England. Instead they continued to support the claims of James II and then his son, [[w:James Francis Edward Stuart|James Francis Edward Stuart]] (the "Old Pretender", a brother of Queen Anne), after James II died in 1701. During the War of the Spanish Succession, each side tried to exploit the internal divisions of the other. Owen (1938, ch. 8) said France almost succeeded in an attempt in 1708 to [[w:Planned French invasion of Britain (1708)|land between 5 and 6 thousand soldiers in northeast Scotland]]. [[w:Louis XIV|Louis XIV]] had hoped to get the Old Protector named as King James VIII of Scotland. There were still [[w:Jacobitism|Jacobites]] in Scotland, who supported the Old Pretender and likely would have supported the invaders if they had landed. If they had succeeded, it would have created major problems for Queen Anne and her supporters. However, the Royal Navy prevented the invaders from disembarking, and they returned to France. When Queen Anne died in 1702, she was succeeded by George I. His Parliament was dominated by the Whig party, which represented the new mercantile and economic interests and had been the main organization behind the Glorious Revolution. Once in power, they were tempted to use their newly found position to prey on the rights of others but were constrained both by competing groups in Parliament, "and by the very institutions that they had fought to introduce to strengthen Parliament and to prevent the emergence of a new absolutism and the return of the Stuarts",<ref>After James II fled England, the new government worked to block [[w:Jacobitism|Jacobitism]], which attempted to restore the [[w:House of Stuart|House of Stuart]] to the thrones of Scotland and Britain. When James II died, his son became titular head of this movement, who became known as "the [[w:James Francis Edward Stuart|Old Pretender]]". After he died, he was replaced by his son, who became known as "the [[w:Charles Edward Stuart|Young Pretender]]".</ref> according to Acemoglu and Robinson (2012, pp. 302-303). The Whigs gave "[[w:Common land|common land]]" to nobles, confiscating it from commoners. Some commoners, who had traditionally shared that land, joined "Blacks", so called, because they blackened their faces when hunting on former commons or destroying the property of the nobility or extorting money from nobles or their employees. Parliament retaliated by passing the [[w:Black Act 1723|Black Act 1723]], under which someone accused of [[w:Poaching|poaching]] a [[w:Hare|hare]], for example, could under certain circumstances receive a summary judgment of guilty and then be executed without delay if apprehended later.<ref>Newman and Brown, eds. (1997, p. 58). See also the Wikipedia article on "[[w:Black Act 1723|Black Act 1723]]", accessed 2026-10-17.</ref> But not all accused were summarily executed. In 1724 "John Huntridge was accused of aiding deer stealers and abetting known Blacks, both crimes punishable by hanging. The prosecution of Huntridge came right from the top, initiated by the Regency Council of Lords Justices ... . After a trial of eight or nine hours, the jury found Huntridge innocent, partly on procedural grounds, since there were irregularities with the way the evidence had been collected. ... The jury was made up not of Huntridge’s peers, but of major landowners and gentry, who ought to have sympathized" with the noble, whose property had been attacked by Blacks allegedly aided by Huntridge.<ref>Acemoglu and Robinson (2012, pp. 304-305).</ref> These kinds of actions by the Parliaments of George I, II, and III might explain the slower growth during that period compared to that of the last half of the seventeenth century. That slower growth ended abruptly at the time of the [[w:Peterloo massacre|Peterloo Massacre]], 1819-08-16, visible in Figure 1. The growth of factories implied growth of cities, where workers lived closer together and could more easily organize. A campaign arose with multiple demands: They opposed the [[w:Corn Laws|Corn Laws]], which kept prices high for cereal grains and products made from them. They also demanded "universal [adult male] suffrage" and secret ballots in elections. A series of public meetings discussing these grievances led to one 1819-08-16, which attracted roughly 60,000. Acemoglu and Robinson (2012, p. 207) reported that, {{quote| The authorities were very nervous about the meeting, and a force of six hundred cavalry of the [[w:15th The King's Hussars|Fifteenth Hussars]] had been assembled. As the speeches began, a local magistrate decided to issue a warrant for the arrest of the speakers. As police tried to enforce the warrant, they met with the opposition of the crowd, and fighting broke out. At this point the Hussars charged the crowd. Within a few chaotic minutes, eleven people were dead and probably six hundred wounded. The ''[[w:Manchester Observer|Manchester Observer]]'' called it the Peterloo Massacre. But given the changes that had already taken place in economic and political institutions, long-run repression was not a solution in England. The Peterloo Massacre would remain an isolated incident. Following the riot, the political institutions in England gave way to the pressure, and the destabilizing threat of much wider social unrest, particularly after the 1830 revolution in France against [[w:Charles X of France|Charles X]], who had tried to restore the absolutism destroyed by the French Revolution of 1789. In 1832 the government passed the [[w:Reform Act 1832|First Reform Act]]. It enfranchised [[w:Birmingham|Birmingham]], [[w:Leeds|Leeds]], [[w:Manchester|Manchester]], and [[w:Sheffield|Sheffield]], and broadened the base of voting so that manufacturers could be represented in Parliament. ... [I]n 1846 they managed to get the hated Corn Laws repealed ... .}} The ''Manchester Observer'' closed roughly two years later after [[w:Manchester Observer#Closure by repeated prosecution|repeated prosecutions]] for [[w:Seditious libel|seditious libel]]. However, the British government placed enough other limits on its power that enough other commoners were allowed to become successful entrepreneurs that the rate of economic growth in the UK increased immediately, as indicated by the increase in the slope of the UK line in Figure 1 until World Wars I and II. == In sum == The above suggests that the rate of economic growth changes with either (a) a change in administration -- a new head of state -- or (b) some other major event like war, the Peterloo Massacre, or the [[w:Wall Street crash of 1929|Wall Street crash of 1929]]. Policy changes that can increase growth include (1) diverse ownership and control of the major media and (2) immigrants. However, correlation does not imply causation. [[w:Randomized controlled trial|Randomized controlled trial]]s have made important contributions to human understanding of economic growth, as documented by Banerjee, Duflo, and Kramer, cited above and recommended in the Wikiversity article on "[[Information is a public good: Designing experiments to improve government]]". Failing that, more analyses of data from different political jurisdictions with different media policies could be useful, as described by Neff and Pickard (2024).<ref>Graves (2025) includes an analysis of the data in Neff and Pickard (2024).</ref> == Caveat == The above attempts to accurately reflect the perspectives of all sources cited. Readers are invited to revise this article, subject to the standard Wikimedia rules of writing from a [[Wikiversity:Disclosures|neutral point of view]], [[Wikiversity:Cite sources|citing credible sources]] -- while raising other questions and concerns on the associated [[Wikiversity:FAQ|'''{{'}}“Discuss”{{'}}''' page]] -- and treating others with respect. == Notes == {{reflist}} == Bibliography == * <!-- Daron Acemoglu (2026-08-11) What Happened to Liberal Democracy?: Remaking a Politics of Shared Prosperity (Function)-->{{cite Q|Q141241291}} * <!--Daron Acemoglu and Simon Johnson (2023) Power and Progress: Our Thousand-Year Struggle Over Technology and Prosperity (PublicAffairs)-->{{cite Q|Q125292212}} * <!--Daron Acemoglu and James A. Robinson (2012) Why Nations Fail (Crown)-->{{cite Q|Q7997840}} * <!--Philippe Aghion, Céline Antonin, and Simon Bunel (2021) The Power of Creative Destruction: Economic Upheaval and the Wealth of Nations (Harvard University Press)-->{{cite Q|Q137641358}} * <!--Ajzenman, Nicolas (29 January 2023). "Migrants don't cause crime rates to increase — but false perceptions endure anyway". The Conversation.-->{{cite Q|Q141758664}} * <!--Costas Arkolakis, Michael Peters, and Sun Kyoung Lee, (2020) "European Immigrants and the United States’ Rise to the Technological Frontier"-->{{cite Q|Q141678287}} * <!--Dean Baker (2016) Rigged: How globalization and the rules to the modern economy were structured to make the rich richer-->{{cite Q|Q100216001}} * <!--Abhijit V. Banerjee and Esther Duflo (2019) Good Economics for Hard Times (PublicAffairs)-->{{cite Q|Q85764011}} * <!--Jutta Bolt and Jan Luiten van Zanden (2024-09-18) "Maddison style estimates of the evolution of the world economy: A new 2023 update", Journal of Economic Surveys, 1-41-->{{cite Q|Q126723821}} * <!--S. N. Broadberry, B. Campbell, A. Klein, M. Overton and B. van Leeuwen (2015) ''British Economic Growth 1270-1870'' (Cambridge University Press)__>{{cite Q|Q57945943}} * <!--Donald S. L. Cardwell (1972) Turning Points in Western Technology (Neale Watson, Science History Pubs)-->{{cite Q|Q141795083}} * <!--Renée DiResta (2026) Invisible Rulers: The People Who Turn Lies into Reality, updated edition-->{{cite Q|Q141156783}} * <!--Renée DiResta and Rachel Kleinfeld (2025-09-10) For Expertise to Matter, Nonpartisan Institutions Need New Communications Strategies-->{{cite Q|Q141158362}} * <!--Katarzyna Dośpiał-Borysiak (2017) "Model of state management of petroleum sector -- Case of Norway-->{{cite Q|Q141774179}} * <!--Rachel Foxley (2013) The Levellers: Radical Political Thought in the English Revolution (Oxford University Press)-->{{cite Q|Q136227988}} * <!--Benjamin Gonzalez O’Brien, Loren Collingwood, Stephen Omar El-Khatib (2017-05-07) "The Politics of Refuge: Sanctuary Cities, Crime, and Undocumented Immigration-->{{cite Q|Q55932800}} * <!--Graves (2025-01-08) Defend Free Speech For All (Town Hall)-->{{cite Q|Q132855526}} * <!--Graves (2023-03-21) Do copyrights and paywalls on academic journals violate the US Constitution?-->{{cite Q|Q126116187}} * <!--Christopher Hibbert, Ben Weinreb, John Keay, Julia Keay, and Matthew Weinreb (2011-09-09) The London Encyclopaedia, 3rd ed.-->{{cite Q|Q137188993}} * <!-- Historian, USPS (2010) Postage Rates for Periodicals -->{{cite Q|Q112167199}} * <!--Richard R. John (1995) Spreading the News: The American Postal System from Franklin to Morse-->{{cite Q|Q54641943}} * <!--Volker Leppin and Timothy J. Wengert (2015) "Sources for and against the Posting of the Ninety-Five Theses"-->{{cite Q|Q141737347}} * <!--Lawrence Lessig (2004) Free Culture-->{{cite Q|Q830526}} * <!--Olivier Marie and Paolo Pinotti (2024-02-01) "Immigration and Crime: An International Perspective", Journal of Economic Perspectives. 38 (1): 181–200-->{{cite Q|Q128434874}} * <!--Martínez, Daniel E.; Martínez-Schuldt, Ricardo D.; Cantor, Guillermo (2017). " Providing Sanctuary or Fostering Crime? A Review of the Research on 'Sanctuary Cities' and Crime" . Sociology Compass. 12 e12547. doi:10.1111/soc4.12547. ISSN 1751-9020.-->{{cite Q|Q55932802}} * <!-- Robert W. McChesney; John Nichols (2010). The Death and Life of American Journalism (Bold Type Books) -->{{cite Q|Q104888067}}. * <!-- Robert W. McChesney; John Nichols (2016). People get ready-->{{cite Q|Q87619174}} * <!--Robert Michels (1911) Les Partis politiques, essai sur les tendances oligarchiques des démocraties-->{{cite Q|Q7210216}} * <!--Joel Mokyr (2017) A Culture of Growth: The Origins of the Modern Economy (Princeton University Press)-->{{cite Q|Q136118590}} * <!--Joel Mokyr (1994) "Cardwell's Law and the political economy of technological progress", Research Policy 23(5)561-574-->{{cite Q|Q139772607}} * <!--Ashley N. Muchow & Taisiia Stanishevska (2026) "From Headlines to Handcuffs: Local News Coverage of Immigrant Crime and Hispanic Arrest", Justice Quarterly, vol. 43(3) (https://doi.org/10.1080/07418825.2025.2550307)-->{{cite Q|Q141760694}} * <!--Neff and Pickard (2024) "Funding Democracy: Public Media and Democratic Health in 33 Countries"-->{{cite Q|Q131468289}} * <!--Gerald G. Newman and Leslie Ellen Brown, eds. (1997) Britain in the Hanoverian Age, 1714–1837: An Encyclopedia-->{{cite Q|Q141759196|authors = Gerald G. Newman and Leslie Ellen Brown, eds.}} * <!--John Hely Owen (1938) War at Sea Under Queen Anne 1702–1708-->{{cite Q|Q141757101}} * <!--Andrew Pettegree (2015-10-27) Brand Luther: How an Unheralded Monk Turned His Small Town into a Center of Publishing, Made Himself the Most Famous Man in Europe--and Started the Protestant Reformation-->{{cite Q|Q141761223}} * <!--Thomas Piketty (2022) A Brief History of Equality-->{{cite Q|Q139886018}} * <!--Joad Raymond (2006-11-02) Pamphlets and Pamphleteering in Early Modern Britain-->{{cite Q|Q141761849}} * <!--Nick Roll (2017-07-16) "Correcting Jeff Sessions: Attorney general railed against crime in sanctuary cities, citing two professors’ work. But his claims are contrary to their conclusions.-->{{cite Q|Q141735231}} * <!--J.P. Smits, E. Horlings and J.L. van Zanden (2000) Dutch GDP and its Components 1800-1913 (Groningen Growth and Development Centre)-->{{cite Q|Q137669960}} * <!--I. Sugimoto (2011) ''Economic growth of Singapore in the twentieth century: historical GDP estimates and empirical investigations'' ( World Scientific Publishing)-->{{cite Q|Q137669987}} * <!--Nik Usher and Sanghoon Kim-Leffingwell (2022-01) How Loud Does the Watchdog Bark? A Reconsideration of Local Journalism, News Non-profits, and Political Corruption -->{{Cite Q|Q134715465}} * <!--J. L. van Zanden and B. van Leeuwen (2012), ‘Persistent but not consistent: the growth of national income in Holland 1347–1807’, Explorations in Economic History, 49, pp. 119–30-->{{cite Q|Q137670058}} * <!--Brittney Washington (2018-06-19) " A Pamphlet War in England, 1641-1643"-->{{cite Q|Q141822147}} * <!--Tom K. Wong (2017-01-26) "The Effects of Sanctuary Policies on Crime and the Economy"-->{{cite Q|Q98775875}} [[Category:Original research]] [[Category:Research]] [[Category:Economics]] [[Category:Macroeconomics]] [[Category:Gross domestic product]] [[Category:Economic growth]] [[Category:Modern Monetary Theory]] [[Category:Political science]] [[Category:Freedom and abundance]] <!-- https://en.wikiversity.org/wiki/Category_Review --> 7ofxeypw8704u3fw7yel2prpb1x6zlh 2842454 2842450 2026-10-24T20:09:10Z DavidMCEddy 218607 wdsmth 2842454 wikitext text/x-wiki {{Research project}} :''This article invites all to contribute references and interpretations to the growth in average annual income, Gross Domestic Product (GDP) per capita at purchasing power parity, as applied to leading countries in the [[w:Maddison Project|Maddison Historical Statistics Project]].'' :''This essay is on Wikiversity to encourage a wide discussion of the issues it raises moderated by the Wikimedia rules that invite contributors to [[w:Wikipedia:Be bold|“be bold but not reckless,”]] contributing revisions written from a [[Wikiversity:Disclosures|neutral point of view]], [[Wikiversity:Cite sources|citing credible sources]] -- while raising other questions and concerns on the associated [[Wikiversity:FAQ|'''{{'}}“Discuss”'''{{'}} page]] -- and treating others with respect.'' ::''This article uses [[w:ISO 8601|ISO 8601]] dates except for References, which are controlled by standard Wikidata formatting, and direct quotes. In the initial author's experience, [[ISO 8601 and computing differences between dates|ISO 8601 dates seem to make it easier to remember dates and to compute differences between them.]]'' == Abstract == This article first identifies and plots GDP per capita for leading countries in the Maddison data then discusses comments by Nobel prize winning economists and others to interpret what appears in said plot. No one really knows how to make economies grow. However, growth at the technology frontier requires creative destruction. Also, creative destruction is facilitated and inequality is limited by diversity of ownership and control of the major media. == Maddison Historical Statistics == [[File:MaddisonDataLeaders1349-2022.svg|thumb|''Figure 1. World leaders in GDP per capita 1349-2022 (NLD = initially Holland to 1809 and the Netherlands after; GBR = initially England to 1700, Great Britain to 1870, then the UK; USA starting with the 13 colonies that became the US; SGP = Singapore).<ref>See Bolt and van Zanden (2024) for the Maddison Data generally, Broadberry et al. (2015) for the data on England 1252–1700 and on Great Britain until 1870, Sugimoto (2011) for Singapore to 2007, van Zanden and van Leeuwen (2012) for the data on Holland 1349–1807, and Smits et al. (2000) for the data on the Netherlands 1808-1913. The dramatic drop in the orange line between 1807 and 1808 could be due to the change in reference from [[w:Holland|Holland]] to the Netherlands, as Holland at that time was more prosperous than the rest of the Netherlands.</ref>'']] The [[w:Maddison Project|Maddison Historical Statistics Project]] (aka Maddison Project) is widely recognized as the premier data source on historical macroeconomic data. It is named after [[w:Angus Maddison|Angus Maddison]] (1926-2010), who was a pioneer in [[w:National accounts|national accounts]], especially estimating [[w:Gros Domestic Product|gross domestic product]] (GDP) per capita (GDPpc) and population (pop) at various times dating back to the year 1 of the [[w:Common Era|Common Era]] (CE / AD). After Maddison's death, the Maddison Project has continued to update the database to include the best sources available on any country for any historical period since year 1. When accessed 2026-10-10 the version of the data available had been released 2024-09-18 containing data to 2022.<ref>Bolt and van Zanden (2024).</ref> This dataset includes 23,280 observations on at least one of GDPpc and pop for an average of 137 years for 169 countries. Only 18 countries had the highest GDPpc for at least one year. Annual data starts in 1252, and that's only for one country, GBR = England / Great Britain / the United Kingdom. NLD (Holland / the Netherlands) had the highest GDPpc for 97 percent of the years between 1349 and 1807, and the Dutch leadership in Figure 1 may have ended then, not because of the devastation from the [[w:Napoleonic Wars|Napoleonic Wars]] (which negatively impacted them) but because the available data prior to 1807 are for [[w:Holland|Holland]], while more recent data are for the Netherlands, of which Holland is only a part, though probably the most prosperous part at that time. This analysis focuses on data since 1349, in part because data were available for at most 4 countries for most years prior to that date. (The 3-letter country codes used here are [[w:ISO 3166-1 alpha-3|ISO 3166-1 alpha-3]], used in the Maddison data.) We are interested in technology leaders, so we eliminate the [[w:petrostates|petrostates]] of Qatar (QAT), Kuwait (KWT), and United Arab Emirates (ARE), plus Norway (NOR), which owes a substantial portion of their wealth to democratic management of [[w:North Sea oil|North Sea oil]].<ref>Dośpiał-Borysiak (2017).</ref> With these restrictions 12 countries had the highest GDPpc for at least one year. However, FRA (France), ITA (Italy), SWE (Sweden), BEL (Belgium), NZL (New Zealand) and CHE (Switzerland) led for at most 7 of the 675 years between 1349 and 2022; we eliminate those, because including them would seem to add more clutter than clarity. We also eliminate AUS (Australia), which led for 17 of the 39 years between 1853 and 1891, when it was competing for the leadership with GBR and USA, for the same reason. And we eliminate LUX (Luxembourg), which has a population under a million. This leaves NLD, GBR, USA, and SGP (Singapore), plotted in Figure 1. This plot has some interesting features, which we discuss in the remainder of this research report. == Nobel economists and others == Banerjee and Duflo (2019, p. 207) said, "despite the best efforts of generations of economists, the deep mechanisms of persistent economic growth remain elusive." That appeared shortly before they shared the 2019 [[w:List of Nobel Memorial Prize laureates in Economic Sciences|Nobel Memorial Prize in Economics]] with [[w:Michael Kremer|Michael Kremer]] for documenting how [[w:Randomized controlled trial|randomized controlled trial]]s could be used to alleviating poverty in [[w:Development economics|development economics]]. While Banerjee and Duflo insist that no one knows how to make an economy grow, other leading economists recommend partial solutions. For example, [[w:Joel Mokyr|Mokyr]] (2017, p. 53) said, "The willingness of the Netherlands and later Britain and the United States to tolerate Jews and dissenting Christians contributed a great deal to their economies, especially in high-skilled manufacturing and financial services." In other words, immigrants made an outsized contribution to the growth of NLD, GBR, and US, visible in Figure 1. Mokyr's comment appeared over a year before Banerjee and Duflo (2019) and roughy eight years before Mokyr shared the 2025 Nobel Memorial Prize in Economics with [[w:Philippe Aghion|Aghion]] and [[w:Peter Howitt (economist)|Howitt]]. Mokyr (2017, p. 255) also said that, "intellectual innovation will eventually end up being increasingly resisted by an incumbent technology; if that resistance is not overcome, technological progress may be extinguished." That resistance, he suggested, drives {{quote| the empirical regularity that no society remains at the cutting edge of technological creativity for very long<ref>Mokyr (2017, p. 260). [[w:Robert Michels|Michels]] (1911) called this "the [[w:Iron law of oligarchy|iron law of oligarchy]]".</ref>.}} Mokyr (2017, p. 260; 1994) calls this "Cardwell's Law" after Cardwell (1972), which describes {{quote| the interaction between technological creativity and political competition ... based on the observation that technological progress encounters resistance from various groups that believe they stand to lose from innovation. These pressure groups will try to manipulate the political system to suppress successful innovation.}} Further documentation of this appears in Acemoglu and Johnson (2023) and Acemoglu and Robinson (2012). This provides one explanation for the end of growth of the economy of Holland shortly before 1600 and for the slowing of economic growth in recent decades in the US, the UK, and the Netherlands, visible in Figure 1. It can also explain [[w:History of the Roman Empire|the rise and fall of the Roman Empire]], of the [[w:Maya civilization|Maya civilization]], and other societies that became very rich from allowing commoners to become entrepreneurs and then declined after previous winners changed the rules of the political economy to block further [[w:creative destruction|creative destruction]]. This includes how [[w:History of the Republic of Venice|Venice]] became "possibly the richest place in the world, with the most advanced set of inclusive economic institutions", dominating trade in the [[w:Mediterranean Sea|Mediterranean]] based on a system that encouraged entrepreneurial commoners to undertake risky trading ventures starting after 810 CE and ending when leaders made wealthy by such ventures changed the rules to end that system starting in 1297, destroying their dominance of Mediterranean trade "and converting Venice into a museum", in the words of Acemoglu and Robinson (2012, pp. 152-155). Mokyr (2017, p. 60) also mentioned "entrepreneurs of hate", who convince others to dislike some group in a way that may benefit those "entrepreneurs" and major media organizations, who promote such "entrepreneurs" to please those who control much of the money for the media. For-profit social media have been particularly effective in herding their audiences into echo chambers, reinforcing their preconceptions in ways that increase political polarization and violence and threaten democracy.<ref>See, e.g., DiResta (2026), DiResta and Keinfeld (2025), and the Wikiversity article on "[[Managing the good and the evils of social media]]", accessed 2026-10-14.</ref> When successful this protects current elites from competition that would otherwise drive [[w:creative destruction|creative destruction]], benefitting the public at large while threatening existing elites. For example, most research on immigrants suggest that on average immigrants are more entrepreneurial and no more likely to commit crime than native born,<ref>e.g., Martínez et al. (2017); Muchow and Stanishevska (2026); Ajzenman(2023).</ref> and some studies report that [[w:Sanctuary city|sanctuary jurisdiction]]s have on average higher median incomes and less crime.<ref>Martínez et al. (2017); Wong (2017). See also the Wikipedia article on "[[w:Sanctuary city|sanctuary city]]", accessed 2026-10-12.</ref> Where immigrant crime is higher, it typically result from anti-immigrant labor market restrictions or from young less-educated males with irregular status.<ref>Marie and Paolo (2024). See also the Wikipedia article on "[[w:Immigration and crime|Immigration and crime]]", accessed 2026-10-12.</ref> But major media outlets make money from featuring charismatic figures, who attract an audience without offending people who control much of the money for the media. In 2017 President [[w:Immigration policy of the first Trump administration|Trump]]'s Attorney General [[w:Jeff Sessions|Jeff Sessions]] quoted a research study that supposedly showed that sanctuary cities had more crime than cities without sanctuary policies. The authors of that study insisted that Sessions had ''misrepresented'' their work, which concluded "that sanctuary policies have no effect on crime rates, despite narratives to the contrary."<ref>Roll (2017); Gonzalez O’Brien, et al. (2017).</ref> To the extent that these comments provide a balanced summary of the research in this area, it suggests that leaders like Trump and Sessions, to name only two, can be fairly describes as "entrepreneurs of hate". We next discuss competition in media markets before discussing immigrants further. === Diversity of ownership of the major media === Acemoglu and Johnson (2023, esp. ch. 4) describe how religious leaders in hierarchical societies prior to [[w:James VI and I|King James]] of the [[w:King James Version|King James Bible]] convinced commoners to live in poverty while giving increasing shares of what they produced, so religious and secular elites could live in opulence. Prior to King James the major media were religious organizations. In such hierarchical societies some religious leaders were entrepreneurs of hate, exhorting the faithful to fear and hate designated evil others and give increasing shares of what they produce so secular elites would "protect" them in wars against designated evil others. Acemoglu and Robinson (2012, p. 461) said, "Empowerment of society at large is difficult to coordinate and maintain without widespread information about whether there are economic and political abuses by those in power. ... Pamphlets and books informing and galvanizing people played an important role during the [[w:Glorious Revolution|Glorious Revolution]] [1688] in England, the French Revolution [1789-1799]" and in the US. ([[w:Daron Acemoglu|Acemoglu]], [[w:Simon Johnson (economist)|Johnson]] and [[w:James A. Robinson|Robinson]] shared the 2024 [[w:List of Nobel Memorial Prize laureates in Economic Sciences|Nobel Memorial Prize in Economics]].) But pamplets started challenging established power structures much earlier. The publication of [[w:Martin Luther|Martin Luther]]'s ''[[w:Ninety-five Theses|Ninety-five Theses]]'' in Latin 1517-10-31 was quickly condemned by [[w:Pope Leo X|Pope Leo X]] and his hierarchy. [[w:Johann Tetzel|Johann Tetzel]], who had been selling [[w:Indulgence|indulgence]]s, published a pamphlet calling for Luther to be burnt for [[w:heresy|heresy]]. Eight hundred copies were reportedly sent to [[w:Wittenberg|Wittenberg]] to be sold, but students seized and burned them.<ref>Hendrix (2015, p. 64), quoted from the Wikipedia article on the ''[[w:Ninety-five Theses|Ninety-five Theses]]'', accessed 2026-10-14.</ref> This led to a pamphlet war. Luther's brief German-language reply was short and easy for commoners to understand<ref>Leppin and Wengert (2015, p. 389), cited from the Wikipedia article on the ''[[w:Ninety-five Theses|Ninety-five Theses]]'', accessed 2026-10-14.</ref> and was much more popular than Tetzel's.<ref>Pettegree (2015, p. 145), cited from the Wikipedia article on the ''[[w:Ninety-five Theses|Ninety-five Theses]]'', accessed 2026-10-14.</ref> Less than twenty years later, in 1534, King [[w:Henry VIII|Henry VIII]] declared himself head of the [[w:Church of England|Church of England]], denying the authority of the Pope. To secure the support of his subjects, he drove a propaganda war that included print literature<ref>Raymond (2006), cited from the Wikipedia article on "[[w:Pamphlet war|pamphlet war]]", accessed 2026-10-15.</ref> and theatre and minstrel troupes funded by the king. The latter traveled around the land to promote the new religious practices. Henry was portrayed as the glorious king of England, bravely and heroically defending the true faith against foreign devils, especially the Pope and Catholic priests, monks, etc.<ref>Hibbert et al. (2010, p. 928), cited from the Wikipedia article on "[[w:Henry VIII|Henry VIII]]", accessed 2026-10-15.</ref> Henry died in 1547. Less than a dozen years later, the crown passed to [[w:Elizabeth I|Elizabeth I]]. Her limited personal wealth and English tradition required her to negotiate with Parliament for tax money. Acemoglu and Robinson (2012, pp. 105-111) noted that she was forced to limit the granting of monopolies and allow a class of merchants to compete in international trade. Many English merchants became wealthy, helped by the defeat of the [[w:Spanish Armada|Spanish Armada]] in 1588. When Elizabeth died in 1603, she was replaced by her cousin, King [[w:James VI and I|James VI of Scotland]], who then became also James I of England. He claimed [[w:Divine right of kings|divine right of kings]], threatening this class of merchants. To raise money without asking Parliament, he granted patents and monopolies to those who would pay. Parliament responded by passing the [[w:Statute of Monopolies|Statute of Monopolies]] on 1624-05-29. This statute repealed some past and future patents and monopolies except for novel inventions. But that Statute of Monopolies did not end the conflict. It ultimately went public, described as, "A Pamphlet War in England, 1641-1643".<ref>Washington (2018).</ref> Some pamphlets supported [[w:Charles I of England|Charles I]], who became King after James died in 1625. Others supported Parliamentarians and [[w:Levellers|Levellers]]. The latter supported radical ideas like [[w:popular sovereignty|popular sovereignty]], extended suffrage, equality before the law and religious tolerance. In 1642 war broke out between Parliament and Charles I, though many in Parliament sided with Charles -- especially those, who had received monopolies from him.<ref>Acemoglu and Robinson (2012, p. 189).</ref> The war was won by the [[w:New Model Army|New Model Army]], which represented many in Parliament plus Levellers. The latter were excluded from the government that followed the [[w:Execution of Charles I of England|beheading of King Charles I]] in 1649.<ref>Foxley (2013, p. 207), cited from the Wikipedia article on [[w:Levellers|Levellers]], accessed 2026-10-15.</ref> However, that new government stopped granting as many monopolies and allowed so many commoners to become entrepreneurs that the GDP per capita began to grow at rates that are key features of Figure 1 for England and the 13 English colonies that became the US over a century later. One explanation for that economic growth was the diversity of independent publishers of pamphlets, books and newspapers. [[w:Robert W. McChesney|McChesney]] and [[w:John Nichols (journalist)|Nichols]] (2010, 2016) suggested that the US [[w:Postal Service Act|Postal Service Act]] of 1792<ref>Wikipedia "[[w:Postal Service Act|Postal Service Act]]", accessed 2023-07-11.</ref> made a major contribution to making the US what it is today. Under that act, newspapers were delivered up to 100 miles for a penny, when postage for letters was between 6 and 25 cents depending on distance.<ref>Historian, USPS (2010).</ref> McChesney and Nichols estimated that between 1840 and 1844, the US postal subsidy was a little over 0.2% of GDP.<ref name=McC-N2010>McChesney and Nichols (2010, pp. 310-311, note 88).</ref> During the first half of the nineteenth century the US had more independent newspaper publishers (more independent media organizations) per million population than at any other time or place in human history.<ref>John (1995). See also the interview with him at "[[Media concentration per Columbia History Professor Richard John]]" on Wikiversity, accessed 2026-10-12.</ref> In 1932 with official unemployment over 20 percent, the conservative argument that blames the poor for their poverty did not sell newspapers, and the US still had sufficient competition in media markets that presidential candidate Franklin Roosevelt (FDR) got a mandate to spend the money to put humans back to work while taxing the ultra-wealthy like they had never been taxed before or since and during World War II to institute wage and price controls under the [[w:Office of Price Administration|Office of Price Administration]]. The latter prevented price gouging by government contractors that had generated inflation and stifled economic growth in previous major wars (World War I, the Civil War, and the War of 1812). Instead, the US averaged 7 percent per year growth for FDR's 12 years.<ref>The "6 percent per year" growth during FDR's 12 years was computed as the average of log(GDPpc) in the Maddison Data. To be precise, that computation produced 0.0597; converting that from a log scale to percent increase gives 0.0615. Using data from [[w:MeasuringWorth|MeasuringWorth]] instead of Maddison, we get 0.783 and 0.815, respectively. More documentation of this is available in "[[Media Literacy and You/Fox, the Great Depression, the Great Recession, and our future]]" on Wikiversity, accessed 2026-10-13.</ref> One of the most interesting features of Figure 1 is the behavior of the US economy in the administrations of [[w:Herbert Hoover|Herbert Hoover]] and FDR: Under Hoover, the economy declined at 9 percent per year for 4 years, a financial disaster that created the public demand for change that gave FDR the mandate to do what he did. Economic growth under two other presidents are numerically comparable to FDR's numbers but for much shorter periods: (1) [[w: Warren G. Harding|Warren G. Harding]] averaged 7 percent annual growth for two years before he died in 1923 during the [[w:Roaring Twenties|Roaring Twenties]]. (2) [[w:Rutherford B Hayes|Rutherford B Hayes]], who took office in 1877 at the end of the [[w:Panic of 1873|Panic of 1873]], oversaw 5.5 percent growth per year for his 4 years as president. Those impressive sounding average growth figures are so short that they are not noticeably different from other minor features visible in Figure 1. FDR also dramatically reduced inequality. For over a century between about 1820 and 1932 the top 1 percent of the US population owned roughly 45 percent of net personal wealth in the US, while the bottom 90 percent owned roughly 22 percent. Shortly after FDR died, the wealth of the top 1 percent and bottom 90 percent were even at roughly 30 percent. FDR also oversaw dramatic growth in the power of organized labor. Their successes continued to reduce inequality to roughly 22 percent for the top 1 percent vs. 37 percent for the bottom 90 percent when Ronald Reagan became president. Since then, [[w:Reaganomics|Reaganomics]] have been rebuilding the [[w:Robber baron (industrialist)|Robber Baron]]s.<ref>This history of inequality is documented in [[:File:Share of US wealth 90p99.svg|Figure 7. Shares of US wealth - bottom 90 and top 1 percent, 1820-2023]] in the Wikiversity article on "[[w:Media Literacy and You/Fox, the Great Depression, the Great Recession, and our future|Media Literacy and You/Fox, the Great Depression, the Great Recession, and our future]]" in Wikiversity, accessed 2026-10-13.</ref> FDR made enemies among the wealthiest and most powerful. In a [[q:Franklin D. Roosevelt#Address at Madison Square Garden (1936)|speech in Maddison Square Garden in 1936]], FDR mentioned, "the old enemies of peace: business and financial monopoly, speculation, reckless banking, class antagonism, sectionalism, war profiteering. ... They are unanimous in their hate for me -- and I welcome their hatred." They doubtless could see that they could no longer afford as many personal servants -- maids, butlers, cooks, chauffeurs, etc. -- as they had before FDR. Broadcasting was relatively new when FDR took office. The economics of broadcasting naturally favored a very few networks, whose power concerned FDR. In 1941 his [[w:Federal Communications Commission| Federal Communications Commission]] (FCC) adopted the [[w:Mayflower doctrine|Mayflower doctrine]], which prohibited broadcasters from advocacy. When FDR died, people with power were able to get the Mayflower doctrine replaced by the more lenient [[w:Fairness Doctrine|Fairness Doctrine]], which opened the floodgates for entrepreneurs of hate, who drove the [[w:McCarthyism|McCarthy Red Scare]] in the US and many other problems since. Over time this became worse, facilitated by increasing concentration of ownership of the media. That contributed to the end of the Fairness Doctrine starting in 1985. That revocation and especially the more recent rise of for-profit social media have amplified the ability of entrepreneurs of hate to make money both for commercial social media and for themselves by disseminating lies about others, e.g., immigrants.<ref>On how for-profit social media companies promote entrepreneurs of hate, see, e.g., DiResta (2026) and the Wikiversity article on, "[[Managing the good and the evils of social media]]", accessed 2026-10-12. DiResta (2026, p. 1) described the promotion of "influencers" by for-profit social media as essentially unleashed "a war for power, influence, and reality itself—a conflict in which the ends justify any means, and one that has ripped America apart, threatened our very democracy, and consumed my own life for the past decade, leading to death threats, congressional subpoenas, lawsuits, and legal bills running into seven figures." In the interview on Wikiversity, DiResta said her legal bills to that point had exceeded $7 million.</ref> And the rate of economic growth in the US (and the Netherlands and the UK) have been slowing in recent decades, as is visible in Figure 1. Inequality expert [[w:Thomas Piketty|Piketty]] (2022, p. 139) noted, {{quote| In the United States, the national income per inhabitant rose at a rate ... of 2.2% between 1950 to 1990 when the top tax rate reached on average 72%. The top rate was then cut in half, with the announced objective of boosting growth. But in fact, growth fell by half, to 1.1% per annum between 1990 and 2020 ... . Beyond a certain level of inequality, repeatedly increasing differences in income and wealth has clearly had no positive effect on economic dynamism. In sum, all the data at our disposal today suggest that virtually confiscatory tax rates have been an immense historical success. They have made it possible to greatly reduce the divergences of fortunes and incomes, while at the same time improving the situation of the middle and lower classes, developing the welfare state, and stimulating better economic and social performance overall. Historically, it is the battle for equality and education that has made economic development and human progress possible, and not the veneration of property, stability, and inequality.}} Table 1 compares Piketty's growth numbers with computations based on the data from both Maddison and [[w:MeasuringWorth|MeasuringWorth]]: The numbers are close but not exact. {| class="wikitable sortable" style="text-align: center;" |+ Table 1. US growth rate 1950 to 1990 to 2020 |- ! ! colspan=3 | US GDP per capita !! colspan=2 | Growth (% / yr) |- ! source || 1950 !! 1990 !! 2020 !! 1950 - 1990 !! 1990 - 2020 |- | Piketty || - || - || - || 2.2% || 1.1% |- | Maddison || $15,240 || $36,982 || $54,379 || 2.24% || 1.29% |- | MeasuringWorth || $16,208 || $40,191 || $61,127 || 2.30% || 1.41% |} Acemoglu and Robinson (2012, pp. 105-106) provide a historical analysis supporting Piketty's concern about inequality slowing growth. They note that in 1588, England, Spain and France {{quote| were ruled by relatively absolutist monarchs: [[w:Elizabeth I|Elizabeth I]] in England, [[w:Philip II of Spain|Philip II]] in Spain, and [[w:Henry III of France|Henry III]] in France. All were battling with assemblies of citizens -- such as the [[w:Parliament of England#Tudor era (1485–1603)|Parliament]] in England, the [[w:Cortes Generales#Habsburg rule (16th–17th centuries)|Cortes]] in Spain, and the [[w:Estates General (France)#Revival in 1560–1614|Estates-General]] in France -- that were demanding more rights and control over the monarchy. ... [T]he English Parliament and the Spanish Cortes had power over taxation, while the Estates-General did not. In Spain this mattered little, because after 1492 the Spanish Crown had a vast American empire and benefited massively from the gold and silver found there. In England ... Elizabeth I was far less financially independent, so she had to beg Parliament for more taxes. In exchange, Parliament demanded concessions, in particular restrictions of the right of Elizabeth to create monopolies. ... In Spain the Cortes lost a similar conflict.}} Consistent with Acemoglu and Johnson (2023) and the discussion above, inequality tends to increase elite control of the media, which it makes it harder for commoners to get a fair hearing of their concerns. === Media conflicts of interest in IP law === Major media have [[w:Conflict of interest|conflicts of interest]] in honestly reporting on anything involving [[w:Intellectual property|intellectual property]] (IP) law. An example of Cardwell's law, mentioned above, is the [[w:Copyright Term Extension Act|Sonny Bono Copyright Term Extension Act]], also known as the Mickey Mouse Protection Act of 1998. The copyright term on ''[[w:Steamboat Willie|Steamboat Willie]]'', the first [[w:Mickey Mouse|Mickey Mouse]] animated film that Disney distributed, was extended multiple times by the US Congress following extensive lobbying by the film industry led by [[w:The Walt Disney Company|the Walt Disney Company]]. ''Steamboat Willie'', which debuted in 1928, finally entered the [[w:Public domain in the United States|US public domain]] on January 1, 2024. Eric Eldred and others filed suit, claiming that allowing any number of retroactive extensions violated the [[w:Copyright Clause|Copyright Clause]] of the US Constitution, which says, {{quote| [the United States Congress shall have power] To promote the Progress of Science and useful Arts, by securing for limited Times to Authors and Inventors the exclusive Right to their respective Writings and Discoveries.}} In ''[[w:Eldred v. Ashcroft|Eldred v. Ashcroft]]'' (2003), the [[w:Supreme Court of the United States|US Supreme Court]] disagreed with Eldred and allowed the statute to remain in effect. However, that case drew enough public attention that copyright terms have not (yet) been extended further, and ''Steamboat Willie'' is currently in the pubic domain. [[w:Lawrence Lessig|Lessig]] (2004) ''[[w:Free Culture (book)|Free Culture]]'' describes other examples, where US copyright law seems to ''limit'' rather than "promote the Progress of Science and useful Arts". This includes a case where a 2002 freshman, Jesse Jordan, modified a search engine to make it easier for other student to find pictures, research, notes, movie clips, etc., on an [[w:intranet|intranet]] at [[w:Rensselaer Polytechnic Institute|Rensselaer Polytechnic Institute]] (RPI). The [[w:Recording Industry Association of America|Recording Industry Association of America]] (RIAA) sued Jordan and three other students for close to $100 billion for copyright violations -- six times the total profit of the film industry in 2001. Jordan had an uncle, who was an attorney, who explained that the legal bills for fighting that case would be at least $250,000. That would bankrupt his parents; they could not recover that money, even if they won. Jordan had saved $12,000 to cover part of his college expenses. RIAA accepted that and settled the case.<ref>Lessig (2004, p. 42).</ref> This is an example of a [[w:Strategic lawsuit against public participation|strategic lawsuit against public participation]] (SLAPP), which in this case rested on questionable interpretations of copyright law. The major media have conflicts of interest in honestly reporting on anything involving (a) copyright laws, because much of their income is derived from copyright royalties, or (b) anything that might offend their major accounts, including major corporations. And anti-SLAPP laws, which exist in some jurisdictions but not others,<ref>[[w:Robert Corn-Revere|Robert Corn-Revere]], Chief Counsel for the [[w:Foundation for Individual Rights and Expression|Foundation for Individual Rights and Expression]], supports anti-SLAPP laws and the "JAWBONE act", to limit the power of the government to threaten questionable actions, like the Trump administration's threats against the ABC network over a Jimmy Kimmel monolog. See the Wikiversity article on "[[Foundation for Individual Rights and Expression]]", accessed 2026-10-16.</ref> limit the ability of major corporations to abuse their power against potential challengers without such deep pockets. Similarly Graves (2023) asked, "Do copyrights and paywalls on refereed academic journals violate the US constitution?" No one expects to make money from copyright royalties for drafts submitted to refereed academic journals. They do it out of a felt need (a) to contribute to the future of humanity or (b) to build their reputation or (c) because promotions at many universities are based partly on publication records. Graves is not alone in his concern about this: Many European countries do not allow reports of research they fund to be published behind paywalls.<ref>Dimi Zagorski is concerned that copyright exceptions for scientific research are different in different European countries, and that's a problem for the Wikimedia Foundation. See the discussion of this in the Wikiversity article on "[[Wikimedia concerns with European copyright rules including AI and scientific research]]". That's the opposite of US law, which grants patents to universities, research institutions, private companies, and individuals for innovations developed under government contracts, per the Bayh-Dole Act of 1980, described by Baker (2016, p. 80).</ref> Regarding patent law, the major media have conflicts of interest in discussing proposed changes affecting major accounts, but they make little use of patent law themselves. This seems to have contributed to the claim that, "[O]ver the last four decades ... stronger patent protection seems to be associated with slower growth" in the US, according to [[w:Dean Baker|Baker]] (2016, pp. 77-78). A potentially similar phenomenon appears in a comparison of 21 different European countries in an analysis that displays "a positive relationship between anticorruption measures in a country and rate of innovation in that country as measured by patents per capita", discussed by Aghion et al. (2021, Figure 15.1b, pp. 292-293).<ref>Usher and Kim-Leffingwell (2022) documented how each member of the [[w:Institute for Nonprofit News|Institute for Nonprofit News]] (INN), relatively small nonprofits independent of major media organizations and not funded by advertising, in a US federal court district was associated with on average 1.4 additional prosecutions for political corruption per year between 2003 and 2019. They found on average 10 federal prosecutions for political corruption for each federal court district per year. Thus, 1.4 additional prosecutions per year per district represents an increase of 14 percent per member of INN.</ref> === Immigrants === We now return to discussing the initiation of economic growth following the beheading of Charles I in 1649 visible in Figure 1. Charles I was replaced by [[w:English Council of State|Lord Protector Cromwell]], who refused to authorize as many monopolies as his predecessors, as mentioned above. He also encouraged Jews and dissenting Christians to settle in England, as mentioned by Mokyr (2017, p. 53), cited above. But Mokyr is not alone in claiming that immigrants contribute to economic growth. Aghion et al. (2021, p. 266) cite Arkolakis et al. (2020) for explaining how the period 1880–1920 is noteworthy for two parallel evolutions. First, the United States caught up with England and France to become the most technologically advanced country in the world as well as the wealthiest in terms of GDP per capita. Second, the United States welcomed large numbers of immigrants, most notably from Europe. The purpose of the cited study was to demonstrate a causal connection between these two phenomena: in other words, to show that European immigration was responsible for the United States’ taking the lead in technology. ... More precisely, if the United States had completely closed its borders to all foreign immigrants in 1880, its per capita GDP would have been 30 percent lower than the level actually attained in 1920". === Industrial Revolution === There seems to be a broad consensus that the [[w:Industrial Revolution|Industrial Revolution]] began in England. However, there is less agreement on when it began. Mokyr (2017, p. 3) says, "The British Industrial Revolution of the late eighteenth century unleashed a phenomenon never before even remotely experienced by any society." He described "three cultural elements" required for "sustainable economic growth propelled by technological progress": (1, 2) The belief that it was both ''possible'' and ''desirable''. (3) Supportive policy measures had to be implemented. "[B]efore 1750 they did not produce anything like an Industrial Revolution or sustainable economic growth propelled by technological progress.<ref>Mokyr (2017, pp. 19-20).</ref> Acemoglu (2026, p. 53) similarly mentioned the eighteenth century in saying, "The breakthroughs in production technology that started in the middle of the eighteenth century in Britain unleashed the Industrial Revolution." Aghion et al. (2021, esp. pp. 29-30) offer a ''later'' date for the start of the Industrial Revolution: They report that, "European economies prior to 1820 experienced a series of phases of growth and decline. This raises a question: How did the pre-1820 period differ from the takeoff that started with the first Industrial Revolution?" Aghion et al. (2021, p. 20) describe the "first Industrial Revolution" as having been triggered by "the invention of the steam engine by James Watt in the 1770s ... . The invention of electricity set off the second Industrial Revolution, whose golden age was in the first half of the twentieth century. This second revolution originated with the invention of the [[w:Incandescent light bulb|light bulb]] by [[w:Thomas Edison|Thomas Edison]] in 1879 and of the [[w:dynamo|dynamo]] by [[w:Werner von Siemens|Werner von Siemens]] in 1866."<ref>The dynamo was independently invented by [[w:Henry Wilde (engineer)|Henry Wilde]], [[w:Charles Wheatstone|Charles Wheatstone]], [[w:Samuel Alfred Varley|Samuel Alfred Varley]] and Siemens, all about the same time, according to the Wikipedia article on [[w:Dynamo#Practical designs|dynamo]], accessed 2026-10-18.</ref> The image in Figure 1 raises questions about these claims by Mokyr, Acemogu, and Aghion et al.: It shows growth followed by stagnation in Holland but not "phases of growth and decline" with the exception of wars and [[w:Herbert Hoover|Herbert Hoover]]. It also does ''not'' show the slope (i.e., rate of growth) increasing starting around 1750 or the 1770s or 1879 or 1886. Instead it shows sustained growth in GDP per capita for England / Great Britain / the UK starting ''abruptly'' after 1649 with economic disruptions during the reign of [[w:Anne, Queen of Great Britain|Queen Anne]] (1702-1714) and the [[w:War of the Spanish Succession|War of the Spanish Succession]] (1701-1714). This was followed by relatively steady but slower growth as England was ruled by Kings [[w:House of Hanover#Monarchs of Great Britain, Ireland, and Hanover|George I, II, and III]] (1714-1820). Another dramatic increase in the slope started abruptly about 1819. Acemoglu and Johnson (2023, chapter 1) are more consistent with Figure 1. They said that, "the Industrial Revolution ... [was] rooted in institutional changes that started empowering the middling sort of English people from the sixteenth and seventeenth centuries onward." ... However, "the quality of life of a European peasant circa 1700 was not much different from that of an Egyptian peasant two thousand or seven thousand years previously." (chapter 5) They also claim that, "English national production per capita doubled from 1500 to 1700", (chapter 5), and "Growth emerged slowly in the second half of the eighteenth century as the Industrial Revolution, based on major technological breakthroughs and their application in industry, took root." (p. 45, chapter 2) Indeed, Figure 1 shows English GDP per capita doubling between 1500 and 1700, but it appears to have been fairly constant from 1500 to 1649, then roughly doubled in the next half century. [The data on England up to 1700 in Figure 1 came from Broadberry et al. (2015), which was not available when Acemoglu and Robinson were writing ''Why Nations Fail'', which appeared in 2012.] Acemoglu and Robinson (2012, ch. 11) describe the financial instability and slow growth during the reigns of Queen Anne and George I, II, and III. After the Glorious Revolution, France, Spain, the pope, and others refused to recognize [[w:William III of England|William and Mary]] (1689-1702) and then Queen Anne as heads of state of England. Instead they continued to support the claims of James II and then his son, [[w:James Francis Edward Stuart|James Francis Edward Stuart]] (the "Old Pretender", a brother of Queen Anne), after James II died in 1701. During the War of the Spanish Succession, each side tried to exploit the internal divisions of the other. Owen (1938, ch. 8) said France almost succeeded in an attempt in 1708 to [[w:Planned French invasion of Britain (1708)|land between 5 and 6 thousand soldiers in northeast Scotland]]. [[w:Louis XIV|Louis XIV]] had hoped to get the Old Protector named as King James VIII of Scotland. There were still [[w:Jacobitism|Jacobites]] in Scotland, who supported the Old Pretender and likely would have supported the invaders if they had landed. If they had succeeded, it would have created major problems for Queen Anne and her supporters. However, the Royal Navy prevented the invaders from disembarking, and they returned to France. When Queen Anne died in 1702, she was succeeded by George I. His Parliament was dominated by the Whig party, which represented the new mercantile and economic interests and had been the main organization behind the Glorious Revolution. Once in power, they were tempted to use their newly found position to prey on the rights of others but were constrained both by competing groups in Parliament, "and by the very institutions that they had fought to introduce to strengthen Parliament and to prevent the emergence of a new absolutism and the return of the Stuarts",<ref>After James II fled England, the new government worked to block [[w:Jacobitism|Jacobitism]], which attempted to restore the [[w:House of Stuart|House of Stuart]] to the thrones of Scotland and Britain. When James II died, his son became titular head of this movement, who became known as "the [[w:James Francis Edward Stuart|Old Pretender]]". After he died, he was replaced by his son, who became known as "the [[w:Charles Edward Stuart|Young Pretender]]".</ref> according to Acemoglu and Robinson (2012, pp. 302-303). The Whigs gave "[[w:Common land|common land]]" to nobles, confiscating it from commoners. Some commoners, who had traditionally shared that land, joined "Blacks", so called, because they blackened their faces when hunting on former commons or destroying the property of the nobility or extorting money from nobles or their employees. Parliament retaliated by passing the [[w:Black Act 1723|Black Act 1723]], under which someone accused of [[w:Poaching|poaching]] a [[w:Hare|hare]], for example, could under certain circumstances receive a summary judgment of guilty and then be executed without delay if apprehended later.<ref>Newman and Brown, eds. (1997, p. 58). See also the Wikipedia article on "[[w:Black Act 1723|Black Act 1723]]", accessed 2026-10-17.</ref> But not all accused were summarily executed. In 1724 "John Huntridge was accused of aiding deer stealers and abetting known Blacks, both crimes punishable by hanging. The prosecution of Huntridge came right from the top, initiated by the Regency Council of Lords Justices ... . After a trial of eight or nine hours, the jury found Huntridge innocent, partly on procedural grounds, since there were irregularities with the way the evidence had been collected. ... The jury was made up not of Huntridge’s peers, but of major landowners and gentry, who ought to have sympathized" with the noble, whose property had been attacked by Blacks allegedly aided by Huntridge.<ref>Acemoglu and Robinson (2012, pp. 304-305).</ref> These kinds of actions by the Parliaments of George I, II, and III might explain the slower growth during that period compared to that of the last half of the seventeenth century. That slower growth ended abruptly at the time of the [[w:Peterloo massacre|Peterloo Massacre]], 1819-08-16, visible in Figure 1. The growth of factories implied growth of cities, where workers lived closer together and could more easily organize. A campaign arose with multiple demands: They opposed the [[w:Corn Laws|Corn Laws]], which kept prices high for cereal grains and products made from them. They also demanded "universal [adult male] suffrage" and secret ballots in elections. A series of public meetings discussing these grievances led to one 1819-08-16, which attracted roughly 60,000. Acemoglu and Robinson (2012, p. 207) reported that, {{quote| The authorities were very nervous about the meeting, and a force of six hundred cavalry of the [[w:15th The King's Hussars|Fifteenth Hussars]] had been assembled. As the speeches began, a local magistrate decided to issue a warrant for the arrest of the speakers. As police tried to enforce the warrant, they met with the opposition of the crowd, and fighting broke out. At this point the Hussars charged the crowd. Within a few chaotic minutes, eleven people were dead and probably six hundred wounded. The ''[[w:Manchester Observer|Manchester Observer]]'' called it the Peterloo Massacre. But given the changes that had already taken place in economic and political institutions, long-run repression was not a solution in England. The Peterloo Massacre would remain an isolated incident. Following the riot, the political institutions in England gave way to the pressure, and the destabilizing threat of much wider social unrest, particularly after the 1830 revolution in France against [[w:Charles X of France|Charles X]], who had tried to restore the absolutism destroyed by the French Revolution of 1789. In 1832 the government passed the [[w:Reform Act 1832|First Reform Act]]. It enfranchised [[w:Birmingham|Birmingham]], [[w:Leeds|Leeds]], [[w:Manchester|Manchester]], and [[w:Sheffield|Sheffield]], and broadened the base of voting so that manufacturers could be represented in Parliament. ... [I]n 1846 they managed to get the hated Corn Laws repealed ... .}} The ''Manchester Observer'' closed roughly two years later after [[w:Manchester Observer#Closure by repeated prosecution|repeated prosecutions]] for [[w:Seditious libel|seditious libel]]. However, the British government placed enough other limits on its power that enough other commoners were allowed to become successful entrepreneurs that the rate of economic growth in the UK increased immediately, as indicated by the increase in the slope of the UK line in Figure 1 until World Wars I and II. == In sum == The above suggests that the rate of economic growth changes with either (a) a change in administration -- a new head of state -- or (b) some other major event like war, the Peterloo Massacre, or the [[w:Wall Street crash of 1929|Wall Street crash of 1929]]. Policy changes that can increase growth include (1) limits on grants of monopolies, (2) diverse ownership and control of the major media and (3) immigrants. However, correlation does not imply causation. [[w:Randomized controlled trial|Randomized controlled trial]]s have made important contributions to human understanding of economic growth, as documented by Banerjee, Duflo, and Kramer, cited above and recommended in the Wikiversity article on "[[Information is a public good: Designing experiments to improve government]]". Failing that, more analyses of data from different political jurisdictions with different media policies could be useful, as described by Neff and Pickard (2024).<ref>Graves (2025) includes an analysis of the data in Neff and Pickard (2024).</ref> == Caveat == The above attempts to accurately reflect the perspectives of all sources cited. Readers are invited to revise this article, subject to the standard Wikimedia rules of writing from a [[Wikiversity:Disclosures|neutral point of view]], [[Wikiversity:Cite sources|citing credible sources]] -- while raising other questions and concerns on the associated [[Wikiversity:FAQ|'''{{'}}“Discuss”{{'}}''' page]] -- and treating others with respect. == Notes == {{reflist}} == Bibliography == * <!-- Daron Acemoglu (2026-08-11) What Happened to Liberal Democracy?: Remaking a Politics of Shared Prosperity (Function)-->{{cite Q|Q141241291}} * <!--Daron Acemoglu and Simon Johnson (2023) Power and Progress: Our Thousand-Year Struggle Over Technology and Prosperity (PublicAffairs)-->{{cite Q|Q125292212}} * <!--Daron Acemoglu and James A. Robinson (2012) Why Nations Fail (Crown)-->{{cite Q|Q7997840}} * <!--Philippe Aghion, Céline Antonin, and Simon Bunel (2021) The Power of Creative Destruction: Economic Upheaval and the Wealth of Nations (Harvard University Press)-->{{cite Q|Q137641358}} * <!--Ajzenman, Nicolas (29 January 2023). "Migrants don't cause crime rates to increase — but false perceptions endure anyway". The Conversation.-->{{cite Q|Q141758664}} * <!--Costas Arkolakis, Michael Peters, and Sun Kyoung Lee, (2020) "European Immigrants and the United States’ Rise to the Technological Frontier"-->{{cite Q|Q141678287}} * <!--Dean Baker (2016) Rigged: How globalization and the rules to the modern economy were structured to make the rich richer-->{{cite Q|Q100216001}} * <!--Abhijit V. Banerjee and Esther Duflo (2019) Good Economics for Hard Times (PublicAffairs)-->{{cite Q|Q85764011}} * <!--Jutta Bolt and Jan Luiten van Zanden (2024-09-18) "Maddison style estimates of the evolution of the world economy: A new 2023 update", Journal of Economic Surveys, 1-41-->{{cite Q|Q126723821}} * <!--S. N. Broadberry, B. Campbell, A. Klein, M. Overton and B. van Leeuwen (2015) ''British Economic Growth 1270-1870'' (Cambridge University Press)__>{{cite Q|Q57945943}} * <!--Donald S. L. Cardwell (1972) Turning Points in Western Technology (Neale Watson, Science History Pubs)-->{{cite Q|Q141795083}} * <!--Renée DiResta (2026) Invisible Rulers: The People Who Turn Lies into Reality, updated edition-->{{cite Q|Q141156783}} * <!--Renée DiResta and Rachel Kleinfeld (2025-09-10) For Expertise to Matter, Nonpartisan Institutions Need New Communications Strategies-->{{cite Q|Q141158362}} * <!--Katarzyna Dośpiał-Borysiak (2017) "Model of state management of petroleum sector -- Case of Norway-->{{cite Q|Q141774179}} * <!--Rachel Foxley (2013) The Levellers: Radical Political Thought in the English Revolution (Oxford University Press)-->{{cite Q|Q136227988}} * <!--Benjamin Gonzalez O’Brien, Loren Collingwood, Stephen Omar El-Khatib (2017-05-07) "The Politics of Refuge: Sanctuary Cities, Crime, and Undocumented Immigration-->{{cite Q|Q55932800}} * <!--Graves (2025-01-08) Defend Free Speech For All (Town Hall)-->{{cite Q|Q132855526}} * <!--Graves (2023-03-21) Do copyrights and paywalls on academic journals violate the US Constitution?-->{{cite Q|Q126116187}} * <!--Christopher Hibbert, Ben Weinreb, John Keay, Julia Keay, and Matthew Weinreb (2011-09-09) The London Encyclopaedia, 3rd ed.-->{{cite Q|Q137188993}} * <!-- Historian, USPS (2010) Postage Rates for Periodicals -->{{cite Q|Q112167199}} * <!--Richard R. John (1995) Spreading the News: The American Postal System from Franklin to Morse-->{{cite Q|Q54641943}} * <!--Volker Leppin and Timothy J. Wengert (2015) "Sources for and against the Posting of the Ninety-Five Theses"-->{{cite Q|Q141737347}} * <!--Lawrence Lessig (2004) Free Culture-->{{cite Q|Q830526}} * <!--Olivier Marie and Paolo Pinotti (2024-02-01) "Immigration and Crime: An International Perspective", Journal of Economic Perspectives. 38 (1): 181–200-->{{cite Q|Q128434874}} * <!--Martínez, Daniel E.; Martínez-Schuldt, Ricardo D.; Cantor, Guillermo (2017). " Providing Sanctuary or Fostering Crime? A Review of the Research on 'Sanctuary Cities' and Crime" . Sociology Compass. 12 e12547. doi:10.1111/soc4.12547. ISSN 1751-9020.-->{{cite Q|Q55932802}} * <!-- Robert W. McChesney; John Nichols (2010). The Death and Life of American Journalism (Bold Type Books) -->{{cite Q|Q104888067}}. * <!-- Robert W. McChesney; John Nichols (2016). People get ready-->{{cite Q|Q87619174}} * <!--Robert Michels (1911) Les Partis politiques, essai sur les tendances oligarchiques des démocraties-->{{cite Q|Q7210216}} * <!--Joel Mokyr (2017) A Culture of Growth: The Origins of the Modern Economy (Princeton University Press)-->{{cite Q|Q136118590}} * <!--Joel Mokyr (1994) "Cardwell's Law and the political economy of technological progress", Research Policy 23(5)561-574-->{{cite Q|Q139772607}} * <!--Ashley N. Muchow & Taisiia Stanishevska (2026) "From Headlines to Handcuffs: Local News Coverage of Immigrant Crime and Hispanic Arrest", Justice Quarterly, vol. 43(3) (https://doi.org/10.1080/07418825.2025.2550307)-->{{cite Q|Q141760694}} * <!--Neff and Pickard (2024) "Funding Democracy: Public Media and Democratic Health in 33 Countries"-->{{cite Q|Q131468289}} * <!--Gerald G. Newman and Leslie Ellen Brown, eds. (1997) Britain in the Hanoverian Age, 1714–1837: An Encyclopedia-->{{cite Q|Q141759196|authors = Gerald G. Newman and Leslie Ellen Brown, eds.}} * <!--John Hely Owen (1938) War at Sea Under Queen Anne 1702–1708-->{{cite Q|Q141757101}} * <!--Andrew Pettegree (2015-10-27) Brand Luther: How an Unheralded Monk Turned His Small Town into a Center of Publishing, Made Himself the Most Famous Man in Europe--and Started the Protestant Reformation-->{{cite Q|Q141761223}} * <!--Thomas Piketty (2022) A Brief History of Equality-->{{cite Q|Q139886018}} * <!--Joad Raymond (2006-11-02) Pamphlets and Pamphleteering in Early Modern Britain-->{{cite Q|Q141761849}} * <!--Nick Roll (2017-07-16) "Correcting Jeff Sessions: Attorney general railed against crime in sanctuary cities, citing two professors’ work. But his claims are contrary to their conclusions.-->{{cite Q|Q141735231}} * <!--J.P. Smits, E. Horlings and J.L. van Zanden (2000) Dutch GDP and its Components 1800-1913 (Groningen Growth and Development Centre)-->{{cite Q|Q137669960}} * <!--I. Sugimoto (2011) ''Economic growth of Singapore in the twentieth century: historical GDP estimates and empirical investigations'' ( World Scientific Publishing)-->{{cite Q|Q137669987}} * <!--Nik Usher and Sanghoon Kim-Leffingwell (2022-01) How Loud Does the Watchdog Bark? A Reconsideration of Local Journalism, News Non-profits, and Political Corruption -->{{Cite Q|Q134715465}} * <!--J. L. van Zanden and B. van Leeuwen (2012), ‘Persistent but not consistent: the growth of national income in Holland 1347–1807’, Explorations in Economic History, 49, pp. 119–30-->{{cite Q|Q137670058}} * <!--Brittney Washington (2018-06-19) " A Pamphlet War in England, 1641-1643"-->{{cite Q|Q141822147}} * <!--Tom K. Wong (2017-01-26) "The Effects of Sanctuary Policies on Crime and the Economy"-->{{cite Q|Q98775875}} [[Category:Original research]] [[Category:Research]] [[Category:Economics]] [[Category:Macroeconomics]] [[Category:Gross domestic product]] [[Category:Economic growth]] [[Category:Modern Monetary Theory]] [[Category:Political science]] [[Category:Freedom and abundance]] <!-- https://en.wikiversity.org/wiki/Category_Review --> s709eyi0k4chiu2xwwd9rert1cq7phl 2842457 2842454 2026-10-24T20:45:20Z DavidMCEddy 218607 wdsmth 2842457 wikitext text/x-wiki {{Research project}} :''This article invites all to contribute references and interpretations to the growth in average annual income, Gross Domestic Product (GDP) per capita at purchasing power parity, as applied to leading countries in the [[w:Maddison Project|Maddison Historical Statistics Project]].'' :''This essay is on Wikiversity to encourage a wide discussion of the issues it raises moderated by the Wikimedia rules that invite contributors to [[w:Wikipedia:Be bold|“be bold but not reckless,”]] contributing revisions written from a [[Wikiversity:Disclosures|neutral point of view]], [[Wikiversity:Cite sources|citing credible sources]] -- while raising other questions and concerns on the associated [[Wikiversity:FAQ|'''{{'}}“Discuss”'''{{'}} page]] -- and treating others with respect.'' ::''This article uses [[w:ISO 8601|ISO 8601]] dates except for References, which are controlled by standard Wikidata formatting, and direct quotes. In the initial author's experience, [[ISO 8601 and computing differences between dates|ISO 8601 dates seem to make it easier to remember dates and to compute differences between them.]]'' == Abstract == This article first identifies and plots GDP per capita for leading countries in the Maddison data then discusses comments by Nobel prize winning economists and others to interpret what appears in said plot. No one really knows how to make economies grow. However, growth at the technology frontier requires creative destruction, and immigrants tend to be more entrepreneurial than native born. Also, creative destruction is facilitated and inequality is limited by diversity of ownership and control of the major media. == Maddison Historical Statistics == [[File:MaddisonDataLeaders1349-2022.svg|thumb|''Figure 1. World leaders in GDP per capita 1349-2022 (NLD = initially Holland to 1809 and the Netherlands after; GBR = initially England to 1700, Great Britain to 1870, then the UK; USA starting with the 13 colonies that became the US; SGP = Singapore).<ref>See Bolt and van Zanden (2024) for the Maddison Data generally, Broadberry et al. (2015) for the data on England 1252–1700 and on Great Britain until 1870, Sugimoto (2011) for Singapore to 2007, van Zanden and van Leeuwen (2012) for the data on Holland 1349–1807, and Smits et al. (2000) for the data on the Netherlands 1808-1913. The dramatic drop in the orange line between 1807 and 1808 could be due to the change in reference from [[w:Holland|Holland]] to the Netherlands, as Holland at that time was more prosperous than the rest of the Netherlands.</ref>'']] The [[w:Maddison Project|Maddison Historical Statistics Project]] (aka Maddison Project) is widely recognized as the premier data source on historical macroeconomic data. It is named after [[w:Angus Maddison|Angus Maddison]] (1926-2010), who was a pioneer in [[w:National accounts|national accounts]], especially estimating [[w:Gros Domestic Product|gross domestic product]] (GDP) per capita (GDPpc) and population (pop) at various times dating back to the year 1 of the [[w:Common Era|Common Era]] (CE / AD). After Maddison's death, the Maddison Project has continued to update the database to include the best sources available on any country for any historical period since year 1. When accessed 2026-10-10 the version of the data available had been released 2024-09-18 containing data to 2022.<ref>Bolt and van Zanden (2024).</ref> This dataset includes 23,280 observations on at least one of GDPpc and pop for an average of 137 years for 169 countries. Only 18 countries had the highest GDPpc for at least one year. Annual data starts in 1252, and that's only for one country, GBR = England / Great Britain / the United Kingdom. NLD (Holland / the Netherlands) had the highest GDPpc for 97 percent of the years between 1349 and 1807, and the Dutch leadership in Figure 1 may have ended then, not because of the devastation from the [[w:Napoleonic Wars|Napoleonic Wars]] (which negatively impacted them) but because the available data prior to 1807 are for [[w:Holland|Holland]], while more recent data are for the Netherlands, of which Holland is only a part, though probably the most prosperous part at that time. This analysis focuses on data since 1349, in part because data were available for at most 4 countries for most years prior to that date. (The 3-letter country codes used here are [[w:ISO 3166-1 alpha-3|ISO 3166-1 alpha-3]], used in the Maddison data.) We are interested in technology leaders, so we eliminate the [[w:petrostates|petrostates]] of Qatar (QAT), Kuwait (KWT), and United Arab Emirates (ARE), plus Norway (NOR), which owes a substantial portion of their wealth to democratic management of [[w:North Sea oil|North Sea oil]].<ref>Dośpiał-Borysiak (2017).</ref> With these restrictions 12 countries had the highest GDPpc for at least one year. However, FRA (France), ITA (Italy), SWE (Sweden), BEL (Belgium), NZL (New Zealand) and CHE (Switzerland) led for at most 7 of the 675 years between 1349 and 2022; we eliminate those, because including them would seem to add more clutter than clarity. We also eliminate AUS (Australia), which led for 17 of the 39 years between 1853 and 1891, when it was competing for the leadership with GBR and USA, for the same reason. And we eliminate LUX (Luxembourg), which has a population under a million. This leaves NLD, GBR, USA, and SGP (Singapore), plotted in Figure 1. This plot has some interesting features, which we discuss in the remainder of this research report. == Nobel economists and others == Banerjee and Duflo (2019, p. 207) said, "despite the best efforts of generations of economists, the deep mechanisms of persistent economic growth remain elusive." That appeared shortly before they shared the 2019 [[w:List of Nobel Memorial Prize laureates in Economic Sciences|Nobel Memorial Prize in Economics]] with [[w:Michael Kremer|Michael Kremer]] for documenting how [[w:Randomized controlled trial|randomized controlled trial]]s could be used to alleviating poverty in [[w:Development economics|development economics]]. While Banerjee and Duflo insist that no one knows how to make an economy grow, other leading economists recommend partial solutions. For example, [[w:Joel Mokyr|Mokyr]] (2017, p. 53) said, "The willingness of the Netherlands and later Britain and the United States to tolerate Jews and dissenting Christians contributed a great deal to their economies, especially in high-skilled manufacturing and financial services." In other words, immigrants made an outsized contribution to the growth of NLD, GBR, and US, visible in Figure 1. Mokyr's comment appeared over a year before Banerjee and Duflo (2019) and roughy eight years before Mokyr shared the 2025 Nobel Memorial Prize in Economics with [[w:Philippe Aghion|Aghion]] and [[w:Peter Howitt (economist)|Howitt]]. Mokyr (2017, p. 255) also said that, "intellectual innovation will eventually end up being increasingly resisted by an incumbent technology; if that resistance is not overcome, technological progress may be extinguished." That resistance, he suggested, drives {{quote| the empirical regularity that no society remains at the cutting edge of technological creativity for very long<ref>Mokyr (2017, p. 260). [[w:Robert Michels|Michels]] (1911) called this "the [[w:Iron law of oligarchy|iron law of oligarchy]]".</ref>.}} Mokyr (2017, p. 260; 1994) calls this "Cardwell's Law" after Cardwell (1972), which describes {{quote| the interaction between technological creativity and political competition ... based on the observation that technological progress encounters resistance from various groups that believe they stand to lose from innovation. These pressure groups will try to manipulate the political system to suppress successful innovation.}} Further documentation of this appears in Acemoglu and Johnson (2023) and Acemoglu and Robinson (2012). This provides one explanation for the end of growth of the economy of Holland shortly before 1600 and for the slowing of economic growth in recent decades in the US, the UK, and the Netherlands, visible in Figure 1. It can also explain [[w:History of the Roman Empire|the rise and fall of the Roman Empire]], of the [[w:Maya civilization|Maya civilization]], and other societies that became very rich from allowing commoners to become entrepreneurs and then declined after previous winners changed the rules of the political economy to block further [[w:creative destruction|creative destruction]]. This includes how [[w:History of the Republic of Venice|Venice]] became "possibly the richest place in the world, with the most advanced set of inclusive economic institutions", dominating trade in the [[w:Mediterranean Sea|Mediterranean]] based on a system that encouraged entrepreneurial commoners to undertake risky trading ventures starting after 810 CE and ending when leaders made wealthy by such ventures changed the rules to end that system starting in 1297, destroying their dominance of Mediterranean trade "and converting Venice into a museum", in the words of Acemoglu and Robinson (2012, pp. 152-155). Mokyr (2017, p. 60) also mentioned "entrepreneurs of hate", who convince others to dislike some group in a way that may benefit those "entrepreneurs" and major media organizations, who promote such "entrepreneurs" to please those who control much of the money for the media. For-profit social media have been particularly effective in herding their audiences into echo chambers, reinforcing their preconceptions in ways that increase political polarization and violence and threaten democracy.<ref>See, e.g., DiResta (2026), DiResta and Keinfeld (2025), and the Wikiversity article on "[[Managing the good and the evils of social media]]", accessed 2026-10-14.</ref> When successful this protects current elites from competition that would otherwise drive [[w:creative destruction|creative destruction]], benefitting the public at large while threatening existing elites. For example, most research on immigrants suggest that on average immigrants are more entrepreneurial and no more likely to commit crime than native born,<ref>e.g., Martínez et al. (2017); Muchow and Stanishevska (2026); Ajzenman(2023).</ref> and some studies report that [[w:Sanctuary city|sanctuary jurisdiction]]s have on average higher median incomes and less crime.<ref>Martínez et al. (2017); Wong (2017). See also the Wikipedia article on "[[w:Sanctuary city|sanctuary city]]", accessed 2026-10-12.</ref> Where immigrant crime is higher, it typically result from anti-immigrant labor market restrictions or from young less-educated males with irregular status.<ref>Marie and Paolo (2024). See also the Wikipedia article on "[[w:Immigration and crime|Immigration and crime]]", accessed 2026-10-12.</ref> But major media outlets make money from featuring charismatic figures, who attract an audience without offending people who control much of the money for the media. In 2017 President [[w:Immigration policy of the first Trump administration|Trump]]'s Attorney General [[w:Jeff Sessions|Jeff Sessions]] quoted a research study that supposedly showed that sanctuary cities had more crime than cities without sanctuary policies. The authors of that study insisted that Sessions had ''misrepresented'' their work, which concluded "that sanctuary policies have no effect on crime rates, despite narratives to the contrary."<ref>Roll (2017); Gonzalez O’Brien, et al. (2017).</ref> To the extent that these comments provide a balanced summary of the research in this area, it suggests that leaders like Trump and Sessions, to name only two, can be fairly describes as "entrepreneurs of hate". We next discuss competition in media markets before discussing immigrants further. === Diversity of ownership of the major media === Acemoglu and Johnson (2023, esp. ch. 4) describe how religious leaders in hierarchical societies prior to [[w:James VI and I|King James]] of the [[w:King James Version|King James Bible]] convinced commoners to live in poverty while giving increasing shares of what they produced, so religious and secular elites could live in opulence. Prior to King James the major media were religious organizations. In such hierarchical societies some religious leaders were entrepreneurs of hate, exhorting the faithful to fear and hate designated evil others and give increasing shares of what they produce so secular elites would "protect" them in wars against designated evil others. Acemoglu and Robinson (2012, p. 461) said, "Empowerment of society at large is difficult to coordinate and maintain without widespread information about whether there are economic and political abuses by those in power. ... Pamphlets and books informing and galvanizing people played an important role during the [[w:Glorious Revolution|Glorious Revolution]] [1688] in England, the French Revolution [1789-1799]" and in the US. ([[w:Daron Acemoglu|Acemoglu]], [[w:Simon Johnson (economist)|Johnson]] and [[w:James A. Robinson|Robinson]] shared the 2024 [[w:List of Nobel Memorial Prize laureates in Economic Sciences|Nobel Memorial Prize in Economics]].) But pamplets started challenging established power structures much earlier. The publication of [[w:Martin Luther|Martin Luther]]'s ''[[w:Ninety-five Theses|Ninety-five Theses]]'' in Latin 1517-10-31 was quickly condemned by [[w:Pope Leo X|Pope Leo X]] and his hierarchy. [[w:Johann Tetzel|Johann Tetzel]], who had been selling [[w:Indulgence|indulgence]]s, published a pamphlet calling for Luther to be burnt for [[w:heresy|heresy]]. Eight hundred copies were reportedly sent to [[w:Wittenberg|Wittenberg]] to be sold, but students seized and burned them.<ref>Hendrix (2015, p. 64), quoted from the Wikipedia article on the ''[[w:Ninety-five Theses|Ninety-five Theses]]'', accessed 2026-10-14.</ref> This led to a pamphlet war. Luther's brief German-language reply was short and easy for commoners to understand<ref>Leppin and Wengert (2015, p. 389), cited from the Wikipedia article on the ''[[w:Ninety-five Theses|Ninety-five Theses]]'', accessed 2026-10-14.</ref> and was much more popular than Tetzel's.<ref>Pettegree (2015, p. 145), cited from the Wikipedia article on the ''[[w:Ninety-five Theses|Ninety-five Theses]]'', accessed 2026-10-14.</ref> Less than twenty years later, in 1534, King [[w:Henry VIII|Henry VIII]] declared himself head of the [[w:Church of England|Church of England]], denying the authority of the Pope. To secure the support of his subjects, he drove a propaganda war that included print literature<ref>Raymond (2006), cited from the Wikipedia article on "[[w:Pamphlet war|pamphlet war]]", accessed 2026-10-15.</ref> and theatre and minstrel troupes funded by the king. The latter traveled around the land to promote the new religious practices. Henry was portrayed as the glorious king of England, bravely and heroically defending the true faith against foreign devils, especially the Pope and Catholic priests, monks, etc.<ref>Hibbert et al. (2010, p. 928), cited from the Wikipedia article on "[[w:Henry VIII|Henry VIII]]", accessed 2026-10-15.</ref> Henry died in 1547. Less than a dozen years later, the crown passed to [[w:Elizabeth I|Elizabeth I]]. Her limited personal wealth and English tradition required her to negotiate with Parliament for tax money. Acemoglu and Robinson (2012, pp. 105-111) noted that she was forced to limit the granting of monopolies and allow a class of merchants to compete in international trade. Many English merchants became wealthy, helped by the defeat of the [[w:Spanish Armada|Spanish Armada]] in 1588. When Elizabeth died in 1603, she was replaced by her cousin, King [[w:James VI and I|James VI of Scotland]], who then became also James I of England. He claimed [[w:Divine right of kings|divine right of kings]], threatening this class of merchants. To raise money without asking Parliament, he granted patents and monopolies to those who would pay. Parliament responded by passing the [[w:Statute of Monopolies|Statute of Monopolies]] on 1624-05-29. This statute repealed some past and future patents and monopolies except for novel inventions. But that Statute of Monopolies did not end the conflict. It ultimately went public, described as, "A Pamphlet War in England, 1641-1643".<ref>Washington (2018).</ref> Some pamphlets supported [[w:Charles I of England|Charles I]], who became King after James died in 1625. Others supported Parliamentarians and [[w:Levellers|Levellers]]. The latter supported radical ideas like [[w:popular sovereignty|popular sovereignty]], extended suffrage, equality before the law and religious tolerance. In 1642 war broke out between Parliament and Charles I, though many in Parliament sided with Charles -- especially those, who had received monopolies from him.<ref>Acemoglu and Robinson (2012, p. 189).</ref> The war was won by the [[w:New Model Army|New Model Army]], which represented many in Parliament plus Levellers. The latter were excluded from the government that followed the [[w:Execution of Charles I of England|beheading of King Charles I]] in 1649.<ref>Foxley (2013, p. 207), cited from the Wikipedia article on [[w:Levellers|Levellers]], accessed 2026-10-15.</ref> However, that new government stopped granting as many monopolies and allowed so many commoners to become entrepreneurs that the GDP per capita began to grow at rates that are key features of Figure 1 for England and the 13 English colonies that became the US over a century later. One explanation for that economic growth was the diversity of independent publishers of pamphlets, books and newspapers. [[w:Robert W. McChesney|McChesney]] and [[w:John Nichols (journalist)|Nichols]] (2010, 2016) suggested that the US [[w:Postal Service Act|Postal Service Act]] of 1792<ref>Wikipedia "[[w:Postal Service Act|Postal Service Act]]", accessed 2023-07-11.</ref> made a major contribution to making the US what it is today. Under that act, newspapers were delivered up to 100 miles for a penny, when postage for letters was between 6 and 25 cents depending on distance.<ref>Historian, USPS (2010).</ref> McChesney and Nichols estimated that between 1840 and 1844, the US postal subsidy was a little over 0.2% of GDP.<ref name=McC-N2010>McChesney and Nichols (2010, pp. 310-311, note 88).</ref> During the first half of the nineteenth century the US had more independent newspaper publishers (more independent media organizations) per million population than at any other time or place in human history.<ref>John (1995). See also the interview with him at "[[Media concentration per Columbia History Professor Richard John]]" on Wikiversity, accessed 2026-10-12.</ref> In 1932 with official unemployment over 20 percent, the conservative argument that blames the poor for their poverty did not sell newspapers, and the US still had sufficient competition in media markets that presidential candidate Franklin Roosevelt (FDR) got a mandate to spend the money to put humans back to work while taxing the ultra-wealthy like they had never been taxed before or since and during World War II to institute wage and price controls under the [[w:Office of Price Administration|Office of Price Administration]]. The latter prevented price gouging by government contractors that had generated inflation and stifled economic growth in previous major wars (World War I, the Civil War, and the War of 1812). Instead, the US averaged 7 percent per year growth for FDR's 12 years.<ref>The "6 percent per year" growth during FDR's 12 years was computed as the average of log(GDPpc) in the Maddison Data. To be precise, that computation produced 0.0597; converting that from a log scale to percent increase gives 0.0615. Using data from [[w:MeasuringWorth|MeasuringWorth]] instead of Maddison, we get 0.783 and 0.815, respectively. More documentation of this is available in "[[Media Literacy and You/Fox, the Great Depression, the Great Recession, and our future]]" on Wikiversity, accessed 2026-10-13.</ref> One of the most interesting features of Figure 1 is the behavior of the US economy in the administrations of [[w:Herbert Hoover|Herbert Hoover]] and FDR: Under Hoover, the economy declined at 9 percent per year for 4 years, a financial disaster that created the public demand for change that gave FDR the mandate to do what he did. Economic growth under two other presidents are numerically comparable to FDR's numbers but for much shorter periods: (1) [[w: Warren G. Harding|Warren G. Harding]] averaged 7 percent annual growth for two years before he died in 1923 during the [[w:Roaring Twenties|Roaring Twenties]]. (2) [[w:Rutherford B Hayes|Rutherford B Hayes]], who took office in 1877 at the end of the [[w:Panic of 1873|Panic of 1873]], oversaw 5.5 percent growth per year for his 4 years as president. Those impressive sounding average growth figures are so short that they are not noticeably different from other minor features visible in Figure 1. FDR also dramatically reduced inequality. For over a century between about 1820 and 1932 the top 1 percent of the US population owned roughly 45 percent of net personal wealth in the US, while the bottom 90 percent owned roughly 22 percent. Shortly after FDR died, the wealth of the top 1 percent and bottom 90 percent were even at roughly 30 percent. FDR also oversaw dramatic growth in the power of organized labor. Their successes continued to reduce inequality to roughly 22 percent for the top 1 percent vs. 37 percent for the bottom 90 percent when Ronald Reagan became president. Since then, [[w:Reaganomics|Reaganomics]] have been rebuilding the [[w:Robber baron (industrialist)|Robber Baron]]s.<ref>This history of inequality is documented in [[:File:Share of US wealth 90p99.svg|Figure 7. Shares of US wealth - bottom 90 and top 1 percent, 1820-2023]] in the Wikiversity article on "[[w:Media Literacy and You/Fox, the Great Depression, the Great Recession, and our future|Media Literacy and You/Fox, the Great Depression, the Great Recession, and our future]]" in Wikiversity, accessed 2026-10-13.</ref> FDR made enemies among the wealthiest and most powerful. In a [[q:Franklin D. Roosevelt#Address at Madison Square Garden (1936)|speech in Maddison Square Garden in 1936]], FDR mentioned, "the old enemies of peace: business and financial monopoly, speculation, reckless banking, class antagonism, sectionalism, war profiteering. ... They are unanimous in their hate for me -- and I welcome their hatred." They doubtless could see that they could no longer afford as many personal servants -- maids, butlers, cooks, chauffeurs, etc. -- as they had before FDR. Broadcasting was relatively new when FDR took office. The economics of broadcasting naturally favored a very few networks, whose power concerned FDR. In 1941 his [[w:Federal Communications Commission| Federal Communications Commission]] (FCC) adopted the [[w:Mayflower doctrine|Mayflower doctrine]], which prohibited broadcasters from advocacy. When FDR died, people with power were able to get the Mayflower doctrine replaced by the more lenient [[w:Fairness Doctrine|Fairness Doctrine]], which opened the floodgates for entrepreneurs of hate, who drove the [[w:McCarthyism|McCarthy Red Scare]] in the US and many other problems since. Over time this became worse, facilitated by increasing concentration of ownership of the media. That contributed to the end of the Fairness Doctrine starting in 1985. That revocation and especially the more recent rise of for-profit social media have amplified the ability of entrepreneurs of hate to make money both for commercial social media and for themselves by disseminating lies about others, e.g., immigrants.<ref>On how for-profit social media companies promote entrepreneurs of hate, see, e.g., DiResta (2026) and the Wikiversity article on, "[[Managing the good and the evils of social media]]", accessed 2026-10-12. DiResta (2026, p. 1) described the promotion of "influencers" by for-profit social media as essentially unleashed "a war for power, influence, and reality itself—a conflict in which the ends justify any means, and one that has ripped America apart, threatened our very democracy, and consumed my own life for the past decade, leading to death threats, congressional subpoenas, lawsuits, and legal bills running into seven figures." In the interview on Wikiversity, DiResta said her legal bills to that point had exceeded $7 million.</ref> And the rate of economic growth in the US (and the Netherlands and the UK) have been slowing in recent decades, as is visible in Figure 1. Inequality expert [[w:Thomas Piketty|Piketty]] (2022, p. 139) noted, {{quote| In the United States, the national income per inhabitant rose at a rate ... of 2.2% between 1950 to 1990 when the top tax rate reached on average 72%. The top rate was then cut in half, with the announced objective of boosting growth. But in fact, growth fell by half, to 1.1% per annum between 1990 and 2020 ... . Beyond a certain level of inequality, repeatedly increasing differences in income and wealth has clearly had no positive effect on economic dynamism. In sum, all the data at our disposal today suggest that virtually confiscatory tax rates have been an immense historical success. They have made it possible to greatly reduce the divergences of fortunes and incomes, while at the same time improving the situation of the middle and lower classes, developing the welfare state, and stimulating better economic and social performance overall. Historically, it is the battle for equality and education that has made economic development and human progress possible, and not the veneration of property, stability, and inequality.}} Table 1 compares Piketty's growth numbers with computations based on the data from both Maddison and [[w:MeasuringWorth|MeasuringWorth]]: The numbers are close but not exact. {| class="wikitable sortable" style="text-align: center;" |+ Table 1. US growth rate 1950 to 1990 to 2020 |- ! ! colspan=3 | US GDP per capita !! colspan=2 | Growth (% / yr) |- ! source || 1950 !! 1990 !! 2020 !! 1950 - 1990 !! 1990 - 2020 |- | Piketty || - || - || - || 2.2% || 1.1% |- | Maddison || $15,240 || $36,982 || $54,379 || 2.24% || 1.29% |- | MeasuringWorth || $16,208 || $40,191 || $61,127 || 2.30% || 1.41% |} Acemoglu and Robinson (2012, pp. 105-106) provide a historical analysis supporting Piketty's concern about inequality slowing growth. They note that in 1588, England, Spain and France {{quote| were ruled by relatively absolutist monarchs: [[w:Elizabeth I|Elizabeth I]] in England, [[w:Philip II of Spain|Philip II]] in Spain, and [[w:Henry III of France|Henry III]] in France. All were battling with assemblies of citizens -- such as the [[w:Parliament of England#Tudor era (1485–1603)|Parliament]] in England, the [[w:Cortes Generales#Habsburg rule (16th–17th centuries)|Cortes]] in Spain, and the [[w:Estates General (France)#Revival in 1560–1614|Estates-General]] in France -- that were demanding more rights and control over the monarchy. ... [T]he English Parliament and the Spanish Cortes had power over taxation, while the Estates-General did not. In Spain this mattered little, because after 1492 the Spanish Crown had a vast American empire and benefited massively from the gold and silver found there. In England ... Elizabeth I was far less financially independent, so she had to beg Parliament for more taxes. In exchange, Parliament demanded concessions, in particular restrictions of the right of Elizabeth to create monopolies. ... In Spain the Cortes lost a similar conflict.}} Consistent with Acemoglu and Johnson (2023) and the discussion above, inequality tends to increase elite control of the media, which it makes it harder for commoners to get a fair hearing of their concerns. === Media conflicts of interest in IP law === Major media have [[w:Conflict of interest|conflicts of interest]] in honestly reporting on anything involving [[w:Intellectual property|intellectual property]] (IP) law. An example of Cardwell's law, mentioned above, is the [[w:Copyright Term Extension Act|Sonny Bono Copyright Term Extension Act]], also known as the Mickey Mouse Protection Act of 1998. The copyright term on ''[[w:Steamboat Willie|Steamboat Willie]]'', the first [[w:Mickey Mouse|Mickey Mouse]] animated film that Disney distributed, was extended multiple times by the US Congress following extensive lobbying by the film industry led by [[w:The Walt Disney Company|the Walt Disney Company]]. ''Steamboat Willie'', which debuted in 1928, finally entered the [[w:Public domain in the United States|US public domain]] on January 1, 2024. Eric Eldred and others filed suit, claiming that allowing any number of retroactive extensions violated the [[w:Copyright Clause|Copyright Clause]] of the US Constitution, which says, {{quote| [the United States Congress shall have power] To promote the Progress of Science and useful Arts, by securing for limited Times to Authors and Inventors the exclusive Right to their respective Writings and Discoveries.}} In ''[[w:Eldred v. Ashcroft|Eldred v. Ashcroft]]'' (2003), the [[w:Supreme Court of the United States|US Supreme Court]] disagreed with Eldred and allowed the statute to remain in effect. However, that case drew enough public attention that copyright terms have not (yet) been extended further, and ''Steamboat Willie'' is currently in the pubic domain. [[w:Lawrence Lessig|Lessig]] (2004) ''[[w:Free Culture (book)|Free Culture]]'' describes other examples, where US copyright law seems to ''limit'' rather than "promote the Progress of Science and useful Arts". This includes a case where a 2002 freshman, Jesse Jordan, modified a search engine to make it easier for other student to find pictures, research, notes, movie clips, etc., on an [[w:intranet|intranet]] at [[w:Rensselaer Polytechnic Institute|Rensselaer Polytechnic Institute]] (RPI). The [[w:Recording Industry Association of America|Recording Industry Association of America]] (RIAA) sued Jordan and three other students for close to $100 billion for copyright violations -- six times the total profit of the film industry in 2001. Jordan had an uncle, who was an attorney, who explained that the legal bills for fighting that case would be at least $250,000. That would bankrupt his parents; they could not recover that money, even if they won. Jordan had saved $12,000 to cover part of his college expenses. RIAA accepted that and settled the case.<ref>Lessig (2004, p. 42).</ref> This is an example of a [[w:Strategic lawsuit against public participation|strategic lawsuit against public participation]] (SLAPP), which in this case rested on questionable interpretations of copyright law. The major media have conflicts of interest in honestly reporting on anything involving (a) copyright laws, because much of their income is derived from copyright royalties, or (b) anything that might offend their major accounts, including major corporations. And anti-SLAPP laws, which exist in some jurisdictions but not others,<ref>[[w:Robert Corn-Revere|Robert Corn-Revere]], Chief Counsel for the [[w:Foundation for Individual Rights and Expression|Foundation for Individual Rights and Expression]], supports anti-SLAPP laws and the "JAWBONE act", to limit the power of the government to threaten questionable actions, like the Trump administration's threats against the ABC network over a Jimmy Kimmel monolog. See the Wikiversity article on "[[Foundation for Individual Rights and Expression]]", accessed 2026-10-16.</ref> limit the ability of major corporations to abuse their power against potential challengers without such deep pockets. Similarly Graves (2023) asked, "Do copyrights and paywalls on refereed academic journals violate the US constitution?" No one expects to make money from copyright royalties for drafts submitted to refereed academic journals. They do it out of a felt need (a) to contribute to the future of humanity or (b) to build their reputation or (c) because promotions at many universities are based partly on publication records. Graves is not alone in his concern about this: Many European countries do not allow reports of research they fund to be published behind paywalls.<ref>Dimi Zagorski is concerned that copyright exceptions for scientific research are different in different European countries, and that's a problem for the Wikimedia Foundation. See the discussion of this in the Wikiversity article on "[[Wikimedia concerns with European copyright rules including AI and scientific research]]". That's the opposite of US law, which grants patents to universities, research institutions, private companies, and individuals for innovations developed under government contracts, per the Bayh-Dole Act of 1980, described by Baker (2016, p. 80).</ref> Regarding patent law, the major media have conflicts of interest in discussing proposed changes affecting major accounts, but they make little use of patent law themselves. This seems to have contributed to the claim that, "[O]ver the last four decades ... stronger patent protection seems to be associated with slower growth" in the US, according to [[w:Dean Baker|Baker]] (2016, pp. 77-78). A potentially similar phenomenon appears in a comparison of 21 different European countries in an analysis that displays "a positive relationship between anticorruption measures in a country and rate of innovation in that country as measured by patents per capita", discussed by Aghion et al. (2021, Figure 15.1b, pp. 292-293).<ref>Usher and Kim-Leffingwell (2022) documented how each member of the [[w:Institute for Nonprofit News|Institute for Nonprofit News]] (INN), relatively small nonprofits independent of major media organizations and not funded by advertising, in a US federal court district was associated with on average 1.4 additional prosecutions for political corruption per year between 2003 and 2019. They found on average 10 federal prosecutions for political corruption for each federal court district per year. Thus, 1.4 additional prosecutions per year per district represents an increase of 14 percent per member of INN.</ref> === Immigrants === We now return to discussing the initiation of economic growth following the beheading of Charles I in 1649 visible in Figure 1. Charles I was replaced by [[w:English Council of State|Lord Protector Cromwell]], who refused to authorize as many monopolies as his predecessors, as mentioned above. He also encouraged Jews and dissenting Christians to settle in England, as mentioned by Mokyr (2017, p. 53), cited above. But Mokyr is not alone in claiming that immigrants contribute to economic growth. Aghion et al. (2021, p. 266) cite Arkolakis et al. (2020) for explaining how the period 1880–1920 is noteworthy for two parallel evolutions. First, the United States caught up with England and France to become the most technologically advanced country in the world as well as the wealthiest in terms of GDP per capita. Second, the United States welcomed large numbers of immigrants, most notably from Europe. The purpose of the cited study was to demonstrate a causal connection between these two phenomena: in other words, to show that European immigration was responsible for the United States’ taking the lead in technology. ... More precisely, if the United States had completely closed its borders to all foreign immigrants in 1880, its per capita GDP would have been 30 percent lower than the level actually attained in 1920". === Industrial Revolution === There seems to be a broad consensus that the [[w:Industrial Revolution|Industrial Revolution]] began in England. However, there is less agreement on when it began. Mokyr (2017, p. 3) says, "The British Industrial Revolution of the late eighteenth century unleashed a phenomenon never before even remotely experienced by any society." He described "three cultural elements" required for "sustainable economic growth propelled by technological progress": (1, 2) The belief that it was both ''possible'' and ''desirable''. (3) Supportive policy measures had to be implemented. "[B]efore 1750 they did not produce anything like an Industrial Revolution or sustainable economic growth propelled by technological progress.<ref>Mokyr (2017, pp. 19-20).</ref> Acemoglu (2026, p. 53) similarly mentioned the eighteenth century in saying, "The breakthroughs in production technology that started in the middle of the eighteenth century in Britain unleashed the Industrial Revolution." Aghion et al. (2021, esp. pp. 29-30) offer a ''later'' date for the start of the Industrial Revolution: They report that, "European economies prior to 1820 experienced a series of phases of growth and decline. This raises a question: How did the pre-1820 period differ from the takeoff that started with the first Industrial Revolution?" Aghion et al. (2021, p. 20) describe the "first Industrial Revolution" as having been triggered by "the invention of the steam engine by James Watt in the 1770s ... . The invention of electricity set off the second Industrial Revolution, whose golden age was in the first half of the twentieth century. This second revolution originated with the invention of the [[w:Incandescent light bulb|light bulb]] by [[w:Thomas Edison|Thomas Edison]] in 1879 and of the [[w:dynamo|dynamo]] by [[w:Werner von Siemens|Werner von Siemens]] in 1866."<ref>The dynamo was independently invented by [[w:Henry Wilde (engineer)|Henry Wilde]], [[w:Charles Wheatstone|Charles Wheatstone]], [[w:Samuel Alfred Varley|Samuel Alfred Varley]] and Siemens, all about the same time, according to the Wikipedia article on [[w:Dynamo#Practical designs|dynamo]], accessed 2026-10-18.</ref> The image in Figure 1 raises questions about these claims by Mokyr, Acemogu, and Aghion et al.: It shows growth followed by stagnation in Holland but not "phases of growth and decline" with the exception of wars and [[w:Herbert Hoover|Herbert Hoover]]. It also does ''not'' show the slope (i.e., rate of growth) increasing starting around 1750 or the 1770s or 1879 or 1886. Instead it shows sustained growth in GDP per capita for England / Great Britain / the UK starting ''abruptly'' after 1649 with economic disruptions during the reign of [[w:Anne, Queen of Great Britain|Queen Anne]] (1702-1714) and the [[w:War of the Spanish Succession|War of the Spanish Succession]] (1701-1714). This was followed by relatively steady but slower growth as England was ruled by Kings [[w:House of Hanover#Monarchs of Great Britain, Ireland, and Hanover|George I, II, and III]] (1714-1820). Another dramatic increase in the slope started abruptly about 1819. Acemoglu and Johnson (2023, chapter 1) are more consistent with Figure 1. They said that, "the Industrial Revolution ... [was] rooted in institutional changes that started empowering the middling sort of English people from the sixteenth and seventeenth centuries onward." ... However, "the quality of life of a European peasant circa 1700 was not much different from that of an Egyptian peasant two thousand or seven thousand years previously." (chapter 5) They also claim that, "English national production per capita doubled from 1500 to 1700", (chapter 5), and "Growth emerged slowly in the second half of the eighteenth century as the Industrial Revolution, based on major technological breakthroughs and their application in industry, took root." (p. 45, chapter 2) Indeed, Figure 1 shows English GDP per capita doubling between 1500 and 1700, but it appears to have been fairly constant from 1500 to 1649, then roughly doubled in the next half century. [The data on England up to 1700 in Figure 1 came from Broadberry et al. (2015), which was not available when Acemoglu and Robinson were writing ''Why Nations Fail'', which appeared in 2012.] Acemoglu and Robinson (2012, ch. 11) describe the financial instability and slow growth during the reigns of Queen Anne and George I, II, and III. After the Glorious Revolution, France, Spain, the pope, and others refused to recognize [[w:William III of England|William and Mary]] (1689-1702) and then Queen Anne as heads of state of England. Instead they continued to support the claims of James II and then his son, [[w:James Francis Edward Stuart|James Francis Edward Stuart]] (the "Old Pretender", a brother of Queen Anne), after James II died in 1701. During the War of the Spanish Succession, each side tried to exploit the internal divisions of the other. Owen (1938, ch. 8) said France almost succeeded in an attempt in 1708 to [[w:Planned French invasion of Britain (1708)|land between 5 and 6 thousand soldiers in northeast Scotland]]. [[w:Louis XIV|Louis XIV]] had hoped to get the Old Protector named as King James VIII of Scotland. There were still [[w:Jacobitism|Jacobites]] in Scotland, who supported the Old Pretender and likely would have supported the invaders if they had landed. If they had succeeded, it would have created major problems for Queen Anne and her supporters. However, the Royal Navy prevented the invaders from disembarking, and they returned to France. When Queen Anne died in 1702, she was succeeded by George I. His Parliament was dominated by the Whig party, which represented the new mercantile and economic interests and had been the main organization behind the Glorious Revolution. Once in power, they were tempted to use their newly found position to prey on the rights of others but were constrained both by competing groups in Parliament, "and by the very institutions that they had fought to introduce to strengthen Parliament and to prevent the emergence of a new absolutism and the return of the Stuarts",<ref>After James II fled England, the new government worked to block [[w:Jacobitism|Jacobitism]], which attempted to restore the [[w:House of Stuart|House of Stuart]] to the thrones of Scotland and Britain. When James II died, his son became titular head of this movement, who became known as "the [[w:James Francis Edward Stuart|Old Pretender]]". After he died, he was replaced by his son, who became known as "the [[w:Charles Edward Stuart|Young Pretender]]".</ref> according to Acemoglu and Robinson (2012, pp. 302-303). The Whigs gave "[[w:Common land|common land]]" to nobles, confiscating it from commoners. Some commoners, who had traditionally shared that land, joined "Blacks", so called, because they blackened their faces when hunting on former commons or destroying the property of the nobility or extorting money from nobles or their employees. Parliament retaliated by passing the [[w:Black Act 1723|Black Act 1723]], under which someone accused of [[w:Poaching|poaching]] a [[w:Hare|hare]], for example, could under certain circumstances receive a summary judgment of guilty and then be executed without delay if apprehended later.<ref>Newman and Brown, eds. (1997, p. 58). See also the Wikipedia article on "[[w:Black Act 1723|Black Act 1723]]", accessed 2026-10-17.</ref> But not all accused were summarily executed. In 1724 "John Huntridge was accused of aiding deer stealers and abetting known Blacks, both crimes punishable by hanging. The prosecution of Huntridge came right from the top, initiated by the Regency Council of Lords Justices ... . After a trial of eight or nine hours, the jury found Huntridge innocent, partly on procedural grounds, since there were irregularities with the way the evidence had been collected. ... The jury was made up not of Huntridge’s peers, but of major landowners and gentry, who ought to have sympathized" with the noble, whose property had been attacked by Blacks allegedly aided by Huntridge.<ref>Acemoglu and Robinson (2012, pp. 304-305).</ref> These kinds of actions by the Parliaments of George I, II, and III might explain the slower growth during that period compared to that of the last half of the seventeenth century. That slower growth ended abruptly at the time of the [[w:Peterloo massacre|Peterloo Massacre]], 1819-08-16, visible in Figure 1. The growth of factories implied growth of cities, where workers lived closer together and could more easily organize. A campaign arose with multiple demands: They opposed the [[w:Corn Laws|Corn Laws]], which kept prices high for cereal grains and products made from them. They also demanded "universal [adult male] suffrage" and secret ballots in elections. A series of public meetings discussing these grievances led to one 1819-08-16, which attracted roughly 60,000. Acemoglu and Robinson (2012, p. 207) reported that, {{quote| The authorities were very nervous about the meeting, and a force of six hundred cavalry of the [[w:15th The King's Hussars|Fifteenth Hussars]] had been assembled. As the speeches began, a local magistrate decided to issue a warrant for the arrest of the speakers. As police tried to enforce the warrant, they met with the opposition of the crowd, and fighting broke out. At this point the Hussars charged the crowd. Within a few chaotic minutes, eleven people were dead and probably six hundred wounded. The ''[[w:Manchester Observer|Manchester Observer]]'' called it the Peterloo Massacre. But given the changes that had already taken place in economic and political institutions, long-run repression was not a solution in England. The Peterloo Massacre would remain an isolated incident. Following the riot, the political institutions in England gave way to the pressure, and the destabilizing threat of much wider social unrest, particularly after the 1830 revolution in France against [[w:Charles X of France|Charles X]], who had tried to restore the absolutism destroyed by the French Revolution of 1789. In 1832 the government passed the [[w:Reform Act 1832|First Reform Act]]. It enfranchised [[w:Birmingham|Birmingham]], [[w:Leeds|Leeds]], [[w:Manchester|Manchester]], and [[w:Sheffield|Sheffield]], and broadened the base of voting so that manufacturers could be represented in Parliament. ... [I]n 1846 they managed to get the hated Corn Laws repealed ... .}} The ''Manchester Observer'' closed roughly two years later after [[w:Manchester Observer#Closure by repeated prosecution|repeated prosecutions]] for [[w:Seditious libel|seditious libel]]. However, the British government placed enough other limits on its power that enough other commoners were allowed to become successful entrepreneurs that the rate of economic growth in the UK increased immediately, as indicated by the increase in the slope of the UK line in Figure 1 until World Wars I and II. == In sum == The above suggests that the rate of economic growth changes with either (a) a change in administration -- a new head of state -- or (b) some other major event like war, the Peterloo Massacre, or the [[w:Wall Street crash of 1929|Wall Street crash of 1929]]. Policy changes that can increase growth include (1) limits on grants of monopolies, (2) diverse ownership and control of the major media and (3) support for immigrants. However, correlation does not imply causation. [[w:Randomized controlled trial|Randomized controlled trial]]s have made important contributions to human understanding of economic growth, as documented by Banerjee, Duflo, and Kramer, cited above and recommended in the Wikiversity article on "[[Information is a public good: Designing experiments to improve government]]". Failing that, more analyses of data from different political jurisdictions with different media policies could be useful, as described by Neff and Pickard (2024).<ref>Graves (2025) includes an analysis of the data in Neff and Pickard (2024).</ref> == Caveat == The above attempts to accurately reflect the perspectives of all sources cited. Readers are invited to revise this article, subject to the standard Wikimedia rules of writing from a [[Wikiversity:Disclosures|neutral point of view]], [[Wikiversity:Cite sources|citing credible sources]] -- while raising other questions and concerns on the associated [[Wikiversity:FAQ|'''{{'}}“Discuss”{{'}}''' page]] -- and treating others with respect. == Notes == {{reflist}} == Bibliography == * <!-- Daron Acemoglu (2026-08-11) What Happened to Liberal Democracy?: Remaking a Politics of Shared Prosperity (Function)-->{{cite Q|Q141241291}} * <!--Daron Acemoglu and Simon Johnson (2023) Power and Progress: Our Thousand-Year Struggle Over Technology and Prosperity (PublicAffairs)-->{{cite Q|Q125292212}} * <!--Daron Acemoglu and James A. Robinson (2012) Why Nations Fail (Crown)-->{{cite Q|Q7997840}} * <!--Philippe Aghion, Céline Antonin, and Simon Bunel (2021) The Power of Creative Destruction: Economic Upheaval and the Wealth of Nations (Harvard University Press)-->{{cite Q|Q137641358}} * <!--Ajzenman, Nicolas (29 January 2023). "Migrants don't cause crime rates to increase — but false perceptions endure anyway". The Conversation.-->{{cite Q|Q141758664}} * <!--Costas Arkolakis, Michael Peters, and Sun Kyoung Lee, (2020) "European Immigrants and the United States’ Rise to the Technological Frontier"-->{{cite Q|Q141678287}} * <!--Dean Baker (2016) Rigged: How globalization and the rules to the modern economy were structured to make the rich richer-->{{cite Q|Q100216001}} * <!--Abhijit V. Banerjee and Esther Duflo (2019) Good Economics for Hard Times (PublicAffairs)-->{{cite Q|Q85764011}} * <!--Jutta Bolt and Jan Luiten van Zanden (2024-09-18) "Maddison style estimates of the evolution of the world economy: A new 2023 update", Journal of Economic Surveys, 1-41-->{{cite Q|Q126723821}} * <!--S. N. Broadberry, B. Campbell, A. Klein, M. Overton and B. van Leeuwen (2015) ''British Economic Growth 1270-1870'' (Cambridge University Press)__>{{cite Q|Q57945943}} * <!--Donald S. L. Cardwell (1972) Turning Points in Western Technology (Neale Watson, Science History Pubs)-->{{cite Q|Q141795083}} * <!--Renée DiResta (2026) Invisible Rulers: The People Who Turn Lies into Reality, updated edition-->{{cite Q|Q141156783}} * <!--Renée DiResta and Rachel Kleinfeld (2025-09-10) For Expertise to Matter, Nonpartisan Institutions Need New Communications Strategies-->{{cite Q|Q141158362}} * <!--Katarzyna Dośpiał-Borysiak (2017) "Model of state management of petroleum sector -- Case of Norway-->{{cite Q|Q141774179}} * <!--Rachel Foxley (2013) The Levellers: Radical Political Thought in the English Revolution (Oxford University Press)-->{{cite Q|Q136227988}} * <!--Benjamin Gonzalez O’Brien, Loren Collingwood, Stephen Omar El-Khatib (2017-05-07) "The Politics of Refuge: Sanctuary Cities, Crime, and Undocumented Immigration-->{{cite Q|Q55932800}} * <!--Graves (2025-01-08) Defend Free Speech For All (Town Hall)-->{{cite Q|Q132855526}} * <!--Graves (2023-03-21) Do copyrights and paywalls on academic journals violate the US Constitution?-->{{cite Q|Q126116187}} * <!--Christopher Hibbert, Ben Weinreb, John Keay, Julia Keay, and Matthew Weinreb (2011-09-09) The London Encyclopaedia, 3rd ed.-->{{cite Q|Q137188993}} * <!-- Historian, USPS (2010) Postage Rates for Periodicals -->{{cite Q|Q112167199}} * <!--Richard R. John (1995) Spreading the News: The American Postal System from Franklin to Morse-->{{cite Q|Q54641943}} * <!--Volker Leppin and Timothy J. Wengert (2015) "Sources for and against the Posting of the Ninety-Five Theses"-->{{cite Q|Q141737347}} * <!--Lawrence Lessig (2004) Free Culture-->{{cite Q|Q830526}} * <!--Olivier Marie and Paolo Pinotti (2024-02-01) "Immigration and Crime: An International Perspective", Journal of Economic Perspectives. 38 (1): 181–200-->{{cite Q|Q128434874}} * <!--Martínez, Daniel E.; Martínez-Schuldt, Ricardo D.; Cantor, Guillermo (2017). " Providing Sanctuary or Fostering Crime? A Review of the Research on 'Sanctuary Cities' and Crime" . Sociology Compass. 12 e12547. doi:10.1111/soc4.12547. ISSN 1751-9020.-->{{cite Q|Q55932802}} * <!-- Robert W. McChesney; John Nichols (2010). The Death and Life of American Journalism (Bold Type Books) -->{{cite Q|Q104888067}}. * <!-- Robert W. McChesney; John Nichols (2016). People get ready-->{{cite Q|Q87619174}} * <!--Robert Michels (1911) Les Partis politiques, essai sur les tendances oligarchiques des démocraties-->{{cite Q|Q7210216}} * <!--Joel Mokyr (2017) A Culture of Growth: The Origins of the Modern Economy (Princeton University Press)-->{{cite Q|Q136118590}} * <!--Joel Mokyr (1994) "Cardwell's Law and the political economy of technological progress", Research Policy 23(5)561-574-->{{cite Q|Q139772607}} * <!--Ashley N. Muchow & Taisiia Stanishevska (2026) "From Headlines to Handcuffs: Local News Coverage of Immigrant Crime and Hispanic Arrest", Justice Quarterly, vol. 43(3) (https://doi.org/10.1080/07418825.2025.2550307)-->{{cite Q|Q141760694}} * <!--Neff and Pickard (2024) "Funding Democracy: Public Media and Democratic Health in 33 Countries"-->{{cite Q|Q131468289}} * <!--Gerald G. Newman and Leslie Ellen Brown, eds. (1997) Britain in the Hanoverian Age, 1714–1837: An Encyclopedia-->{{cite Q|Q141759196|authors = Gerald G. Newman and Leslie Ellen Brown, eds.}} * <!--John Hely Owen (1938) War at Sea Under Queen Anne 1702–1708-->{{cite Q|Q141757101}} * <!--Andrew Pettegree (2015-10-27) Brand Luther: How an Unheralded Monk Turned His Small Town into a Center of Publishing, Made Himself the Most Famous Man in Europe--and Started the Protestant Reformation-->{{cite Q|Q141761223}} * <!--Thomas Piketty (2022) A Brief History of Equality-->{{cite Q|Q139886018}} * <!--Joad Raymond (2006-11-02) Pamphlets and Pamphleteering in Early Modern Britain-->{{cite Q|Q141761849}} * <!--Nick Roll (2017-07-16) "Correcting Jeff Sessions: Attorney general railed against crime in sanctuary cities, citing two professors’ work. But his claims are contrary to their conclusions.-->{{cite Q|Q141735231}} * <!--J.P. Smits, E. Horlings and J.L. van Zanden (2000) Dutch GDP and its Components 1800-1913 (Groningen Growth and Development Centre)-->{{cite Q|Q137669960}} * <!--I. Sugimoto (2011) ''Economic growth of Singapore in the twentieth century: historical GDP estimates and empirical investigations'' ( World Scientific Publishing)-->{{cite Q|Q137669987}} * <!--Nik Usher and Sanghoon Kim-Leffingwell (2022-01) How Loud Does the Watchdog Bark? A Reconsideration of Local Journalism, News Non-profits, and Political Corruption -->{{Cite Q|Q134715465}} * <!--J. L. van Zanden and B. van Leeuwen (2012), ‘Persistent but not consistent: the growth of national income in Holland 1347–1807’, Explorations in Economic History, 49, pp. 119–30-->{{cite Q|Q137670058}} * <!--Brittney Washington (2018-06-19) " A Pamphlet War in England, 1641-1643"-->{{cite Q|Q141822147}} * <!--Tom K. Wong (2017-01-26) "The Effects of Sanctuary Policies on Crime and the Economy"-->{{cite Q|Q98775875}} [[Category:Original research]] [[Category:Research]] [[Category:Economics]] [[Category:Macroeconomics]] [[Category:Gross domestic product]] [[Category:Economic growth]] [[Category:Modern Monetary Theory]] [[Category:Political science]] [[Category:Freedom and abundance]] <!-- https://en.wikiversity.org/wiki/Category_Review --> g88q26digofrpd4v9x1t8hupd0w52ws 0 332587 2842528 2842239 2026-10-25T06:25:51Z Wb525 3112992 Wb525 moved page [[User:Wb525/Cosmology: UG.NUG a coherent model]] to [[User:Wb525/Cosmology: UG.NUG an Observation-Based Model]]: meaning change 2842239 wikitext text/x-wiki {{Research project}} ''Original research project by Willy Bohane — UG.NUG.'' = Cosmology: UG.NUG an alternative coherent model = == Abstract == This article is a reading of what, in short, may have been the cosmogony of our universe and, consequently, how its structure allows [[/%CE%9BCDM|flaws and added patches]] to vanish. = UG = This term stands for French '''''Uniformité Genrée''''' (''Gendered Uniformity''). We coined it to describe a phenomenon that, strangely, has gone unnoticed until now. In short, everything produced by the universe in its earliest stages — particles, atoms, nuclei, molecules — none of which possess geometry, is subject to this regime. Its objects have no individuality, no history, no variation. This uniformity is the hidden foundation of everything, the silent order governing the birth of the universe. For now, let’s simply understand that the world of the small — below the molecule or chemistry — has no geometry, even if textbooks erroneously suggest otherwise. = NUG = This term stands for French '''''Non-Uniformité Genrée''''' ''(Non-Gendered Uniformity''). It represents everything else: stars, planets, meteors, and of course, all that derives from them — humans, artifacts, flora, fauna. All these objects are to be considered geometric because they have a form perceptible to humans without sophisticated optical equipment. From this, we can already deduce that humanity is an integral part of its own physics. Without humans, this text — and others like it — would have no reason to exist, and its understanding, or rather, its reading, would not have been necessary. This realization is of capital importance because time — the kind we count on our wristwatches — is also subject to the presence of humanity. = UG.NUG = This is the name of this model. It encompasses these two regimes that govern the entire universe. We will see that NUG (geometry) appears much later than UG, but once present, it becomes an integral part of the universe. = the Arrow of Emergence = This is the principle that decides the order of the elements' emergence or modification. It is part of the universe's properties, or even prior to it — it makes no difference. When a glass slips from our hands and, by misfortune, the floor is tiled rather than carpeted, it shatters. And if we read the universe simply, we catch its message — the message of the irreversible cosmic order: the one that prevents the glass from reassembling, the one that prevents events from denying their own existence. Thus, the universe gives us the direction from past to future, what current physics commonly calls the Arrow of Time. But no chronometric time. = Time = In encyclopedias, one cannot really say that [[User:Wb525/Cosmology: UG.NUG a coherent model/Cosmic Time|time]] is defined, and when it is, it is closer to a sequence of events than to the measurement of [[/Cosmic Time|time]] — its chronometry. A year, a day, a minute… This chronometry is only possible relative to precise references. References that are not other than the circadian pace of day and night that impact and rhythm the life of living beings. References that now exist throughout the universe, since the cesium atom is used to measure the second — even though its origin remains a human convention and the second itself expresses a duration, not human chronometric time. So one could live on Mars using the same time we use on the Earth by convention. A watch, based on cesium pace, could tell him when to stop working or watch TV. The only absolute necessity would be to keep contact with Earth in order to check, from time to time if their time systems are correctly synchronized to the circadian time. It is then evident that it is technically realizable as long as a logical limit of communication delay is not reached — delay that could affect the precision of the synchronization. This shows that this little interplanetary arrangement is not a universal solution and that still, cosmic time does not exist. We can add that time remains a human interpretation and chronometry of duration — and that, without Man, time exists nowhere. Not even on Earth. = Universe Formation = Faraday intuited the possibility of a substrate. It was randomly discovered in 1964: what the standard model calls [[/the CMB|CMB]], is not a fossil, but this invisible substrate, an omnipresent quantum form of the universe, called here the [[/the GRID|GRID]], that plays many roles: * The support of the whole universe. * The quantum matrix of what is to become the universe and its geometry. * The  counter energy that will, by contortion and pressure, create the [[/gravity|gravity]]. Therefore, and clearly, the universe — at least as far as humans are concerned — has always existed in its first primordial form, the one of the [[/the CMB|CMB]]. Then, somehow, without entering into a detailed description of an unknown so-called nucleosynthesis, the Arrow of Emergence triggers the evolution of the elements into more complex forms to finally emerge as geometry — the NUG regime, * Where [[/gravity|gravity]], as said, is expressed by the pressure of geometry on the [[/the GRID|GRID]]. * Where the so-called [[/black holes|black holes]] are none other than permanent centers of geometry emergence, which explains the spiral form of galaxies * Where gravitational waves are ripples in the [[/the GRID|GRID]], produced by extreme geometric distortions. LIGO/Virgo/KAGRA detect them without contradiction and certainly with greater probability than a mathematical construct such as spacetime. __FORCETOC__ 9bwc8v1jokxrzojnx34x04yv11mezd2 2842530 2842528 2026-10-25T06:31:26Z Wb525 3112992 Wb525 moved page [[User:Wb525/Cosmology: UG.NUG an Observation-Based Model]] to [[User:Cosmology: UG.NUG an Observation-Based Model]] 2842239 wikitext text/x-wiki {{Research project}} ''Original research project by Willy Bohane — UG.NUG.'' = Cosmology: UG.NUG an alternative coherent model = == Abstract == This article is a reading of what, in short, may have been the cosmogony of our universe and, consequently, how its structure allows [[/%CE%9BCDM|flaws and added patches]] to vanish. = UG = This term stands for French '''''Uniformité Genrée''''' (''Gendered Uniformity''). We coined it to describe a phenomenon that, strangely, has gone unnoticed until now. In short, everything produced by the universe in its earliest stages — particles, atoms, nuclei, molecules — none of which possess geometry, is subject to this regime. Its objects have no individuality, no history, no variation. This uniformity is the hidden foundation of everything, the silent order governing the birth of the universe. For now, let’s simply understand that the world of the small — below the molecule or chemistry — has no geometry, even if textbooks erroneously suggest otherwise. = NUG = This term stands for French '''''Non-Uniformité Genrée''''' ''(Non-Gendered Uniformity''). It represents everything else: stars, planets, meteors, and of course, all that derives from them — humans, artifacts, flora, fauna. All these objects are to be considered geometric because they have a form perceptible to humans without sophisticated optical equipment. From this, we can already deduce that humanity is an integral part of its own physics. Without humans, this text — and others like it — would have no reason to exist, and its understanding, or rather, its reading, would not have been necessary. This realization is of capital importance because time — the kind we count on our wristwatches — is also subject to the presence of humanity. = UG.NUG = This is the name of this model. It encompasses these two regimes that govern the entire universe. We will see that NUG (geometry) appears much later than UG, but once present, it becomes an integral part of the universe. = the Arrow of Emergence = This is the principle that decides the order of the elements' emergence or modification. It is part of the universe's properties, or even prior to it — it makes no difference. When a glass slips from our hands and, by misfortune, the floor is tiled rather than carpeted, it shatters. And if we read the universe simply, we catch its message — the message of the irreversible cosmic order: the one that prevents the glass from reassembling, the one that prevents events from denying their own existence. Thus, the universe gives us the direction from past to future, what current physics commonly calls the Arrow of Time. But no chronometric time. = Time = In encyclopedias, one cannot really say that [[User:Wb525/Cosmology: UG.NUG a coherent model/Cosmic Time|time]] is defined, and when it is, it is closer to a sequence of events than to the measurement of [[/Cosmic Time|time]] — its chronometry. A year, a day, a minute… This chronometry is only possible relative to precise references. References that are not other than the circadian pace of day and night that impact and rhythm the life of living beings. References that now exist throughout the universe, since the cesium atom is used to measure the second — even though its origin remains a human convention and the second itself expresses a duration, not human chronometric time. So one could live on Mars using the same time we use on the Earth by convention. A watch, based on cesium pace, could tell him when to stop working or watch TV. The only absolute necessity would be to keep contact with Earth in order to check, from time to time if their time systems are correctly synchronized to the circadian time. It is then evident that it is technically realizable as long as a logical limit of communication delay is not reached — delay that could affect the precision of the synchronization. This shows that this little interplanetary arrangement is not a universal solution and that still, cosmic time does not exist. We can add that time remains a human interpretation and chronometry of duration — and that, without Man, time exists nowhere. Not even on Earth. = Universe Formation = Faraday intuited the possibility of a substrate. It was randomly discovered in 1964: what the standard model calls [[/the CMB|CMB]], is not a fossil, but this invisible substrate, an omnipresent quantum form of the universe, called here the [[/the GRID|GRID]], that plays many roles: * The support of the whole universe. * The quantum matrix of what is to become the universe and its geometry. * The  counter energy that will, by contortion and pressure, create the [[/gravity|gravity]]. Therefore, and clearly, the universe — at least as far as humans are concerned — has always existed in its first primordial form, the one of the [[/the CMB|CMB]]. Then, somehow, without entering into a detailed description of an unknown so-called nucleosynthesis, the Arrow of Emergence triggers the evolution of the elements into more complex forms to finally emerge as geometry — the NUG regime, * Where [[/gravity|gravity]], as said, is expressed by the pressure of geometry on the [[/the GRID|GRID]]. * Where the so-called [[/black holes|black holes]] are none other than permanent centers of geometry emergence, which explains the spiral form of galaxies * Where gravitational waves are ripples in the [[/the GRID|GRID]], produced by extreme geometric distortions. LIGO/Virgo/KAGRA detect them without contradiction and certainly with greater probability than a mathematical construct such as spacetime. __FORCETOC__ 9bwc8v1jokxrzojnx34x04yv11mezd2 2842532 2842530 2026-10-25T06:42:44Z Wb525 3112992 Wb525 moved page [[User:Cosmology: UG.NUG an Observation-Based Model]] to [[Cosmology: UG.NUG an Observation-Based Model]]: Move to main space 2842239 wikitext text/x-wiki {{Research project}} ''Original research project by Willy Bohane — UG.NUG.'' = Cosmology: UG.NUG an alternative coherent model = == Abstract == This article is a reading of what, in short, may have been the cosmogony of our universe and, consequently, how its structure allows [[/%CE%9BCDM|flaws and added patches]] to vanish. = UG = This term stands for French '''''Uniformité Genrée''''' (''Gendered Uniformity''). We coined it to describe a phenomenon that, strangely, has gone unnoticed until now. In short, everything produced by the universe in its earliest stages — particles, atoms, nuclei, molecules — none of which possess geometry, is subject to this regime. Its objects have no individuality, no history, no variation. This uniformity is the hidden foundation of everything, the silent order governing the birth of the universe. For now, let’s simply understand that the world of the small — below the molecule or chemistry — has no geometry, even if textbooks erroneously suggest otherwise. = NUG = This term stands for French '''''Non-Uniformité Genrée''''' ''(Non-Gendered Uniformity''). It represents everything else: stars, planets, meteors, and of course, all that derives from them — humans, artifacts, flora, fauna. All these objects are to be considered geometric because they have a form perceptible to humans without sophisticated optical equipment. From this, we can already deduce that humanity is an integral part of its own physics. Without humans, this text — and others like it — would have no reason to exist, and its understanding, or rather, its reading, would not have been necessary. This realization is of capital importance because time — the kind we count on our wristwatches — is also subject to the presence of humanity. = UG.NUG = This is the name of this model. It encompasses these two regimes that govern the entire universe. We will see that NUG (geometry) appears much later than UG, but once present, it becomes an integral part of the universe. = the Arrow of Emergence = This is the principle that decides the order of the elements' emergence or modification. It is part of the universe's properties, or even prior to it — it makes no difference. When a glass slips from our hands and, by misfortune, the floor is tiled rather than carpeted, it shatters. And if we read the universe simply, we catch its message — the message of the irreversible cosmic order: the one that prevents the glass from reassembling, the one that prevents events from denying their own existence. Thus, the universe gives us the direction from past to future, what current physics commonly calls the Arrow of Time. But no chronometric time. = Time = In encyclopedias, one cannot really say that [[User:Wb525/Cosmology: UG.NUG a coherent model/Cosmic Time|time]] is defined, and when it is, it is closer to a sequence of events than to the measurement of [[/Cosmic Time|time]] — its chronometry. A year, a day, a minute… This chronometry is only possible relative to precise references. References that are not other than the circadian pace of day and night that impact and rhythm the life of living beings. References that now exist throughout the universe, since the cesium atom is used to measure the second — even though its origin remains a human convention and the second itself expresses a duration, not human chronometric time. So one could live on Mars using the same time we use on the Earth by convention. A watch, based on cesium pace, could tell him when to stop working or watch TV. The only absolute necessity would be to keep contact with Earth in order to check, from time to time if their time systems are correctly synchronized to the circadian time. It is then evident that it is technically realizable as long as a logical limit of communication delay is not reached — delay that could affect the precision of the synchronization. This shows that this little interplanetary arrangement is not a universal solution and that still, cosmic time does not exist. We can add that time remains a human interpretation and chronometry of duration — and that, without Man, time exists nowhere. Not even on Earth. = Universe Formation = Faraday intuited the possibility of a substrate. It was randomly discovered in 1964: what the standard model calls [[/the CMB|CMB]], is not a fossil, but this invisible substrate, an omnipresent quantum form of the universe, called here the [[/the GRID|GRID]], that plays many roles: * The support of the whole universe. * The quantum matrix of what is to become the universe and its geometry. * The  counter energy that will, by contortion and pressure, create the [[/gravity|gravity]]. Therefore, and clearly, the universe — at least as far as humans are concerned — has always existed in its first primordial form, the one of the [[/the CMB|CMB]]. Then, somehow, without entering into a detailed description of an unknown so-called nucleosynthesis, the Arrow of Emergence triggers the evolution of the elements into more complex forms to finally emerge as geometry — the NUG regime, * Where [[/gravity|gravity]], as said, is expressed by the pressure of geometry on the [[/the GRID|GRID]]. * Where the so-called [[/black holes|black holes]] are none other than permanent centers of geometry emergence, which explains the spiral form of galaxies * Where gravitational waves are ripples in the [[/the GRID|GRID]], produced by extreme geometric distortions. LIGO/Virgo/KAGRA detect them without contradiction and certainly with greater probability than a mathematical construct such as spacetime. __FORCETOC__ 9bwc8v1jokxrzojnx34x04yv11mezd2 0 332591 2842546 2842069 2026-10-25T06:53:10Z Wb525 3112992 Wb525 moved page [[User:Wb525/Cosmology: UG.NUG a coherent model/ΛCDM]] to [[Cosmology: UG.NUG an Observation-Based Model/ΛCDM]]: Move to sub-page 2842069 wikitext text/x-wiki {{Research project}} Original research project by Willy Bohane — UG.NUG. = Abstract = This article deals with the incongruities ΛCDM. = The Long Long list = # ΛCDM says that only the universe exists. Yet it posits a '''singularity''' and a Big Bang outside the universe. # If the singularity is impossible, ''' the Big Bang''' never happened. # If the Big Bang never happened, the age of the universe is not a question. # If '''expansion''' existed, the disks of spiral galaxies would be distorted — elongated along the direction of expansion. This is not observed. If space expands, then it must be a physical entity — something that can stretch. But in general relativity, [[User:Wb525/Cosmology: UG.NUG a coherent model/Cosmic Time|local-spacetime]] is a mathematical structure: a metric, a set of coordinates. An abstraction does not stretch. So either space is a substance, and general relativity is wrong about its nature — or space is not a substance, and the expansion is a mathematical artifact with no physical counterpart. # The [[User:Wb525/Cosmology: UG.NUG a coherent model/the Mass Imbroglio|mass of Newton]] is defined by a definition of the mass. As a result, mass has never been independently defined. # Even Einstein's definition substitutes for [[User:Wb525/Cosmology: UG.NUG a coherent model/the Mass Imbroglio|mass]] a value that defines compressed energy, not mass. # '''Cold dark matter''', the term used in the model's name, is a patch that has never been observed, and nobody knows what it is. # '''Dark energy''' is a patch that has never been observed and nobody knows what it is. # [[User:Wb525/Cosmology: UG.NUG a coherent model/black holes|Black holes]] are a pure mathematical conjecture that has never been observed and nobody knows what they are. What is observed is a compact, luminous source at galactic centres, with young stars orbiting around it. This suggests a process of active formation — not an ogre hungry for matter. # The [[User:Wb525/Cosmology: UG.NUG a coherent model/the CMB|CMB]] is described as a fossil, but a cosmological fossil is not defined. And if it were, its nature would be that of something long dead — not still emitting microwaves. # '''''Model and reality''''' (the mathematical model is taken as an ontological description). # '''Ghosts to Geometry''' (ΛCDM wants to put shoes on ghost particles in order to make them geometric, more "human" and less frightening.). # '''gravity quantization'''  (forced quantization of gravity without conceptual justification). # '''''Symmetries vs. reality''''' (mathematical symmetries treated as properties of the real world). # '''''Horizon problem''''' (regions too distant to have ever ''coordinated''). # '''''Flatness problem''''' (the universe is too flat without fine‑tuning). # '''Homogeneity and Isotropy'''  (initial homogeneity unexplained). # '''''Origin of geometry''''' (the model assumes geometry but does not explain it). # '''''Cosmological constant''''' (value inexplicable, off by 120 orders of magnitude). # '''the Measurement Problem'''  (introduced to fix the model, not predicted by it). # '''Quantum and Relativity'''  (two conceptually incompatible frameworks). # '''''Curvature of spacetime'''''. (Spacetime is an abstract mathematical construct. A physical deformation is therefore attributed to a non‑physical structure). And one can add to this list the new flaws regularly revealed by the JWST. = the Divorce = Criticizing ΛCDM is not criticizing anyone — it is criticizing a reading of the universe. Despite its venerable name, ΛCDM '''is nothing more than an interpretation'''. And since the universe itself remains silent and sparing in its revelations, it is remarkably easy to be right — or wrong. What distinguishes one reading from another is, consequently, its coherence, its agreement with observation, and its capacity to be tested. Coherence alone is not enough — a perfectly coherent system can be false, and history has shown this more than once. As in a police investigation or a computer program, if coherence is missing, the culprit is never found, the program does not run, and the equations meant to describe the universe fail to match observation. It is not dramatic; it is only theory. The current theory — the BBP as it stands — does not account for the universe in a way that satisfies its own criteria. It accumulates unresolved problems and adds new entities to mask them. This may sound extreme, but once you add: ''and yet the world still works'', the statement becomes far less shocking. This is made possible by the existence of two distinct physics. This distinction is rarely made. Yes, it is surprising. No, it is never mentioned. These two physics are: * A '''theoretical physics''', currently the Big Bang paradigm (BBP), which accumulates conceptual flaws. * An '''applied physics''', through which we build lasers, MP3s, smartphones, antennas, MRIs, and satellites — and in this case, everything works perfectly. Without this distinction, it is impossible to understand how the world can function despite a conceptual physics largely disconnected from reality — where many proposed theories have failed to deliver what they promised: * string theory (no experimental evidence, unverifiable extra dimensions), * M-theory (no testable predictions), * supergravity (never observed), * supersymmetry (no superpartners detected at the LHC), * GUTs (no proton decay), * canonical quantum [[User:Wb525/Cosmology: UG.NUG a coherent model/gravity|gravity]] (non-renormalizable), * loop quantum [[User:Wb525/Cosmology: UG.NUG a coherent model/gravity|gravity]] (no testable predictions). Some of these are genuinely ruled out by experiment: * local hidden variables (ruled out by Bell tests), * steady-state cosmology (contradicted by the [[User:Wb525/Cosmology: UG.NUG a coherent model/the CMB|CMB]]), * naïve dark-matter-free cosmologies (invalidated), * naïve emergent [[User:Wb525/Cosmology: UG.NUG a coherent model/gravity|gravity]] (incomplete and non-predictive). Applied physics therefore relies primarily on operational equations of its own making — the ones capable of building machines. But it must be said clearly: applied physics is not a separate physics. It uses quantum mechanics, electromagnetism, and relativity — the same theories that foundational research develops. A transistor is quantum mechanics. A GPS is general relativity. The point is not that the theories differ. It is that cosmological constructs such as dark matter, dark energy, inflation, and the cosmological constant play no role in the design of any instrument. Not one component, not one calculation, not one correction. Physicists concerned with practical reality rarely interact with conceptual physicists who, judging by the distance separating the two communities, no longer seem to read the universe as it truly is. Applied physics coexists with these conceptual models without actually relying on them. Those models remain largely confined to academic settings, and their influence seldom extends beyond the walls of universities. Gradually, this became impossible to ignore. The divergence can be traced back almost to Einstein's General Theory of Relativity, when he introduced into his most famous equation — as we shall examine in detail in the chapter on mass — a profoundly ambiguous factor, *m*, thereby freezing into place a fundamental confusion that has led, over the course of a century, to an ongoing conceptual drift. A tiny ambiguity, yet enough to divert an entire branch of physics from its natural course. Yes, this is precisely the kind of information that is rarely shared with the general public and that, without doubt, many physicists themselves have never even suspected. When a rocket must be sent to the Moon, engineers use Newtonian mechanics, orbital dynamics, and relativity. They do not consult cosmology. This separation is real, but it is not a separation between two physics. It is a separation between the physics that builds things and the physics that speculates about origins. Two cultures coexist, they speak the same language, but they no longer read the same universe — and no one seems particularly concerned by it. Well, perhaps taxpayers should be, although most of them are unaware of it. = Anthropocentrism = The universe provides no anchor, no reference. Without GPS, it is impossible to know a position. Thus, we do not know where Earth is located — especially given its minuscule size in the cosmic expanse, comparable to a pinhead on a trillion football fields. All the measurements we know are therefore anthropocentric or purely terrestrial. Their close connection to humanity and its existence is undeniable. If position does not exist in space, then distance does not exist either, nor acceleration, velocity, weight, mass, or time. When we send a rocket to the Moon, it navigates using GPS stations orbiting Earth, which provide a reference up to a certain distance — the range of communication. Beyond that distance, the rocket is lost. In the case of a space probe like Hubble, which travels millions of kilometers away, it is a radio signal sent back to Earth that informs us of its distance. If communication is lost, the probe is also lost. Nothing can occur in space without a terrestrial reference — or, in the future, a reference from another planet if humanity were to inhabit one. = No References = The lack of scientific rigour is nowhere more visible than in quantum mechanics — this mysterious regime present in every atom of matter, which the [[User:Wb525/Cosmology: UG.NUG a coherent model/ΛCDM|Big Bang paradigm]] (BBP) insists on reducing to mere mathematics out of fear of the unknown. It represents the unique, singular, and indelible signature of the universe. The BBP has constructed mathematical spaces, geometries, and states that no observation has ever confirmed. They work — they predict the right numbers. But they do not describe what exists. Elementary particles are invisible to the naked eye and to the scanning tunneling microscope. No one has ever observed a wave function. This means that any representation of this regime is purely conjectural. These representations are all the more useless because only certain non-geometric parameters allow the use of quantum mechanics in research: * Energy levels via spectrometry, * Transition probabilities in lasers, * Electronic structures via STM, * Conduction bands in solid-state physics, * Bound states in quantum chemistry. 2r5njksqaucl1y2qapdq1iwmcd42vj7 2842550 2842546 2026-10-25T07:05:05Z Wb525 3112992 /* the Divorce */ 2842550 wikitext text/x-wiki {{Research project}} Original research project by Willy Bohane — UG.NUG. = Abstract = This article deals with the incongruities ΛCDM. = The Long Long list = # ΛCDM says that only the universe exists. Yet it posits a '''singularity''' and a Big Bang outside the universe. # If the singularity is impossible, ''' the Big Bang''' never happened. # If the Big Bang never happened, the age of the universe is not a question. # If '''expansion''' existed, the disks of spiral galaxies would be distorted — elongated along the direction of expansion. This is not observed. If space expands, then it must be a physical entity — something that can stretch. But in general relativity, [[User:Wb525/Cosmology: UG.NUG a coherent model/Cosmic Time|local-spacetime]] is a mathematical structure: a metric, a set of coordinates. An abstraction does not stretch. So either space is a substance, and general relativity is wrong about its nature — or space is not a substance, and the expansion is a mathematical artifact with no physical counterpart. # The [[User:Wb525/Cosmology: UG.NUG a coherent model/the Mass Imbroglio|mass of Newton]] is defined by a definition of the mass. As a result, mass has never been independently defined. # Even Einstein's definition substitutes for [[User:Wb525/Cosmology: UG.NUG a coherent model/the Mass Imbroglio|mass]] a value that defines compressed energy, not mass. # '''Cold dark matter''', the term used in the model's name, is a patch that has never been observed, and nobody knows what it is. # '''Dark energy''' is a patch that has never been observed and nobody knows what it is. # [[User:Wb525/Cosmology: UG.NUG a coherent model/black holes|Black holes]] are a pure mathematical conjecture that has never been observed and nobody knows what they are. What is observed is a compact, luminous source at galactic centres, with young stars orbiting around it. This suggests a process of active formation — not an ogre hungry for matter. # The [[User:Wb525/Cosmology: UG.NUG a coherent model/the CMB|CMB]] is described as a fossil, but a cosmological fossil is not defined. And if it were, its nature would be that of something long dead — not still emitting microwaves. # '''''Model and reality''''' (the mathematical model is taken as an ontological description). # '''Ghosts to Geometry''' (ΛCDM wants to put shoes on ghost particles in order to make them geometric, more "human" and less frightening.). # '''gravity quantization'''  (forced quantization of gravity without conceptual justification). # '''''Symmetries vs. reality''''' (mathematical symmetries treated as properties of the real world). # '''''Horizon problem''''' (regions too distant to have ever ''coordinated''). # '''''Flatness problem''''' (the universe is too flat without fine‑tuning). # '''Homogeneity and Isotropy'''  (initial homogeneity unexplained). # '''''Origin of geometry''''' (the model assumes geometry but does not explain it). # '''''Cosmological constant''''' (value inexplicable, off by 120 orders of magnitude). # '''the Measurement Problem'''  (introduced to fix the model, not predicted by it). # '''Quantum and Relativity'''  (two conceptually incompatible frameworks). # '''''Curvature of spacetime'''''. (Spacetime is an abstract mathematical construct. A physical deformation is therefore attributed to a non‑physical structure). And one can add to this list the new flaws regularly revealed by the JWST. = the Divorce = Criticizing ΛCDM is not criticizing anyone but a reading of the universe. Despite its venerable name, ΛCDM '''is nothing more than an interpretation'''. And since the universe itself remains silent and sparing in its revelations, it is remarkably easy to be right — or wrong. What distinguishes one reading from another is, consequently, its coherence, its agreement with observation, and its capacity to be tested. Coherence alone is not enough — a perfectly coherent system can be false, and history has shown this more than once. As in a police investigation or a computer program, if coherence is missing, the culprit is never found, the program does not run, and the equations meant to describe the universe fail to match observation. It is not dramatic; it is only theory. The current theory — the BBP as it stands — does not account for the universe in a way that satisfies its own criteria. It accumulates unresolved problems and adds new entities to mask them. This may sound extreme, but once you add: ''and yet the world still works'', the statement becomes far less shocking. This is made possible by the existence of two distinct physics. This distinction is rarely made. Yes, it is surprising. No, it is never mentioned. These two physics are: * A '''theoretical physics''', currently the Big Bang paradigm (BBP), which accumulates conceptual flaws. * An '''applied physics''', through which we build lasers, MP3s, smartphones, antennas, MRIs, and satellites — and in this case, everything works perfectly. Without this distinction, it is impossible to understand how the world can function despite a conceptual physics largely disconnected from reality — where many proposed theories have failed to deliver what they promised: * string theory (no experimental evidence, unverifiable extra dimensions), * M-theory (no testable predictions), * supergravity (never observed), * supersymmetry (no superpartners detected at the LHC), * GUTs (no proton decay), * canonical quantum [[User:Wb525/Cosmology: UG.NUG a coherent model/gravity|gravity]] (non-renormalizable), * loop quantum [[User:Wb525/Cosmology: UG.NUG a coherent model/gravity|gravity]] (no testable predictions). Some of these are genuinely ruled out by experiment: * local hidden variables (ruled out by Bell tests), * steady-state cosmology (contradicted by the [[User:Wb525/Cosmology: UG.NUG a coherent model/the CMB|CMB]]), * naïve dark-matter-free cosmologies (invalidated), * naïve emergent [[User:Wb525/Cosmology: UG.NUG a coherent model/gravity|gravity]] (incomplete and non-predictive). Applied physics therefore relies primarily on operational equations of its own making — the ones capable of building machines. But it must be said clearly: applied physics is not a separate physics. It uses quantum mechanics, electromagnetism, and relativity — the same theories that foundational research develops. A transistor is quantum mechanics. A GPS is general relativity. The point is not that the theories differ. It is that cosmological constructs such as dark matter, dark energy, inflation, and the cosmological constant play no role in the design of any instrument. Not one component, not one calculation, not one correction. Physicists concerned with practical reality rarely interact with conceptual physicists who, judging by the distance separating the two communities, no longer seem to read the universe as it truly is. Applied physics coexists with these conceptual models without actually relying on them. Those models remain largely confined to academic settings, and their influence seldom extends beyond the walls of universities. Gradually, this became impossible to ignore. The divergence can be traced back almost to Einstein's General Theory of Relativity, when he introduced into his most famous equation — as we shall examine in detail in the chapter on mass — a profoundly ambiguous factor, *m*, thereby freezing into place a fundamental confusion that has led, over the course of a century, to an ongoing conceptual drift. A tiny ambiguity, yet enough to divert an entire branch of physics from its natural course. Yes, this is precisely the kind of information that is rarely shared with the general public and that, without doubt, many physicists themselves have never even suspected. When a rocket must be sent to the Moon, engineers use Newtonian mechanics, orbital dynamics, and relativity. They do not consult cosmology. This separation is real, but it is not a separation between two physics. It is a separation between the physics that builds things and the physics that speculates about origins. Two cultures coexist, they speak the same language, but they no longer read the same universe — and no one seems particularly concerned by it. Well, perhaps taxpayers should be, although most of them are unaware of it. = Anthropocentrism = The universe provides no anchor, no reference. Without GPS, it is impossible to know a position. Thus, we do not know where Earth is located — especially given its minuscule size in the cosmic expanse, comparable to a pinhead on a trillion football fields. All the measurements we know are therefore anthropocentric or purely terrestrial. Their close connection to humanity and its existence is undeniable. If position does not exist in space, then distance does not exist either, nor acceleration, velocity, weight, mass, or time. When we send a rocket to the Moon, it navigates using GPS stations orbiting Earth, which provide a reference up to a certain distance — the range of communication. Beyond that distance, the rocket is lost. In the case of a space probe like Hubble, which travels millions of kilometers away, it is a radio signal sent back to Earth that informs us of its distance. If communication is lost, the probe is also lost. Nothing can occur in space without a terrestrial reference — or, in the future, a reference from another planet if humanity were to inhabit one. = No References = The lack of scientific rigour is nowhere more visible than in quantum mechanics — this mysterious regime present in every atom of matter, which the [[User:Wb525/Cosmology: UG.NUG a coherent model/ΛCDM|Big Bang paradigm]] (BBP) insists on reducing to mere mathematics out of fear of the unknown. It represents the unique, singular, and indelible signature of the universe. The BBP has constructed mathematical spaces, geometries, and states that no observation has ever confirmed. They work — they predict the right numbers. But they do not describe what exists. Elementary particles are invisible to the naked eye and to the scanning tunneling microscope. No one has ever observed a wave function. This means that any representation of this regime is purely conjectural. These representations are all the more useless because only certain non-geometric parameters allow the use of quantum mechanics in research: * Energy levels via spectrometry, * Transition probabilities in lasers, * Electronic structures via STM, * Conduction bands in solid-state physics, * Bound states in quantum chemistry. rvtzpylmgb57k15wjnxa4bkexaz6jgp 0 332595 2842534 2842049 2026-10-25T06:44:50Z Wb525 3112992 Wb525 moved page [[User:Wb525/Cosmology: UG.NUG a coherent model/Cosmic Time]] to [[Cosmology: UG.NUG an Observation-Based Model/Cosmic Time]]: Move to sub-page 2842049 wikitext text/x-wiki {{Research project}} ''Original research project by Willy Bohane — UG.NUG.'' = Current Definition = In encyclopedias, one cannot really say that time is defined, and when it is, it is closer to a sequence of events than to the measurement of time — its chronometry. A year, a day, a minute… This chronometry is only possible relative to precise references. References that are not other than the circadian pace of day and night that impact and rhythm the life of living beings. References that now exist throughout the universe, since the cesium atom is used to measure the second — even though its origin remains a human convention and the second itself expresses a duration, not human chronometric time. So one could live on Mars using the same time we use on the Earth by convention. A watch, based on cesium pace, could tell him when to stop working or watch TV. The only absolute necessity would be to keep contact with Earth in order to check, from time to time if their time systems are correctly synchronized to the circadian time. It is then evident that it is technically realizable as long as a logical limit of communication delay is not reached — delay that could affect the precision of the synchronization. This shows that this little interplanetary arrangement is not a universal solution and that still, cosmic time does not exist. We can add that time remains a human interpretation and chronometry of duration — and that, without Man, time exists nowhere. Not even on Earth. = Succession = What does exist, however, is '''Succession.''' This sequence of events gives the idea of past and future. A glass that breaks cannot reassemble itself. Indeed, once it has broken, we can say: it was intact (past), and now (present/future) it is shattered. This succession gives the impression of time passing, but without chronometry it is not the useful time indicated by our watches; it is simply a '''Succession''' of events. The confusion is easy, but the difference is immense. Our prehistoric ancestors saw that days followed one another, that the dead did not wake up, that water flowed in one direction in streams and rivers. But they had no watches. They did not measure time. They had no appointments at 3:00 p.m. At best, they could agree to meet at sunrise or sunset — nothing more. And it is precisely this reference — '''sunrise or sunset''' — actually '''the circadian periodicity,''' that makes time measurement possible. = the Circadian Periodicity = What we wear on our wrist is the calculation of this periodicity, divided into units of measurement: hours, minutes, seconds. Time zones are interpretations of this periodicity, based on the dance of the Sun and Earth for each latitude. People wake with the day in America, and in France as well — at the same local time, '''but not at the same moment'''. Universal time therefore does not exist. On the contrary, it exists only for this pin‑head called Earth, hidden among trillions of football fields. Worse still, it exists only for the humans who live on it. Animals do not care. Time zones are the best evidence of the circadian periodicity as they measure the sleep and awake time of humans all around the world. = the Misundertanding = Chronometric time is a key value for many physical parameters and equations. It is the basis of acceleration and velocity, which in turn allow us to calculate force, thrust, and so on. Even Einstein misunderstood this, naming his concept ''spacetime'', whereas its true name should have been closer to ''local-space‑time'' since time exists only for humans, locally, and not in the universe.   ΛCDM itself, when it calculates the age of the universe as 13.7 billion years, speaks of years that did not exist before the appearance of the Earth and the Sun. It obviously gives an idea of duration in a human's mind but says nothing about the real age of the universe. = The Light Years = Since distances are expressed in kilometres travelled by light in a year, we assume that cosmic distances are a question and a function of time. This is why this passage is of immense importance for the understanding of cosmology — or rather, for its misunderstanding, because this mistake calls the whole of it into question. The light-year is a human mind unit. The same is true of the speed of light. It is usually presented as a universal constant — 299,792,458 metres per second. But the metre and the second are human units. The second is defined by the circadian periodicity of life on Earth. The metre is defined by the distance light travels in 1/299,792,458 of a second. Each unit refers back to the other, and both refer back to the Earth and its humans. The speed of light is therefore not a constant of the universe, but a human convention — a circular one. '''Nothing is measurable in the universe without a human or any other entity's reference..''' t548s8m2g5lnxwkoh8vw6ww3ob5lqkk 0 332602 2842542 2842053 2026-10-25T06:51:12Z Wb525 3112992 Wb525 moved page [[User:Wb525/Cosmology: UG.NUG a coherent model/the CMB]] to [[Cosmology: UG.NUG an Observation-Based Model/the CMB]]: Move to sub-page 2842053 wikitext text/x-wiki {{Research project}} Original research project by Willy Bohane — UG.NUG. The CMB, which could be renamed Cosmic Matrix Background, is what we call the [[User:Wb525/Cosmology: UG.NUG a coherent model/the GRID|GRID]] (Global Reference in Dynamics). In other words, it is a substrate intuited by Faraday, which he never named. What is observed is simple: one single, omnipresent radiation at 2.7 K, with a perfect blackbody spectrum, in all directions. One temperature. Yet it is presented as fossil light from 380,000 years after the [[User:Wb525/Cosmology: UG.NUG a coherent model/ΛCDM|Big Bang]], before galaxies existed, but already containing the disposition of the ''seeds of the universe''. Here are some observations about what is mistaken in these assertions: = of the Fossil = For an object to be a fossil, the fossil definition must exist. It is not the case for a cosmological fossil. <ref>University of California Museum of Paleontology, "What is a fossil?", Fossils Home Page, https://ucmp.berkeley.edu/paleo/fossils/ </ref> Let us be magnanimous and suppose this noun has been inherited from paleontology as it stands. In this case it is ''the preserved remains, impression, or trace of a plant or animal from a past geologic age, typically found embedded in sedimentary rock. The term can also informally refer to an outdated person or thing.'' But in no way, an object that still emits anything. It is a dead thing. A long dead thing. '''''This definition alone proves that the CMB is not a fossil'''''. = of the Blackbody = The blackbody is observed as being a perfect one. But to claim that this temperature is a cooled remnant is an assumption. And a quite weird one. A perfect blackbody can more likely be an original property of the universe. This temperature of 2.7 K is indispensable to the existence of a structured universe. If space were filled with hot photons, the universe would be a burning plasma, without atoms, without galaxies, without life. The CMB is therefore preferably a property rather than a coincidence: a condition for a ready‑to‑use universe to be able to make geometry appear. Furthermore, such homogeneity in temperature cannot, statistically, be the result of long‑term cooling over such vast distances. The alea of the journey would never have allowed such a feat. Conversely, it is a property the universe masters perfectly, as it is precisely what characterizes the gendered uniformity of quantum objects. More fundamentally, an expanding space cannot maintain thermal equilibrium. The stretching of wavelengths breaks the Planck distribution. A perfect blackbody is the signature of an isolated system at equilibrium — not of a photon that has traveled through an expanding universe. This alone proves that the CMB is not a fossil, but the permanent matrix of the universe. = of the Dated? Noise = It is really preferable not to know how the date has been fixed — especially since Universal Time does not exist. And if this logic is admitted, it is clear that nobody can imagine the CMB is a no‑date fossil. Furthermore, fixing this date at 380,000 years after the [[User:Wb525/Cosmology: UG.NUG a coherent model/ΛCDM|singularity]] is an unpredicted exploit, far more impressive than the COBE mission results themselves. For such a statement, the date would have to be engraved on each photon at the very least. The word "years" is misleading. Its historical origin is tied to the circadian and terrestrial cycle, a cycle that did not exist at that epoch. To project our current clock — defined by 9,192,631,770 vibrations of a cesium atom — onto an epoch where the cesium atom itself did not exist is an extrapolation, not a direct measurement. The tough truth is that the universe's durations are not definable. = of the S''eeds of the'' ''Universe'' = What is called "seeds" cannot exist as a visible image. Before geometry, there is only a ghostly regime, that is to say, an invisible and ungraspable regime that we know today as quantum and that remains ghostly even though we have learned to manipulate some energetic manifestations of elementary particles. These particles have always remained elusive, invisible and unmaterialized. ''Therefore, seeds, if they exist, are by definition not photographable.'' Before showing a photograph of seeds and calling the amplified spots ''seeds of galaxies'', one must first prove what a seed is in the invisible regime. = of the Map = If the CMB were a true fossil from well before the creation of galaxies, then once cleaned of our own galaxy, its map should be smooth. Forever smooth. No structure of the present universe should appear. The real cleaned map (SMICA) is indeed smooth and grey at 2.726K. But once the uniform 2.726K is subtracted and residuals are amplified 100,000 times in false colors, the image we obtain resembles the present large-scale structure. And this is completely incompatible. '''A fossil from before structure cannot have the shape of the structure that was created after.''' = of the Temperature = Today we observe only two kinds of photons in the universe: # Very hot, always found around stars. Proof of a continuous local production. # Very cold at 2.7K, diffuse everywhere. This is the CMB. If the cold came, according to [[User:Wb525/Cosmology: UG.NUG a coherent model/ΛCDM|ΛCDM]], from hot distant photons cooling during their journey, * we should find the whole continuum in between: a uniform diffuse background of lukewarm photons at 50K, 1000K wandering among the cold. We find nothing of the kind. * Furthermore, the hot photons found around stars would have also cooled during the journey. It is not the case. <ref>Penzias, A. A., & Wilson, R. W. (1965). "A Measurement of Excess Antenna Temperature at 4080 Mc/s". The Astrophysical Journal, 142, 419-421. https://ui.adsabs.harvard.edu/abs/1965ApJ...142..419P</ref> = Prediction = If a reanalysis of the Planck data, performed without any expansion model (neither [[User:Wb525/Cosmology: UG.NUG a coherent model/ΛCDM|ΛCDM]] nor a pre‑integrated cosmological background), reveals a persistent acoustic peak, then the interpretation of the CMB as a fundamental state is invalidated. = Conclusion = '''The CMB is not a fossil.''' The CMB is not the echo of a Bang that no one has ever seen. It is the thermal condition of a stable, non-expanding universe: the proof that the universe is unfolded and ready to host structure, not that it was born, just like that, who knows where, on a fine summer day in -13,800,000,000 AD." And the consequence is that we have a universe genesis in 3 points: # The CMB, called [[User:Wb525/Cosmology: UG.NUG a coherent model/the CMB/the GRID|GRID]] in this work, is not a fossil but the unfolded universe in its quantum primordial form. # When the quantum elements are ready, the emergence of their geometry is triggered. # This geometry exerts pressure on the [[User:Wb525/Cosmology: UG.NUG a coherent model/the GRID|GRID]] — [[User:Wb525/Cosmology: UG.NUG a coherent model/gravity|gravity]]. '''That is enough to make all known paradoxes vanish as well as the new JWST revealed anomalies.''' == References == {{reflist}} entna7tcyy7f5l5mzi6mjtlo9ps8qfz 0 332606 2842544 2842067 2026-10-25T06:52:01Z Wb525 3112992 Wb525 moved page [[User:Wb525/Cosmology: UG.NUG a coherent model/the GRID]] to [[Cosmology: UG.NUG an Observation-Based Model/the GRID]]: Move to sub-page 2842067 wikitext text/x-wiki Faraday rejected the idea of empty space, insisting that a minimal presence must always exist. The [[User:Wb525/Cosmology: UG.NUG a coherent model/the CMB|CMB]] <ref name=":1">Penzias, A. A., & Wilson, R. W. (1965). "A Measurement of Excess Antenna Temperature at 4080 Mc/s". The Astrophysical Journal, 142, 419-421</ref> — omnipresent, uniform, and indelible — directly confirms this intuition. Instead of recognizing it as the universe’s permanent substrate, it was interpreted as fossil radiation, a conceptual mistake with lasting consequences.<ref>Faraday's Theories of Matter and Electricity", The British Journal for the History of Science (Cambridge University Press)</ref> This presence plays exactly the role Faraday anticipated: a diffuse, coherent support in which emerging geometric objects float and maintain their stability. Without it, the universe would collapse. Faraday argued that lines of force exist even without matter — not as an ether, but as the minimal presence required for electromagnetic phenomena. His intuition anticipated the non‑trivial quantum vacuum, the need for natural support, and the impossibility of true emptiness. * Primary energy allows everything to exist and imposes simple laws: straight → inertia; curved → gravity; orbit → equilibrium; fall → gradient; collision → interaction. This unnamed presence is what Faraday could not name: the GRID — Global Reference In Dynamics. * The GRID is the persistent, universal presence revealed by the [[User:Wb525/Cosmology: UG.NUG a coherent model/the CMB|CMB]]. <ref name=":1" /> Its roles vary with context: * CMB = GRID when referring to the permanent omnipresent background, * Primary energy when referring to the coherence it maintains, GRID when acting as regulator, engine of emergence, or filter enabling motion and supporting geometry. The GRID is pre‑geometric. It precedes space, has no form, position, or measurable dimension, yet contains the information required for the emergence of objects (particles, atoms). It is a source code that, once executed, generates the geometric world ([[User:Wb525/Cosmology: UG.NUG a coherent model#NUG|NUG]]) we partially know. Everything that moves interacts with the GRID. Fast rotation leaves a dynamic imprint, producing friction — the magnetic field — just like a dynamo. [[User:Wb525/Cosmology: UG.NUG a coherent model/gravity|Gravity]] is the static imprint; inertia is the dynamic one. Modern physics forbids mass for W and Z bosons to preserve gauge symmetry, yet experiments show they are massive. The Higgs mechanism was added to bypass this contradiction. With the GRID, inertia comes from deformation of the GRID around objects, eliminating the need for mass and resolving the paradox. From the primordial stage onward, the GRID enables emergence (UG → NUG). [[User:Wb525/Cosmology: UG.NUG a coherent model#UG|UG]] contains particles and primary energy without geometry; NUG contains geometric objects. The GRID filters UG objects, organizes them into pre‑geometric structures, and provides a non‑geometric phase space allowing them to evolve toward [[User:Wb525/Cosmology: UG.NUG a coherent model#NUG|NUG]]. * The GRID binds the phase of emergence: * Phase 1 → Phase 2: stabilizing nuclei (gravity). * Phase 2 → Phase 3: enabling dynamic interactions (magnetic imprint). Without the GRID, [[User:Wb525/Cosmology: UG.NUG a coherent model#UG|UG]] objects could never assemble into [[User:Wb525/Cosmology: UG.NUG a coherent model#NUG|NUG]] structures. Once objects emerge, the GRID maintains stability through inertia and gravity. Inertia is a deformation of the GRID; gravity is its static imprint. Orbits are equilibria between these deformations. The GRID preserves the Order of emergence — the Arrow of Emergence — acting as the engine of the cosmic sequence without timekeeping. It bridges [[User:Wb525/Cosmology: UG.NUG a coherent model#UG|UG]] and [[User:Wb525/Cosmology: UG.NUG a coherent model#NUG|NUG]] by translating non‑geometric properties into geometric ones. Thus the GRID is: * the physical substrate (primary energy), * the mediator between [[User:Wb525/Cosmology: UG.NUG a coherent model#UG|UG]] and [[User:Wb525/Cosmology: UG.NUG a coherent model#NUG|NUG]] (filter, bridge), * the guarantor of cosmic Order ([[User:Wb525/Cosmology: UG.NUG a coherent model#the Arrow of Emergence|Arrow of Emergence]]). It resolves the paradoxes of the [[User:Wb525/Cosmology: UG.NUG a coherent model/ΛCDM|Big Bang paradigm]] while preserving a non‑anthropocentric reading of the universe. = Prediction — Constancy of c = If a space‑based interferometric mission (at 1 AU from the Sun) were to measure no systematic deviation in the speed of light compared with the terrestrial value, after correcting for known relativistic effects, then the idea of a local dependence of c on the GRID potential would be refuted. == References == {{reflist}} 49lvbwab1jgxuyld031q1gqknexwyst 0 332607 2842538 2842052 2026-10-25T06:47:08Z Wb525 3112992 Wb525 moved page [[User:Wb525/Cosmology: UG.NUG a coherent model/gravity]] to [[Cosmology: UG.NUG an Observation-Based Model/Gravity]]: Move to sub-page 2842052 wikitext text/x-wiki {{Research project}} Original research project by Willy Bohane — UG.NUG. In the ΛCDM, this force would supposedly arise from the curvature of a non‑existent spacetime. The Big Bang paradigm provides no physical substrate to curve. Spacetime is an abstract mathematical object, without minimal presence (unlike Faraday or UG.NUG). A physical deformation is therefore attributed to a non‑physical structure, which constitutes a major fundamental contradiction. In the UG.NUG reading, gravity is the static reaction of the GRID in the presence of geometric objects. It is not a fundamental force, has no quantum carrier, and cannot be reduced to a quantum field theory. It is therefore not quantifiable in the physical sense — not because it is immeasurable, but because it does not belong to the quantum regime. It is measurable only on Earth, through its effect on a physical reference (e.g., one liter of water). And the only real measurement of gravity is simply the pressure it exerts on an everyday object. Take one liter of water. Whatever its container — a thin bottle, a wide one, or a balloon — it will always weigh one kilogram. Well, without any grandiloquent formula, Newtonian or Einsteinian, I can tell you that gravity at Earth’s surface is 1 kilogram‑force (1 kgf). Assuming our water balloon would float in weightlessness and weighs the equivalent of one liter of water on Earth, the result is beyond doubt. In the Big Bang Paradigm, gravity at Earth’s surface is computed in several steps. For non‑specialists, understanding the equations is irrelevant. It only serves to demonstrate an unnecessary complexity to reach the same result than without equations: We assume Earth has a mass M. # This mass is not measured directly: it is inferred from other models (orbits, constants, etc.). # We assume gravity follows Newton’s law: <math>F = G \frac{m_1 m_2}{r^2}</math> where * F is gravitational force, * G the gravitational constant, * M Earth’s mass, * m the object’s mass, * and R Earth’s radius. # We define gravitational acceleration For an object of mass m, we write Take a 1‑kg object. By definition, in this reading, one liter of water has a mass of 1 kg. # The force exerted by gravity on this object is then We introduce a practical unit: the kilogram‑force (kgf). # By convention, we define : Thus, the force exerted by gravity on 1 kg (ou 1 liter of water) Earth’s : Result : For the ΛCDM, gravity on earth, applied to 1 liter of water, corresponds to 1 kgf. QED Gravity is therefore the expression of the GRID’s pressure on Earth’s surface — or on any spatial object. The denser an object is in matter, the more it deforms the GRID. It works like a slingshot: when the elastic is stretched, it thins and loses efficiency in supporting spatial objects approaching Earth. This is why such objects are attracted and, having progressively lost the support the GRID offered them, end up crashing onto its surface. The GRID thus logically resolves many anomalies that trouble the ΛCDM. The problem is that this paradigm is not satisfied with gravity at large scales; it wants gravity at the scale of the infinitely small. The first serious and structured proposal for quantum gravity came from Matvei Bronstein in 1935. He suggested that gravity should also be expressed and calculated in the quantum regime, on the grounds that the gravitational field becomes unobservable below a certain scale. During his lifetime, it would have been useful to reply to him that if gravity disappears at a certain point, it may simply be because it is no longer there. Apparently, he did not care, and his obsession was enough to drag the ΛCDM into a long, hopeless quest that, strangely, is still being pursued. For yes, gravity remains the stone in the shoe of the ΛCDM. It is hard to understand why one refuses to accept that a tension effect cannot act on invisible, non‑geometric objects — which, for non‑specialists, is like being surprised that one cannot pat a ghost on the shoulder. So it seems useless to comment further on this waste of time and resources in such a pursuit. Just for the record, gravity is not the only property that exists only at the macroscopic level: temperature, friction, and pressure belong to the same category and interact only with geometric, perceptible objects. It is also strange to imagine that a force — any force — could act at both levels. Just because you can tap someone on the shoulder does not mean you can high‑five a particle. A force that acts on structured, geometric objects cannot reasonably act on entities devoid of geometry. The only coherent solution is therefore a purely geometric gravity — all the more so since it has resisted for a century every attempt to force it into a quantum reality. = Prediction = If a free‑fall experiment (such as MICROSCOPE) were to measure a deviation in acceleration depending on the composition of the objects, beyond instrumental uncertainties, then the hypothesis of gravity as a tension of the GRID — blind to matter — would be contradicted. aj5d7k89w6q1bbmgbseu4wc1fnb52zl 0 332608 2842536 2842051 2026-10-25T06:45:24Z Wb525 3112992 Wb525 moved page [[User:Wb525/Cosmology: UG.NUG a coherent model/black holes]] to [[Watch source page and target page/black holes]]: Move to sub-page 2842051 wikitext text/x-wiki {{Research project}} Original research project by Willy Bohane — UG.NUG. It is striking that, from the sole observation of intense light and rapid motion at galactic centres, one has built the improbable image of a black, invisible object that swallows matter. This interpretive feat is a remarkable example of how the standard model turns observations into ideology. In the standard model, black holes are described as objects of extreme density, surrounded by an event horizon from which nothing can escape. They are said to be formed by the collapse of massive stars, and they are invoked to explain the intense luminosity and rapid motions observed at the centres of galaxies. But this description is a curious interpretation, not an observation. == What is actually observed: == At the centre of most large galaxies, there is a compact, luminous source <ref>{{Cite journal|last=The Event Horizon Telescope Collaboration|last2=Akiyama|first2=Kazunori|last3=Alberdi|first3=Antxon|last4=Alef|first4=Walter|last5=Asada|first5=Keiichi|last6=Azulay|first6=Rebecca|last7=Baczko|first7=Anne-Kathrin|last8=Ball|first8=David|last9=Baloković|first9=Mislav|date=2019-04-10|title=First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole|url=https://iopscience.iop.org/article/10.3847/2041-8213/ab0ec7|journal=The Astrophysical Journal Letters|volume=875|issue=1|pages=L1|doi=10.3847/2041-8213/ab0ec7|issn=2041-8205}}</ref>. This source emits intense radiation across a wide range of wavelengths: X‑rays, infrared, radio. Around this source, stars move very fast, in tight orbits. Jets of matter and energy are sometimes ejected along a preferred axis. The source itself is extremely small compared to the galaxy that surrounds it. ''That is what we see. No hole. No singularity. No invisible object''. == Observing the CMB/GRID == The [[User:Wb525/Cosmology: UG.NUG a coherent model/the CMB|CMB]] does not look like a flat, peaceful lake, but rather like a collection of intense zones. These regions are natural centres of emergence. And it is certainly why the universe's organisation is merely based on groups of objects like galaxies. Where the [[User:Wb525/Cosmology: UG.NUG a coherent model/the GRID|GRID]] is dense enough, geometry emerges. This emergence is not a passive event: it is an active, energetic process. It produces heat, radiation, and motion. It concentrates energy in a small volume, creating a bright, compact region. This is what we observe at the centres of galaxies. * The light and radiation come from the emergence itself. * The rapid motion of stars is due to the pressure of the [[User:Wb525/Cosmology: UG.NUG a coherent model/the GRID|GRID]] on the emerging geometry. * The jets are the expulsion of energy along the axis of emergence. * There is no singularity, no infinite density, no absorption of matter. * There is no horizon — only a region where geometry is still forming, and which remains compact because the emergence is not yet fully deployed. This reading explains all the observations without invoking invisible objects, without postulating singularities, and without requiring the absorption of matter. It also eliminates the need for quantum gravity, which in any case, was, [[User:Wb525/Cosmology: UG.NUG a coherent model/gravity|is and will always be unconditionally impossible]]. Black holes, therefore, do not exist. These very bright galaxy centres are birthplaces of geometry — centres where the universe emerges by groups of objects, actively, from its own substrate. == Galaxies' persistent activity == The spiral structure of galaxies is the visible trace of a continuous quantum activity due to a probable permanent process of emergence. The direction of rotation is not really understandable and could be just aleatory and depend on the spreading fluctuations of quantum effervescence. What actually happens is that galaxies are, poetically, living organisms with a beating heart that renews — through its quantum source, the [[User:Wb525/Cosmology: UG.NUG a coherent model/the GRID|GRID]] — its energetic capacity by continuously creating new stars, these energy-producing facilities. This cycle keeps the global equilibrium intact despite the end-of-life of existing stars. The universe is not dying. It is closer to what mankind has sought since the dawn of time: perpetual motion. In this case, at the centre, a quantum effervescence gives rise to multiple geometries at once — objects that do not appear in their final form immediately and obviously increase the mass of the centre so a powerful  gravity is exerted. The new objects first take on intermediate, lighter forms, orbiting close to the centre. As they mature, they grow in mass and gradually move outward, shifting to more distant orbits. This progressive migration, driven by the increasing maturity of the emerging objects, naturally produces a spiral pattern — without the need for dark matter or complex gravitational adjustments. == Black Hole Observations == {| class="wikitable" |'''Observation''' |'''Standard Interpretation''' |'''UG.NUG Interpretation''' |- |Gravitational waves (LIGO/Virgo/KAGRA <ref>{{Cite journal|last=Abbott|first=B. P.|last2=Abbott|first2=R.|last3=Abbott|first3=T. D.|last4=Abernathy|first4=M. R.|last5=Acernese|first5=F.|last6=Ackley|first6=K.|last7=Adams|first7=C.|last8=Adams|first8=T.|last9=Addesso|first9=P.|date=2016-02-11|title=Observation of Gravitational Waves from a Binary Black Hole Merger|url=https://link.aps.org/doi/10.1103/PhysRevLett.116.061102|journal=Physical Review Letters|language=en|volume=116|issue=6|doi=10.1103/PhysRevLett.116.061102|issn=0031-9007}}</ref>) |Ripples in spacetime from black hole mergers. |Ripples in the [[User:Wb525/Cosmology: UG.NUG a coherent model/the GRID|GRID]], produced by the dynamics of emergent geometry. |- |Spin (ΩH) |Rotation of the black hole's horizon. |Not a rotation of a horizon. It is a dynamic feature of the emergence zone. |- |Surface gravity (κ) |Gravity at the event horizon. |Not gravity at a horizon. It is the local pressure of the [[User:Wb525/Cosmology: UG.NUG a coherent model/the GRID|GRID]] on emerging geometry. |- |Ringdown |The black hole settling into its final state. |Not a black hole settling. It is the [[User:Wb525/Cosmology: UG.NUG a coherent model/the GRID|GRID]] returning to equilibrium after an emergence event. |- |Event horizon |A "surface of no return." |Not a surface of no return. It is the boundary of a region where geometry is still forming. |- |Pair-instability mass gap |Gap in black hole masses due to stellar physics; filled by hierarchical mergers. |There is no mass gap. Mass is not a property of these regions. What is observed is the distribution of emergence intensity. |- |Spin populations |Low-spin (direct collapse) and high-spin (hierarchical mergers). |No low‑spin or high‑spin populations. Spin is a secondary effect of geometric emergence, not a product of mergers. |- |12C(α,γ)16O reaction rate |Critical for nuclear astrophysics and stellar evolution. |Not relevant. These regions are not formed by stellar evolution. |- |Pair-instability gap (m₂ only) |Evidence for hierarchical mergers; primary components in the gap are merger products. |No gap. The observed distribution reflects different levels of emergence maturity. |- |High spins in gap binaries |Signatures of hierarchical mergers. |Not a signature of mergers. It is a signature of intense emergence. |- |Low black hole masses (10⁵–10⁷ M⊙) |The lowest mass black holes known at high redshift. |Not a measure of [[User:Wb525/Cosmology: UG.NUG a coherent model/the Mass Imbroglio|mass]]. It is a measure of the intensity of the emergence zone. |- |12C(α,γ)16O S-factor constraint |Links gravitational-wave astronomy to nuclear astrophysics. |Not relevant. This region does not involve nuclear astrophysics. |- |Growth of light seed black holes at high redshift |Tiny black holes from Population III stars grow rapidly (~10⁴ M⊙) to become the seeds of supermassive black holes observed by JWST <ref>{{Cite journal|last=Loeb|first=Abraham|date=2024-07-12|title=Little Red Dots from Low-spin Galaxies at High Redshifts|url=https://iopscience.iop.org/article/10.3847/2515-5172/ad614c|journal=Research Notes of the AAS|volume=8|issue=7|pages=182|doi=10.3847/2515-5172/ad614c|issn=2515-5172}}</ref> at z > 8. |No seeds, no growth. The observed objects are simply emergence zones at different levels of geometric maturity. Redshift is not cosmic time; it is a measure of geometric maturity. |} == References == <references /> 9lwfya2dkm96a5ykuldwwh7jzwao39l 2842540 2842536 2026-10-25T06:48:21Z Wb525 3112992 Wb525 moved page [[Watch source page and target page/black holes]] to [[Cosmology: UG.NUG an Observation-Based Model/Black Holes]]: Move to sub-page 2842051 wikitext text/x-wiki {{Research project}} Original research project by Willy Bohane — UG.NUG. It is striking that, from the sole observation of intense light and rapid motion at galactic centres, one has built the improbable image of a black, invisible object that swallows matter. This interpretive feat is a remarkable example of how the standard model turns observations into ideology. In the standard model, black holes are described as objects of extreme density, surrounded by an event horizon from which nothing can escape. They are said to be formed by the collapse of massive stars, and they are invoked to explain the intense luminosity and rapid motions observed at the centres of galaxies. But this description is a curious interpretation, not an observation. == What is actually observed: == At the centre of most large galaxies, there is a compact, luminous source <ref>{{Cite journal|last=The Event Horizon Telescope Collaboration|last2=Akiyama|first2=Kazunori|last3=Alberdi|first3=Antxon|last4=Alef|first4=Walter|last5=Asada|first5=Keiichi|last6=Azulay|first6=Rebecca|last7=Baczko|first7=Anne-Kathrin|last8=Ball|first8=David|last9=Baloković|first9=Mislav|date=2019-04-10|title=First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole|url=https://iopscience.iop.org/article/10.3847/2041-8213/ab0ec7|journal=The Astrophysical Journal Letters|volume=875|issue=1|pages=L1|doi=10.3847/2041-8213/ab0ec7|issn=2041-8205}}</ref>. This source emits intense radiation across a wide range of wavelengths: X‑rays, infrared, radio. Around this source, stars move very fast, in tight orbits. Jets of matter and energy are sometimes ejected along a preferred axis. The source itself is extremely small compared to the galaxy that surrounds it. ''That is what we see. No hole. No singularity. No invisible object''. == Observing the CMB/GRID == The [[User:Wb525/Cosmology: UG.NUG a coherent model/the CMB|CMB]] does not look like a flat, peaceful lake, but rather like a collection of intense zones. These regions are natural centres of emergence. And it is certainly why the universe's organisation is merely based on groups of objects like galaxies. Where the [[User:Wb525/Cosmology: UG.NUG a coherent model/the GRID|GRID]] is dense enough, geometry emerges. This emergence is not a passive event: it is an active, energetic process. It produces heat, radiation, and motion. It concentrates energy in a small volume, creating a bright, compact region. This is what we observe at the centres of galaxies. * The light and radiation come from the emergence itself. * The rapid motion of stars is due to the pressure of the [[User:Wb525/Cosmology: UG.NUG a coherent model/the GRID|GRID]] on the emerging geometry. * The jets are the expulsion of energy along the axis of emergence. * There is no singularity, no infinite density, no absorption of matter. * There is no horizon — only a region where geometry is still forming, and which remains compact because the emergence is not yet fully deployed. This reading explains all the observations without invoking invisible objects, without postulating singularities, and without requiring the absorption of matter. It also eliminates the need for quantum gravity, which in any case, was, [[User:Wb525/Cosmology: UG.NUG a coherent model/gravity|is and will always be unconditionally impossible]]. Black holes, therefore, do not exist. These very bright galaxy centres are birthplaces of geometry — centres where the universe emerges by groups of objects, actively, from its own substrate. == Galaxies' persistent activity == The spiral structure of galaxies is the visible trace of a continuous quantum activity due to a probable permanent process of emergence. The direction of rotation is not really understandable and could be just aleatory and depend on the spreading fluctuations of quantum effervescence. What actually happens is that galaxies are, poetically, living organisms with a beating heart that renews — through its quantum source, the [[User:Wb525/Cosmology: UG.NUG a coherent model/the GRID|GRID]] — its energetic capacity by continuously creating new stars, these energy-producing facilities. This cycle keeps the global equilibrium intact despite the end-of-life of existing stars. The universe is not dying. It is closer to what mankind has sought since the dawn of time: perpetual motion. In this case, at the centre, a quantum effervescence gives rise to multiple geometries at once — objects that do not appear in their final form immediately and obviously increase the mass of the centre so a powerful  gravity is exerted. The new objects first take on intermediate, lighter forms, orbiting close to the centre. As they mature, they grow in mass and gradually move outward, shifting to more distant orbits. This progressive migration, driven by the increasing maturity of the emerging objects, naturally produces a spiral pattern — without the need for dark matter or complex gravitational adjustments. == Black Hole Observations == {| class="wikitable" |'''Observation''' |'''Standard Interpretation''' |'''UG.NUG Interpretation''' |- |Gravitational waves (LIGO/Virgo/KAGRA <ref>{{Cite journal|last=Abbott|first=B. P.|last2=Abbott|first2=R.|last3=Abbott|first3=T. D.|last4=Abernathy|first4=M. R.|last5=Acernese|first5=F.|last6=Ackley|first6=K.|last7=Adams|first7=C.|last8=Adams|first8=T.|last9=Addesso|first9=P.|date=2016-02-11|title=Observation of Gravitational Waves from a Binary Black Hole Merger|url=https://link.aps.org/doi/10.1103/PhysRevLett.116.061102|journal=Physical Review Letters|language=en|volume=116|issue=6|doi=10.1103/PhysRevLett.116.061102|issn=0031-9007}}</ref>) |Ripples in spacetime from black hole mergers. |Ripples in the [[User:Wb525/Cosmology: UG.NUG a coherent model/the GRID|GRID]], produced by the dynamics of emergent geometry. |- |Spin (ΩH) |Rotation of the black hole's horizon. |Not a rotation of a horizon. It is a dynamic feature of the emergence zone. |- |Surface gravity (κ) |Gravity at the event horizon. |Not gravity at a horizon. It is the local pressure of the [[User:Wb525/Cosmology: UG.NUG a coherent model/the GRID|GRID]] on emerging geometry. |- |Ringdown |The black hole settling into its final state. |Not a black hole settling. It is the [[User:Wb525/Cosmology: UG.NUG a coherent model/the GRID|GRID]] returning to equilibrium after an emergence event. |- |Event horizon |A "surface of no return." |Not a surface of no return. It is the boundary of a region where geometry is still forming. |- |Pair-instability mass gap |Gap in black hole masses due to stellar physics; filled by hierarchical mergers. |There is no mass gap. Mass is not a property of these regions. What is observed is the distribution of emergence intensity. |- |Spin populations |Low-spin (direct collapse) and high-spin (hierarchical mergers). |No low‑spin or high‑spin populations. Spin is a secondary effect of geometric emergence, not a product of mergers. |- |12C(α,γ)16O reaction rate |Critical for nuclear astrophysics and stellar evolution. |Not relevant. These regions are not formed by stellar evolution. |- |Pair-instability gap (m₂ only) |Evidence for hierarchical mergers; primary components in the gap are merger products. |No gap. The observed distribution reflects different levels of emergence maturity. |- |High spins in gap binaries |Signatures of hierarchical mergers. |Not a signature of mergers. It is a signature of intense emergence. |- |Low black hole masses (10⁵–10⁷ M⊙) |The lowest mass black holes known at high redshift. |Not a measure of [[User:Wb525/Cosmology: UG.NUG a coherent model/the Mass Imbroglio|mass]]. It is a measure of the intensity of the emergence zone. |- |12C(α,γ)16O S-factor constraint |Links gravitational-wave astronomy to nuclear astrophysics. |Not relevant. This region does not involve nuclear astrophysics. |- |Growth of light seed black holes at high redshift |Tiny black holes from Population III stars grow rapidly (~10⁴ M⊙) to become the seeds of supermassive black holes observed by JWST <ref>{{Cite journal|last=Loeb|first=Abraham|date=2024-07-12|title=Little Red Dots from Low-spin Galaxies at High Redshifts|url=https://iopscience.iop.org/article/10.3847/2515-5172/ad614c|journal=Research Notes of the AAS|volume=8|issue=7|pages=182|doi=10.3847/2515-5172/ad614c|issn=2515-5172}}</ref> at z > 8. |No seeds, no growth. The observed objects are simply emergence zones at different levels of geometric maturity. Redshift is not cosmic time; it is a measure of geometric maturity. |} == References == <references /> 9lwfya2dkm96a5ykuldwwh7jzwao39l 0 332609 2842548 2842073 2026-10-25T06:53:51Z Wb525 3112992 Wb525 moved page [[User:Wb525/the Mass Imbroglio]] to [[Cosmology: UG.NUG an Observation-Based Model/the Mass Imbroglio]]: Move to sub-page 2842073 wikitext text/x-wiki = Abstract = This paper traces the historical failure to define mass as a fundamental physical concept. We show that Newton defines mass circularly <ref>{{Cite book|url=https://doi.org/10.5479/sil.52126.39088015628399|title=Philosophiae naturalis principia mathematica|last=Newton|first=Isaac|date=1687|publisher=Jussu Societatis Regiae ac Typis Josephi Streater. Prostat apud plures bibliopolas}}</ref> through force (m=F/a and F=Gm1m2/r²), a critique initiated by Mach (1883) <ref>{{Cite journal|date=1885-03-06|title=A Recent Discussion of the Axioms of Mechanics: <i>Die mechanik in ihrer entwickelung historisch-kritisch dargestellt</i> . Von E. Mach, professor an der Universität zu Prag. Leipzig, Brockhaus, 1883. 10+483 p., illustr. 8°.|url=https://doi.org/10.1126/science.ns-5.109.201|journal=Science|volume=ns-5|issue=109|pages=201–202|doi=10.1126/science.ns-5.109.201|issn=0036-8075}}</ref>. We then show that Einstein's mass-energy equivalence (E=mc²) operates an ontological substitution where mass becomes an emergent accounting of confined energy, as exemplified by the proton mass paradox (9 MeV vs 938 MeV). Unable to provide a non-referential definition for composite and quantum objects, we propose an operationalist alternative based on a single standard — one liter of pure water — deploying * a static terrestrial weight indicator for any body on Earth * a dynamic microgravity resistance indicator (<math>F_s </math>) — the flick method — for any body in space, * an isobaric volumetric comparison for gases where the flick is inoperative. We show that ''m'' is undefinable anywhere: weight is definable on Earth, <math>F_s </math> in microgravity, and even <math>F_s </math> fails in the quantum regime, for it is not geometric. Keywords: mass, circular definition, Einstein, Mach, matter-density, universal measure. = Introduction = The definition of mass presents a persistent logical difficulty. It is largely understandable because mass is not definable. Mass strangely occupies an unjustly crucial place in physics even though it is neither truly defined, nor truly existent, nor truly useful. Its history is fascinating, because we guess it must be rare, in the so‑called hard sciences, to encounter such an imbroglio. That said, nothing prevents us from trying to untangle it and understand the successive mistakes that led to this highly questionable notion of mass. = The Human Definition of Mass = First of all, we need to get rid of the dictionary definition, which describes the kilogram as a unit of mass rather than weight. A kilo of tomatoes is a weight for every human being. No one ever speaks of a ''mass of tomatoes''. So let us forget that definition and take the kilo as a unit of weight, as common usage does. It will make things simpler for everyone. Weight can be measured on Earth because, in 1795, it was decided that it would correspond to one liter of water at 4 degrees Celsius. Since then, the practical standards have changed for convenience, but the underlying reference has remained the same: 1 kg = the weight of one cubic decimeter of water at 4°C (maximum density), pure water, normal pressure. From there, we can imagine that a kilo of tomatoes drifting through space does not weigh the same as on Earth. So we can deduce that the kilogram expresses a weight produced by [[User:Wb525/Cosmology: UG.NUG a coherent model/gravity|gravity]]. * '''Space''': weight = 0 kg * '''Earth''': weight = 1 kg What, then, should mass be? In pure logic, it ought to represent the density of matter contained in that kilo of tomatoes. This density is an invariant geometric reality, which may — or may not — be influenced by [[User:Wb525/Cosmology: UG.NUG a coherent model/gravity|gravity]]. = Matter-Density = To illustrate this idea of matter-density, let us replace the tomatoes with a pack of water bottles. Whether they contain water or not, their geometry (their volume) remains the same. But if the bottles are empty, [[User:Wb525/Cosmology: UG.NUG a coherent model/gravity|gravity]] will make them weigh, let us say, 400 g. If they are full, they will weigh 9 kg. So [[User:Wb525/Cosmology: UG.NUG a coherent model/gravity|gravity]] acts, on the one hand, on geometry, and on the other hand, on matter-density (the amount of matter inside that volume). Both are crucial in evaluating the pressure that weight produces. It is obvious that if the pack is replaced by a cherry-sized ball made of the densest metal on Earth, its tiny geometry will never allow it to weigh as much as a full pack of water bottles. In the collective imagination, this definition of mass (as matter-density) is far healthier than anything offered by Newton's circular equations. = Mass in Space: Inertia Without Weight = In space, the pack of bottles will float the same way whether it is empty or full. Their weight is zero, but their inertia — that resistance to motion — remains different. Yet this inertia can only be defined relative to an acceleration (a = Δv/Δt), which itself depends on time and velocity. And these two quantities only make sense within a human reference frame. So even if mass (as matter-density) exists, it cannot be defined without a terrestrial reference. Mass, like acceleration, time, or velocity, is a human construction — a necessary illusion to describe our experience, but without any absolute foundation in the universe. The lack of terrestrial references is not the sole reason. The other reason is the mixture of matters. While a shopping bag with potatoes, tomatoes, and fish is still weighable despite the variety of products, mass can absolutely not be described nor calculated. We now have a logical and coherent proof that mass is undefinable without a pure single composition — which is not possible — and, in any case, in an environment without reference. It follows, therefore, that mass is undefinable anywhere. Weight remains definable on Earth because it is a pressure on a support. In space, only the resistance to a flick (<math>F_s </math>) is definable. In the quantum regime, even <math>F_s </math> is inoperative, for one cannot push a ghost. There is no geometry to push against, and no handle to flick. The quantum regime requires other, non-mechanical probes — deflection in a field, cyclotron frequency, atomic recoil — none of which constitute a definition of mass in itself, only measurements of its effects. = The Circular Definition of Newton = Before Newton, people weighed things with stones. Merchants simply compared reactions between two objects. A potato was worth three stones, a piece of metal balanced a sack of grain, and all of this was nothing more than a game of equivalences between pressures — a silent dialogue between objects and their responses. No one spoke of "amount of matter"; no one imagined that an object possessed, within itself, a unit that indicated its weight. Newton was the first to try to define mass — but he did so using… mass itself, creating a definitional loop that no one noticed. Here is his reasoning: To know mass, you must know force. So he writes: m = F / a. Then, to know force (F), he writes: F = G·m₁·m₂ / r². It is clear that Newton uses m to define m. He also uses: * G, a universal gravitational constant with the value 6.674×10⁻¹¹  whose numerical value depends on the unit system chosen, and which is not derived from any deeper principle. It is a measured constant, not a deduced one. * r (distance between bodies), impossible to calculate at the time, yet made dependent on [[User:Wb525/Cosmology: UG.NUG a coherent model/gravity|gravity]], adding another loop. To this day, this interdependence has remained largely unexamined — even though, despite its circular structure, it remains useful for calibrating every scale in the world. = The Quantum Mistake = Three centuries later, mass bursts into quantum physics with a plethora of inventors and a succession of patch-ups: * Einstein made it ambiguous (E = mc²) <ref>{{Cite journal|last=Einstein|first=A.|date=1905-01|title=Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?|url=https://doi.org/10.1002/andp.19053231314|journal=Annalen der Physik|volume=323|issue=13|pages=639–641|doi=10.1002/andp.19053231314|issn=0003-3804}}</ref>. * Dirac introduced it as a free parameter in his equation <ref>{{Cite journal|last=Dirac|first=Paul Adrien Maurice|date=1928-02-01|title=The quantum theory of the electron|url=https://doi.org/10.1098/rspa.1928.0023|journal=Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character|volume=117|issue=778|pages=610–624|doi=10.1098/rspa.1928.0023|issn=0950-1207}}</ref>. * Higgs displaced its origin onto another field. Many physicists — Dirac, Englert <ref>{{Cite journal|last=Englert|first=F.|last2=Brout|first2=R.|date=1964-08-31|title=Broken Symmetry and the Mass of Gauge Vector Mesons|url=https://doi.org/10.1103/physrevlett.13.321|journal=Physical Review Letters|volume=13|issue=9|pages=321–323|doi=10.1103/physrevlett.13.321|issn=0031-9007}}</ref>, Brout, Guralnik <ref>{{Cite journal|last=Guralnik|first=G. S.|last2=Hagen|first2=C. R.|last3=Kibble|first3=T. W. B.|date=1964-11-16|title=Global Conservation Laws and Massless Particles|url=https://doi.org/10.1103/physrevlett.13.585|journal=Physical Review Letters|volume=13|issue=20|pages=585–587|doi=10.1103/physrevlett.13.585|issn=0031-9007}}</ref>, Hagen, Kibble — contributed to its rise. But none ever defined it independently. In the 20th century, Peter Higgs <ref>{{Cite journal|last=Higgs|first=Peter W.|date=1964-10-19|title=Broken Symmetries and the Masses of Gauge Bosons|url=https://doi.org/10.1103/physrevlett.13.508|journal=Physical Review Letters|volume=13|issue=16|pages=508–509|doi=10.1103/physrevlett.13.508|issn=0031-9007}}</ref> and a few others took over, but here, the problems grew with more patches to the model. = Einstein's Effacement = It turns out that the total mass of a stone is far greater than the sum of the masses of the elementary particles that compose it. Nuclear physics and special relativity explain this through Einstein's mass-energy equivalence: E = mc² (or m = E/c²). In this framework, the mass of an object is the sum of its total internal energies divided by c². That is why ordinary matter has a mass far greater than what the tiny masses of quarks would suggest. A striking example: if one adds up the rest masses of the quarks in a proton, one gets about 9 MeV/c². Yet the measured mass of the proton is about 938 MeV/c² — the difference exceeds 99%. Modern physics resolved this paradox by redefining mass not as a sum of tiny bricks of matter, but as an emergent property of internal energy. In doing so, Einstein built a world where the m in his equation has very little to do with the mass humans intuitively expect. Above all, this m is undefinable everywhere. Einstein himself would acknowledge the problem in 1948. In a letter to Lincoln Barnett, he writes <ref>{{Cite journal|last=Einstein|first=Albert|date=1948-12|title=A Letter from Albert Einstein|url=https://doi.org/10.1080/00963402.1948.11460269|journal=Bulletin of the Atomic Scientists|volume=4|issue=12|pages=354–354|doi=10.1080/00963402.1948.11460269|issn=0096-3402}}</ref>: ''"It is not good to introduce the concept of the mass'' <math>M = \frac{m}{\sqrt{1 - v^2/c^2}}</math> ''of a moving body, for which no clear definition can be given. It is better to introduce no other mass concept than the rest mass mm."'' The symbol had already circulated worldwide. It was never recalled. He could have written E = b·c², where b simply represents the amount of "boom" (compressed energy) contained in a system. That would have been clearer and less misleading than using m, a symbol burdened with a false identity inherited from classical mechanics. = Consequences = Thus, mass is widely used by the model, but since it is undefinable in reality, outside that model it exists only under false identities  — often as a mathematical parameter with no direct connection to the tangible world. Physics should have renamed it to avoid this mess. This mass enigma is also one of the reasons for the justified divorce between applied physics and conceptual physics. Without ever being stated explicitly, the inherited conceptual frameworks are insufficient to interpret the universe and send rockets into space. = Toward a Universal Measure = The analysis of the historical trajectory of mass reveals a dead end: theoretical physics has trapped itself in a Newtonian circularity and an Einsteinian ontological substitution to mask its inability to calculate the microscopic essence of a composite object. To escape this mathematical fiction, there is no need to invent new parameters. The universe offers a direct, universal, and pragmatic reading of physical reality through a mirror of measurements based on a single standard material: one liter of pure water. This system deploys two observable and non-anthropocentric realities: '''On Earth''' (The Static Indicator) The only way to apprehend a complex mix of matters is the direct measurement of its pressure on a support. This weight is simply qualified in parts or multiples of the terrestrial standard liter of water. '''In Space''' (The Dynamic Indicator) In weightlessness, this same liter of water does not change its nature, but its weight vanishes. It then reveals a standard resistance force ( F_s ): the exact minimum energy of the flick needed to initiate its motion. From then on, the measurement of any body in the universe frees itself from the abstract concept of mass. The resistance of an object in weightlessness is qualified in strict multiples of this standard flick force ( F_s ), creating a perfect mirror with the terrestrial kilogram-force. '''For Gases''' For gases, measurement is made by pressure compared within a standard volume — for example, by injecting a volume of gas into a liter of water and observing the resulting bubble. In space, a flexible container holding the gas is compared to the same container holding a liter of standard water. The isobaric volumetric comparison serves as the indicator, without any calculation of force. = Final Statement = The universe does not calculate mass — it responds to an interaction. On Earth it responds with weight. In space it responds with resistance to a flick (F_s). In the quantum, it does not even respond to a push. This method of universal measurement restores physics to its primary function: to write what it observes, without projecting its own geometric illusions onto it. = References = <references /> owra1kzv3iungk1gmio3tmuuh2zkaxl 0 332625 2842443 2842340 2026-10-24T16:33:55Z KyleWii 3113193 2842443 wikitext text/x-wiki Wikiversity's Advanced Placement (AP) courses need significant improvement in organization and depth of information. Almost all the learning resources can be difficult to navigate, not have explanations and leave students confused. This is very crucial because take this for example. an AP Biology resource won't be helpful if they introduce concepts like cellular transport without explaining the difference between passive and active transport. These problems can make independent studying extremely difficult, especially for students who need free resources to prepare for AP exams and don't have the money to pay for something like a private tutor. A better course would make Wikivesity an effective learning source. To solve this problem, Wikiversity should reorganize its AP courses around the College Board course frameworks and make subunits for it. Each course should have clearly defined units, learning objectives, sufficient explanations, diagrams, worked examples and practice questions with answer explanations similarly of what will be on the AP exam. Editors should review existing pages to identify what info is missing and outdated content. By working with teachers, edits and students, Wikiversity could make a more useful and accurate resource for studying AP. == Our progress == === AP Courses === Courses marked as ▲ are considered high priority. ==== Arts ==== * AP 2-D Art and Design: 0% (0 units completed, 0 units in progress and 0 units not started) * AP 3-D Art and Design: 0% (0 units completed, 0 units in progress and 0 units not started) * AP Drawing: 0% (0 units completed, 0 units in progress and 0 units not started) * AP Art History: 0% (0 units completed, 0 units in progress and 0 units not started) * AP Music Theory: 0% (0 units completed, 0 units in progress and 0 units not started) ==== English ==== * AP English Language and Composition: 0% (0 units completed, 0 units in progress and 0 units not started) * AP English Literature and Composition: 0% (0 units completed, 0 units in progress and 0 units not started) ==== History and Social Sciences ==== * AP African American Studies: 0% (0 units completed, 0 units in progress and 0 units not started) * AP Comparative Government and Politics: 0% (0 units completed, 0 units in progress and 0 units not started) * AP European History: 0% (0 units completed, 0 units in progress and 0 units not started) * AP Human Geography: 0% (0 units completed, 0 units in progress and 0 units not started) * AP Macroeconomics: 0% (0 units completed, 0 units in progress and 0 units not started) * AP Microeconomics: 0% (0 units completed, 0 units in progress and 0 units not started) * [[AP Psychology]]: 0% (0 units completed, 0 units in progress and 0 units not started) ▲ * AP United States Government and Politics: 0% (0 units completed, 0 units in progress and 0 units not started) * [[AP United States History]]: 0% (0 units completed, 0 units in progress and 0 units not started) ▲ * [[AP World History - Modern]]: 0% (0 units completed, 0 units in progress and 0 units not started) ▲ ==== Math and Computer Science ==== * [[AP Calculus AB]]: 0% (0 units completed, 0 units in progress and 0 units not started) ▲ * [[AP Calculus BC]]: 0% (0 units completed, 0 units in progress and 0 units not started) ▲ * AP Computer Science A: 0% (0 units completed, 0 units in progress and 0 units not started) * AP Computer Science Principles: 0% (0 units completed, 0 units in progress and 0 units not started) * AP Precalculus: 0% (0 units completed, 0 units in progress and 0 units not started) * AP Statistics: 0% (0 units completed, 0 units in progress and 0 units not started) ==== Sciences ==== * [[AP Biology]]: 0% (0 units completed, 0 units in progress and 0 units not started) ▲ * [[AP Chemistry]]: 0% (0 units completed, 0 units in progress and 0 units not started) ▲ * [[AP Environmental Science]]: 0% (0 units completed, 0 units in progress and 0 units not started) * [[AP Physics 1]]: 0% (0 units completed, 0 units in progress and 0 units not started) * [[AP Physics 2]]: 0% (0 units completed, 0 units in progress and 0 units not started) * [[AP Physics C - Electricity and Magnetism]]: 0% (0 units completed, 0 units in progress and 0 units not started) * [[AP Physics C - Mechanics]]: 0% (0 units completed, 0 units in progress and 0 units not started) ==== AP World Languages and Cultures ==== * AP Chinese Language and Culture: 0% (0 units completed, 0 units in progress and 0 units not started) * AP French Language and Culture: 0% (0 units completed, 0 units in progress and 0 units not started) * AP German Language and Culture: 0% (0 units completed, 0 units in progress and 0 units not started) * AP Italian Language and Culture: 0% (0 units completed, 0 units in progress and 0 units not started) * AP Japanese Language and Culture: 0% (0 units completed, 0 units in progress and 0 units not started) * AP Latin: 0% (0 units completed, 0 units in progress and 0 units not started) * [[AP Spanish Language and Culture]]: 0% (0 units completed, 0 units in progress and 0 units not started) * AP Spanish Literature and Culture: 0% (0 units completed, 0 units in progress and 0 units not started) ==== AP Capstone Diploma Program ==== * AP Research: 0% (0 units completed, 0 units in progress and 0 units not started) * AP Seminar: 0% (0 units completed, 0 units in progress and 0 units not started) ==== AP Career Kickstart ==== * AP Business with Personal Finance: 0% (0 units completed, 0 units in progress and 0 units not started) * [[AP Cybersecurity]]: 0% (0 units completed, 0 units in progress and 0 units not started) 0v54bdioybg793tr8v1dm9jx35ls9fz 0 332627 2842434 2026-10-24T12:50:17Z KyleWii 3113193 Created page with "Water is a polar molecule because oxygen attracts shared electrons more strongly than hydrogen, which creates partial charges. The partial charges allow hydrogen bonds to form between water molecules, giving water properties like cohesoin, high specific heat capacity and the ability to dissolve polar or ionic substances. == Terms == You should learn these terms and what their definitons are before studying how they work together. If you get confused with a word, you can..." 2842434 wikitext text/x-wiki Water is a polar molecule because oxygen attracts shared electrons more strongly than hydrogen, which creates partial charges. The partial charges allow hydrogen bonds to form between water molecules, giving water properties like cohesoin, high specific heat capacity and the ability to dissolve polar or ionic substances. == Terms == You should learn these terms and what their definitons are before studying how they work together. If you get confused with a word, you can always look at this table. {| class="wikitable" |+ !Term !Definiton |- |Water (H₂O) |A molecule composed of two hydrogen atoms and one oxygen atom. |- |Electronegativity |The ability of an atom to attract shared electrons toward itself. |- |Polar covalent bond |A covalent bond in which electrons are shared unequally between atoms. |- |Polarity |An uneven distribution of electrical charge within a molecule. |- |Partial charge (δ⁺ or δ⁻) |A slight positive/negative chargre caused by an unequal distribution of electrons |- |Hydrogen bond |A weak attraction between a partially positive hydrogen and an electronegative atom, for example oxygen. |- |Cohesion |Attraction between molecules of the same substance. |- |Adhesion |Attraction between molecules of different substances. |- |Specific heat capacity |The amount of energy required to raise the temperature of a given amount of a substance by one degree. |- |Solvent |A substance that dissolves another substance |- |Solute |A substance dissolved in a solvent. |- |Hydrophilic |Attracted to water, often polar or charged. |- |Hydrophobic |Not attracted to water, often nonpolar. |- |Evaporative cooling |Cooling that occurs when higher-energy molecules escape form a liquid during evaporation. |} == Notes == === The structure of a water molecule === Water has the chemical formula H₂O. Each molecule contains two hydrogen atoms ''covalently bonded'' to one oxygen atom. Oxygen is more electronegative than hydrogen, so it attracts the shared electrons more strongly. Therefore, electrons spend more time near the oxygen atom. The unequal sharing creates partial charges: * Oxygen has a partial negative charge, indicated with <math>\delta^-</math> * Each hydrogen has a partial positive charge, indicated with <math>\delta^+</math> Water's bent molecular shape means that these charges don't cancel out. Therefore, water is a polar molecule. === Hydrogen bonding between water molecules === The partial charges of water molecules attract neighboring molecules. The partially positive hydrogen of one molecule is attracted to the partially negative oxygen of another, forming a hydrogen bond. There are two types of bonds, a polar covalent bond and a hydrogen bond. * Polar covalent bonds hold the hydrogen and oxygen atoms together within a water molecule. * Hydrogen bonds attract separate water molecules to another. If a substance can form hydrogen bonds with water, it may interact favorably with water. This explains why water interacts very well with polar substances. === Cohesion and adhesion === Cohesion is the attraction between molecules of the same substance. In water, hydrogen bonds cause water molecules to attract one another. Cohesion produces surface tension, which permits the water surface to resist disruption. Adhesion is the attraction between molecules of different substances. Water can stick to other polar surfaces, including the walls of plant xylem vessels. Both of these properties help water move through plants. ==== Examples ==== * If a water droplet forms on a surface, it's because of cohesion. * If water interacts with the walls of plant vessels, it's because of adhesion. === Water's high specific heat capacity === === Ice is less dense than liquid water === === Water as a solvent === === Evaporative cooling === == Quiz == If you want to test your knowledge on this subunit, you can click here. Know that the questions you will see are similar to the AP exam, not for a specific teacher. nfu3wyhmjlct4e448jk1awwrr6v8n0f 2842435 2842434 2026-10-24T12:54:40Z KyleWii 3113193 2842435 wikitext text/x-wiki Water is a polar molecule because oxygen attracts shared electrons more strongly than hydrogen, which creates partial charges. The partial charges allow hydrogen bonds to form between water molecules, giving water properties like cohesoin, high specific heat capacity and the ability to dissolve polar or ionic substances. == Terms == You should learn these terms and what their definitons are before studying how they work together. If you get confused with a word, you can always look at this table. {| class="wikitable" |+ !Term !Definiton |- |Water (H₂O) |A molecule composed of two hydrogen atoms and one oxygen atom. |- |Electronegativity |The ability of an atom to attract shared electrons toward itself. |- |Polar covalent bond |A covalent bond in which electrons are shared unequally between atoms. |- |Polarity |An uneven distribution of electrical charge within a molecule. |- |Partial charge (δ⁺ or δ⁻) |A slight positive/negative chargre caused by an unequal distribution of electrons |- |Hydrogen bond |A weak attraction between a partially positive hydrogen and an electronegative atom, for example oxygen. |- |Cohesion |Attraction between molecules of the same substance. |- |Adhesion |Attraction between molecules of different substances. |- |Specific heat capacity |The amount of energy required to raise the temperature of a given amount of a substance by one degree. |- |Solvent |A substance that dissolves another substance |- |Solute |A substance dissolved in a solvent. |- |Hydrophilic |Attracted to water, often polar or charged. |- |Hydrophobic |Not attracted to water, often nonpolar. |- |Evaporative cooling |Cooling that occurs when higher-energy molecules escape form a liquid during evaporation. |} == Notes == === The structure of a water molecule === Water has the chemical formula H₂O. Each molecule contains two hydrogen atoms ''covalently bonded'' to one oxygen atom. Oxygen is more electronegative than hydrogen, so it attracts the shared electrons more strongly. Therefore, electrons spend more time near the oxygen atom. The unequal sharing creates partial charges: * Oxygen has a partial negative charge, indicated with <math>\delta^-</math> * Each hydrogen has a partial positive charge, indicated with <math>\delta^+</math> Water's bent molecular shape means that these charges don't cancel out. Therefore, water is a polar molecule. === Hydrogen bonding between water molecules === The partial charges of water molecules attract neighboring molecules. The partially positive hydrogen of one molecule is attracted to the partially negative oxygen of another, forming a hydrogen bond. There are two types of bonds, a polar covalent bond and a hydrogen bond. * Polar covalent bonds hold the hydrogen and oxygen atoms together within a water molecule. * Hydrogen bonds attract separate water molecules to another. If a substance can form hydrogen bonds with water, it may interact favorably with water. This explains why water interacts very well with polar substances. === Cohesion and adhesion === Cohesion is the attraction between molecules of the same substance. In water, hydrogen bonds cause water molecules to attract one another. Cohesion produces surface tension, which permits the water surface to resist disruption. Adhesion is the attraction between molecules of different substances. Water can stick to other polar surfaces, including the walls of plant xylem vessels. Both of these properties help water move through plants. ==== Examples ==== * If a water droplet forms on a surface, it's because of cohesion. * If water interacts with the walls of plant vessels, it's because of adhesion. === Water's high specific heat capacity === === Ice is less dense than liquid water === === Water as a solvent === === Evaporative cooling === == Quiz == If you want to test your knowledge on this subunit, you can click [[/Quiz/|here]]. Know that the questions you will see are similar to the AP exam, not for a specific teacher. ll5ttsddhc0igk1inqbsx85eyjvp91e 0 332628 2842436 2026-10-24T13:01:56Z KyleWii 3113193 Created page with "This quiz will contain 15 multiple-choice similar to the AP exam. Calculators are allowed, you cannot use your notes. <quiz display=simple> {A researcher measures the temperature of equal volumes of water and vegatable oil after both absorb the same amount of thermal. The water experiences a smaller increase in temperature than the oil. Which property of water best explains this observation. |type="()"} - Water has strong adhesive properties. + Water has a high specific..." 2842436 wikitext text/x-wiki This quiz will contain 15 multiple-choice similar to the AP exam. Calculators are allowed, you cannot use your notes. <quiz display=simple> {A researcher measures the temperature of equal volumes of water and vegatable oil after both absorb the same amount of thermal. The water experiences a smaller increase in temperature than the oil. Which property of water best explains this observation. |type="()"} - Water has strong adhesive properties. + Water has a high specific heat capacity due to hydrogen bnoding. - Water is less dense as a liquid than as a solid. - Water forms covalent bonds with neighboring water molecules. </quiz> not done yet jctbyfabmn1cbfa6rc1v7bqqqlxj8cw 2 332629 2842437 2026-10-24T14:07:03Z Alexeeeeeee 3113080 /* */ 2842437 wikitext text/x-wiki Rank Idea Why I would consider it Revenue potential 1 Small Business Manager Solves recurring problems for business owners High 2 AI Study Assistant Addresses a clear student need High 3 Local Services Finder Connects customers with service providers High 4 AI Career Coach Helps people improve their job prospects Medium–High 5 Student Community App Can grow through campus communities Medium–High 6xz6s1m5skwntmx6qa1eqm8del8px99 0 332630 2842487 2026-10-25T00:41:23Z George Kasey 3113248 Creating learning project page on Natural Procreative Hygiene 2842487 wikitext text/x-wiki #!/usr/bin/env bash # # post-nph-wikiversity.sh # # Publishes the "Natural Procreative Hygiene" learning-project page to # Wikiversity (en.wikiversity.org) through the MediaWiki API, using a # bot password so the login is not tied to the browser session. # # BEFORE RUNNING, two things must exist: # # 1. A bot password, created while logged in as George Kasey at # https://en.wikiversity.org/wiki/Special:BotPasswords # Name it e.g. "nph-upload" with the "Edit existing pages" and # "Create, edit, and move pages" grants. It gives you a login of the # form George Kasey@nph-upload plus a generated 32-character password. # # 2. The page wikitext saved as plain UTF-8 text at: # ~/nph-wikiversity.txt # # Then: # chmod +x ~/scripts/post-nph-wikiversity.sh # ~/scripts/post-nph-wikiversity.sh # # --------------------------------------------------------------------------- set -euo pipefail WIKI="https://en.wikiversity.org/w/api.php" PAGE="Natural Procreative Hygiene" SUMMARY="Creating learning project page on Natural Procreative Hygiene" TEXTFILE="${HOME}/nph-wikiversity.txt" COOKIES="${HOME}/.wikiversity.cookies" UA="GK-NPH-upload/1.0 (contact: georgekasey.org)" # --- small JSON reader: jget query.tokens.csrftoken ------------------------ jget() { python3 -c ' import sys, json d = json.load(sys.stdin) for k in sys.argv[1].split("."): d = d[k] print(d) ' "$1" } # --- 1. credentials -------------------------------------------------------- if [[ -z "${WIKI_USER:-}" ]]; then read -rp "Bot username (Name@BotName): " WIKI_USER fi if [[ -z "${WIKI_PASS:-}" ]]; then read -rsp "Bot password: " WIKI_PASS echo fi # --- 2. sanity checks ------------------------------------------------------ [[ -s "$TEXTFILE" ]] || { echo "ERROR: $TEXTFILE is missing or empty."; exit 1; } command -v curl >/dev/null || { echo "ERROR: curl not installed."; exit 1; } command -v python3 >/dev/null || { echo "ERROR: python3 not installed."; exit 1; } WORDS=$(wc -w < "$TEXTFILE") BYTES=$(wc -c < "$TEXTFILE") echo "Wikitext file OK: ${WORDS} words, ${BYTES} bytes." echo rm -f "$COOKIES" # --- 3. fetch login token -------------------------------------------------- LOGIN_TOKEN=$(curl -s -A "$UA" -c "$COOKIES" \ "${WIKI}?action=query&meta=tokens&type=login&format=json" \ | jget query.tokens.logintoken) # --- 4. log in ------------------------------------------------------------- LOGIN_JSON=$(curl -s -A "$UA" -b "$COOKIES" -c "$COOKIES" -X POST "$WIKI" \ --data-urlencode "action=login" \ --data-urlencode "lgname=${WIKI_USER}" \ --data-urlencode "lgpassword=${WIKI_PASS}" \ --data-urlencode "lgtoken=${LOGIN_TOKEN}" \ --data-urlencode "format=json") STATUS=$(printf '%s' "$LOGIN_JSON" | jget login.result) echo "Login result: ${STATUS}" if [[ "$STATUS" != "Success" ]]; then echo "Login failed. Server said:" printf '%s\n' "$LOGIN_JSON" exit 1 fi echo # --- 5. fetch CSRF (edit) token ------------------------------------------- CSRF=$(curl -s -A "$UA" -b "$COOKIES" -c "$COOKIES" \ "${WIKI}?action=query&meta=tokens&format=json" \ | jget query.tokens.csrftoken) # --- 6. create the page ---------------------------------------------------- EDIT_JSON=$(curl -s -A "$UA" -b "$COOKIES" -c "$COOKIES" -X POST "$WIKI" \ --data-urlencode "action=edit" \ --data-urlencode "title=${PAGE}" \ --data-urlencode "text@${TEXTFILE}" \ --data-urlencode "summary=${SUMMARY}" \ --data-urlencode "token=${CSRF}" \ --data-urlencode "format=json") if printf '%s' "$EDIT_JSON" | grep -q '"result"[[:space:]]*:[[:space:]]*"Success"'; then echo "PUBLISHED." echo "URL: https://en.wikiversity.org/wiki/Natural_Procreative_Hygiene" else echo "EDIT FAILED. Server said:" printf '%s\n' "$EDIT_JSON" exit 1 fi echo # --- 7. read back, no login needed ---------------------------------------- echo "Verifying page exists (anonymous read):" curl -s -A "$UA" \ "${WIKI}?action=query&prop=info&inprop=url&titles=Natural%20Procreative%20Hygiene&format=json" echo echo "Done." okx01rqu6njf0xi6zybn17t92jpg2rx 2842488 2842487 2026-10-25T00:46:45Z George Kasey 3113248 {{Draft article|subject=proposed population and health concept|note=This is a draft for a new article. It documents a term coined and used in the published corpus of the George Kasey Foundation / Media Free Times think tank. Independent third-party coverage is still pending; reviewers may promote to article space when such coverage exists.}} '''Natural Procreative Hygiene''' (sometimes '''Natural-Procreative Hygiene'''; '''NPH''') is a proposed framework holding that human reproduction, like ot 2842488 wikitext text/x-wiki #{{Draft article|subject=proposed population and health concept|note=This is a draft for a new article. It documents a term coined and used in the published corpus of the George Kasey Foundation / Media Free Times think tank. Independent third-party coverage is still pending; reviewers may promote to article space when such coverage exists.}} '''Natural Procreative Hygiene''' (sometimes '''Natural-Procreative Hygiene'''; '''NPH''') is a proposed framework holding that human reproduction, like other aspects of health, requires deliberate hygiene practices, and that the historical abandonment of such practices—particularly following the rise of pronatalist religious and political doctrine—is a root cause of overpopulation, environmental degradation, and related social harms. == Overview == The concept was originated and elaborated over more than five decades by Canadian writer and researcher George Kasey, publisher of the ''Media Free Times'', in an archive of recorded essays and dialogues exceeding 25 million words.<ref>Kasey, George. ''The Point in the Land of Point'' (corpus, 1969–2026). Archived at Zenodo and ResearchGate.</ref> Within this body of work, NPH is presented as a hygiene discipline analogous to sanitation or nutrition: a set of knowledge and practices concerning when and under what conditions to bring new human life into being, understood as a personal, public-health, and ecological responsibility. Proponents contrast NPH with both unrestricted pronatalism and coercive population-control programs, describing it instead as consensual, education-based reproductive stewardship rooted in pre-modern folk knowledge. The framework is cited by its originator in analyses of water scarcity, soil depletion, deforestation, and the commodification of freshwater ("Blue Gold"), in which population pressure is treated as the upstream driver of environmental decline. == Historical claims == The corpus associated with NPH advances several historical claims: * That pre-agricultural and some ancient societies practiced deliberate reproductive hygiene, and that this knowledge was widespread prior to the emergence of pronatalist doctrines. * That the Roman Empire's marriage legislation ({{lang|la|''Lex Julia de maritandis ordinibus''}}, 18 BCE) and associated laws institutionalized compulsory reproduction, displacing earlier practices.<ref>On the Augustan marriage laws generally, see e.g. {{cite book|title=The Roman Family|last=Dixon|first=Susan|publisher=Johns Hopkins University Press|year=1992}}.</ref> * That contraceptive and abortifacient plants known to antiquity—most famously the North African plant ''silphium'', reputedly harvested to extinction—were progressively removed from common use through a combination of overharvesting and institutional suppression.<ref>Riddle, John M. ''Eve's Herbs: A History of Contraception and Abortion in the West''. Harvard University Press, 1997.</ref> Independent scholarship corroborates fragments of these claims (the Augustan marriage laws; the existence and economic importance of ''silphium''; widespread pre-modern contraceptive plant knowledge as surveyed by Riddle), while the stronger thesis—that a coherent, systematic "natural procreative hygiene" was practiced globally and then deliberately suppressed—remains documented chiefly within Kasey's own publications and those of associates. == Contemporary use == The abbreviation '''NPH''' and the spelled-out term appear in the publications and public communications of the George Kasey Foundation and the affiliated ViFrU Virtual Free University think tank.<ref>''Media Free Times'' archive; georgekasey.org.</ref> The framework underlies proposals for a public-education initiative, including a condensed volume of the corpus written at an adolescent reading level, and is advanced as an alternative to growth-oriented population policy. == Reception == As of 2026, Natural Procreative Hygiene has not received substantial coverage in independent academic literature or mainstream media. Discussion is concentrated in the originator's corpus, self-published repositories (Zenodo, ResearchGate), and social media. The framework is therefore considered a neologism of its originator rather than an established term of art in demography, public health, or environmental science. == See also == * [[Demographic transition]] * [[Human overpopulation]] * [[Reproductive rights]] * [[Silphium]] * ''[[Eve's Herbs]]'' * [[Voluntary human extinction movement]] == References == {{reflist}}{{Draft categories|}} # 0npzola0vws0t5zelorpcfz49lyiblt 2842490 2842488 2026-10-25T00:56:13Z George Kasey 3113248 removed draft The Template:Draft article line at the top and The Latin template errored — the {{lang|la|''Lex Julia...''}} 2842490 wikitext text/x-wiki '''Natural Procreative Hygiene''' (sometimes '''Natural-Procreative Hygiene'''; '''NPH''') is a proposed framework holding that human reproduction, like other aspects of health, requires deliberate hygiene practices, and that the historical abandonment of such practices—particularly following the rise of pronatalist religious and political doctrine—is a root cause of overpopulation, environmental degradation, and related social harms. == Overview == The concept was originated and elaborated over more than five decades by Canadian writer and researcher George Kasey, publisher of the ''Media Free Times'', in an archive of recorded essays and dialogues exceeding 25 million words.<ref>Kasey, George. ''The Point in the Land of Point'' (corpus, 1969–2026). Archived at Zenodo and ResearchGate.</ref> Within this body of work, NPH is presented as a hygiene discipline analogous to sanitation or nutrition: a set of knowledge and practices concerning when and under what conditions to bring new human life into being, understood as a personal, public-health, and ecological responsibility. Proponents contrast NPH with both unrestricted pronatalism and coercive population-control programs, describing it instead as consensual, education-based reproductive stewardship rooted in pre-modern folk knowledge. The framework is cited by its originator in analyses of water scarcity, soil depletion, deforestation, and the commodification of freshwater ("Blue Gold"), in which population pressure is treated as the upstream driver of environmental decline. == Historical claims == The corpus associated with NPH advances several historical claims: * That pre-agricultural and some ancient societies practiced deliberate reproductive hygiene, and that this knowledge was widespread prior to the emergence of pronatalist doctrines. * That the Roman Empire's marriage legislation , 18 BCE) and associated laws institutionalized compulsory reproduction, displacing earlier practices.<ref>On the Augustan marriage laws generally, see e.g. {{cite book|title=The Roman Family|last=Dixon|first=Susan|publisher=Johns Hopkins University Press|year=1992}}.</ref> * That contraceptive and abortifacient plants known to antiquity—most famously the North African plant ''silphium'', reputedly harvested to extinction—were progressively removed from common use through a combination of overharvesting and institutional suppression.<ref>Riddle, John M. ''Eve's Herbs: A History of Contraception and Abortion in the West''. Harvard University Press, 1997.</ref> Independent scholarship corroborates fragments of these claims (the Augustan marriage laws; the existence and economic importance of ''silphium''; widespread pre-modern contraceptive plant knowledge as surveyed by Riddle), while the stronger thesis—that a coherent, systematic "natural procreative hygiene" was practiced globally and then deliberately suppressed—remains documented chiefly within Kasey's own publications and those of associates. == Contemporary use == The abbreviation '''NPH''' and the spelled-out term appear in the publications and public communications of the George Kasey Foundation and the affiliated ViFrU Virtual Free University think tank.<ref>''Media Free Times'' archive; georgekasey.org.</ref> The framework underlies proposals for a public-education initiative, including a condensed volume of the corpus written at an adolescent reading level, and is advanced as an alternative to growth-oriented population policy. == Reception == As of 2026, Natural Procreative Hygiene has not received substantial coverage in independent academic literature or mainstream media. Discussion is concentrated in the originator's corpus, self-published repositories (Zenodo, ResearchGate), and social media. The framework is therefore considered a neologism of its originator rather than an established term of art in demography, public health, or environmental science. == See also == * [[Demographic transition]] * [[Human overpopulation]] * [[Reproductive rights]] * [[Silphium]] * ''[[Eve's Herbs]]'' * [[Voluntary human extinction movement]] == References == # q5zoipfptli3cf1e7u95iynvhdkeqrl 2842492 2842490 2026-10-25T01:12:01Z George Kasey 3113248 /* References */ 2842492 wikitext text/x-wiki '''Natural Procreative Hygiene''' (sometimes '''Natural-Procreative Hygiene'''; '''NPH''') is a proposed framework holding that human reproduction, like other aspects of health, requires deliberate hygiene practices, and that the historical abandonment of such practices—particularly following the rise of pronatalist religious and political doctrine—is a root cause of overpopulation, environmental degradation, and related social harms. == Overview == The concept was originated and elaborated over more than five decades by Canadian writer and researcher George Kasey, publisher of the ''Media Free Times'', in an archive of recorded essays and dialogues exceeding 25 million words.<ref>Kasey, George. ''The Point in the Land of Point'' (corpus, 1969–2026). Archived at Zenodo and ResearchGate.</ref> Within this body of work, NPH is presented as a hygiene discipline analogous to sanitation or nutrition: a set of knowledge and practices concerning when and under what conditions to bring new human life into being, understood as a personal, public-health, and ecological responsibility. Proponents contrast NPH with both unrestricted pronatalism and coercive population-control programs, describing it instead as consensual, education-based reproductive stewardship rooted in pre-modern folk knowledge. The framework is cited by its originator in analyses of water scarcity, soil depletion, deforestation, and the commodification of freshwater ("Blue Gold"), in which population pressure is treated as the upstream driver of environmental decline. == Historical claims == The corpus associated with NPH advances several historical claims: * That pre-agricultural and some ancient societies practiced deliberate reproductive hygiene, and that this knowledge was widespread prior to the emergence of pronatalist doctrines. * That the Roman Empire's marriage legislation , 18 BCE) and associated laws institutionalized compulsory reproduction, displacing earlier practices.<ref>On the Augustan marriage laws generally, see e.g. {{cite book|title=The Roman Family|last=Dixon|first=Susan|publisher=Johns Hopkins University Press|year=1992}}.</ref> * That contraceptive and abortifacient plants known to antiquity—most famously the North African plant ''silphium'', reputedly harvested to extinction—were progressively removed from common use through a combination of overharvesting and institutional suppression.<ref>Riddle, John M. ''Eve's Herbs: A History of Contraception and Abortion in the West''. Harvard University Press, 1997.</ref> Independent scholarship corroborates fragments of these claims (the Augustan marriage laws; the existence and economic importance of ''silphium''; widespread pre-modern contraceptive plant knowledge as surveyed by Riddle), while the stronger thesis—that a coherent, systematic "natural procreative hygiene" was practiced globally and then deliberately suppressed—remains documented chiefly within Kasey's own publications and those of associates. == Contemporary use == The abbreviation '''NPH''' and the spelled-out term appear in the publications and public communications of the George Kasey Foundation and the affiliated ViFrU Virtual Free University think tank.<ref>''Media Free Times'' archive; georgekasey.org.</ref> The framework underlies proposals for a public-education initiative, including a condensed volume of the corpus written at an adolescent reading level, and is advanced as an alternative to growth-oriented population policy. == Reception == As of 2026, Natural Procreative Hygiene has not received substantial coverage in independent academic literature or mainstream media. Discussion is concentrated in the originator's corpus, self-published repositories (Zenodo, ResearchGate), and social media. The framework is therefore considered a neologism of its originator rather than an established term of art in demography, public health, or environmental science. == See also == * [[Demographic transition]] * [[Human overpopulation]] * [[Reproductive rights]] * [[Silphium]] * ''[[Eve's Herbs]]'' * [[Voluntary human extinction movement]] == References == # https://www.researchgate.net/publication/414415889_THE_BURNT_WREATH_Natural_Procreative_Hygiene_and_Epistemicide_in_Lithuania_-A_Dialogue_on_the_Destruction_of_Ancestral_Wisdom 9mgy003ef561xjqcus1x26fl3g6db3k 2842495 2842492 2026-10-25T01:21:06Z George Kasey 3113248 /* References */ 2842495 wikitext text/x-wiki '''Natural Procreative Hygiene''' (sometimes '''Natural-Procreative Hygiene'''; '''NPH''') is a proposed framework holding that human reproduction, like other aspects of health, requires deliberate hygiene practices, and that the historical abandonment of such practices—particularly following the rise of pronatalist religious and political doctrine—is a root cause of overpopulation, environmental degradation, and related social harms. == Overview == The concept was originated and elaborated over more than five decades by Canadian writer and researcher George Kasey, publisher of the ''Media Free Times'', in an archive of recorded essays and dialogues exceeding 25 million words.<ref>[https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality The Technethics Gap: Natural Procreative Hygiene, Media Hegemony, and the Suppression of Demographic Reality] [https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality April 2026] [https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality DOI:] [https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality 10.13140/RG.2.2.21878.56649] [https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality License] [https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality CC BY-NC-SA 4.0] [https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality George Kaseyhttps://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality]</ref> Within this body of work, NPH is presented as a hygiene discipline analogous to sanitation or nutrition: a set of knowledge and practices concerning when and under what conditions to bring new human life into being, understood as a personal, public-health, and ecological responsibility. Proponents contrast NPH with both unrestricted pronatalism and coercive population-control programs, describing it instead as consensual, education-based reproductive stewardship rooted in pre-modern folk knowledge. The framework is cited by its originator in analyses of water scarcity, soil depletion, deforestation, and the commodification of freshwater ("Blue Gold"), in which population pressure is treated as the upstream driver of environmental decline. == Historical claims == The corpus associated with NPH advances several historical claims: * That pre-agricultural and some ancient societies practiced deliberate reproductive hygiene, and that this knowledge was widespread prior to the emergence of pronatalist doctrines. * That the Roman Empire's marriage legislation , 18 BCE) and associated laws institutionalized compulsory reproduction, displacing earlier practices.<ref>On the Augustan marriage laws generally, see e.g. {{cite book|title=The Roman Family|last=Dixon|first=Susan|publisher=Johns Hopkins University Press|year=1992}}.</ref> * That contraceptive and abortifacient plants known to antiquity—most famously the North African plant ''silphium'', reputedly harvested to extinction—were progressively removed from common use through a combination of overharvesting and institutional suppression.<ref>Riddle, John M. ''Eve's Herbs: A History of Contraception and Abortion in the West''. Harvard University Press, 1997.</ref> Independent scholarship corroborates fragments of these claims (the Augustan marriage laws; the existence and economic importance of ''silphium''; widespread pre-modern contraceptive plant knowledge as surveyed by Riddle), while the stronger thesis—that a coherent, systematic "natural procreative hygiene" was practiced globally and then deliberately suppressed—remains documented chiefly within Kasey's own publications and those of associates. == Contemporary use == The abbreviation '''NPH''' and the spelled-out term appear in the publications and public communications of the George Kasey Foundation and the affiliated ViFrU Virtual Free University think tank.<ref>''Media Free Times'' archive; georgekasey.org.</ref> The framework underlies proposals for a public-education initiative, including a condensed volume of the corpus written at an adolescent reading level, and is advanced as an alternative to growth-oriented population policy. == Reception == As of 2026, Natural Procreative Hygiene has not received substantial coverage in independent academic literature or mainstream media. Discussion is concentrated in the originator's corpus, self-published repositories (Zenodo, ResearchGate), and social media. The framework is therefore considered a neologism of its originator rather than an established term of art in demography, public health, or environmental science. == See also == * [[Demographic transition]] * [[Human overpopulation]] * [[Reproductive rights]] * [[Silphium]] * ''[[Eve's Herbs]]'' * [[Voluntary human extinction movement]] == References == # https://www.researchgate.net/publication/414415889_THE_BURNT_WREATH_Natural_Procreative_Hygiene_and_Epistemicide_in_Lithuania_-A_Dialogue_on_the_Destruction_of_Ancestral_Wisdom 4s6nhqe7tchc6kdjjvqe71487971bfj 2842496 2842495 2026-10-25T01:24:06Z George Kasey 3113248 /* References */ 2842496 wikitext text/x-wiki '''Natural Procreative Hygiene''' (sometimes '''Natural-Procreative Hygiene'''; '''NPH''') is a proposed framework holding that human reproduction, like other aspects of health, requires deliberate hygiene practices, and that the historical abandonment of such practices—particularly following the rise of pronatalist religious and political doctrine—is a root cause of overpopulation, environmental degradation, and related social harms. == Overview == The concept was originated and elaborated over more than five decades by Canadian writer and researcher George Kasey, publisher of the ''Media Free Times'', in an archive of recorded essays and dialogues exceeding 25 million words.<ref>[https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality The Technethics Gap: Natural Procreative Hygiene, Media Hegemony, and the Suppression of Demographic Reality] [https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality April 2026] [https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality DOI:] [https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality 10.13140/RG.2.2.21878.56649] [https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality License] [https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality CC BY-NC-SA 4.0] [https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality George Kaseyhttps://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality]</ref> Within this body of work, NPH is presented as a hygiene discipline analogous to sanitation or nutrition: a set of knowledge and practices concerning when and under what conditions to bring new human life into being, understood as a personal, public-health, and ecological responsibility. Proponents contrast NPH with both unrestricted pronatalism and coercive population-control programs, describing it instead as consensual, education-based reproductive stewardship rooted in pre-modern folk knowledge. The framework is cited by its originator in analyses of water scarcity, soil depletion, deforestation, and the commodification of freshwater ("Blue Gold"), in which population pressure is treated as the upstream driver of environmental decline. == Historical claims == The corpus associated with NPH advances several historical claims: * That pre-agricultural and some ancient societies practiced deliberate reproductive hygiene, and that this knowledge was widespread prior to the emergence of pronatalist doctrines. * That the Roman Empire's marriage legislation , 18 BCE) and associated laws institutionalized compulsory reproduction, displacing earlier practices.<ref>On the Augustan marriage laws generally, see e.g. {{cite book|title=The Roman Family|last=Dixon|first=Susan|publisher=Johns Hopkins University Press|year=1992}}.</ref> * That contraceptive and abortifacient plants known to antiquity—most famously the North African plant ''silphium'', reputedly harvested to extinction—were progressively removed from common use through a combination of overharvesting and institutional suppression.<ref>Riddle, John M. ''Eve's Herbs: A History of Contraception and Abortion in the West''. Harvard University Press, 1997.</ref> Independent scholarship corroborates fragments of these claims (the Augustan marriage laws; the existence and economic importance of ''silphium''; widespread pre-modern contraceptive plant knowledge as surveyed by Riddle), while the stronger thesis—that a coherent, systematic "natural procreative hygiene" was practiced globally and then deliberately suppressed—remains documented chiefly within Kasey's own publications and those of associates. == Contemporary use == The abbreviation '''NPH''' and the spelled-out term appear in the publications and public communications of the George Kasey Foundation and the affiliated ViFrU Virtual Free University think tank.<ref>''Media Free Times'' archive; georgekasey.org.</ref> The framework underlies proposals for a public-education initiative, including a condensed volume of the corpus written at an adolescent reading level, and is advanced as an alternative to growth-oriented population policy. == Reception == As of 2026, Natural Procreative Hygiene has not received substantial coverage in independent academic literature or mainstream media. Discussion is concentrated in the originator's corpus, self-published repositories (Zenodo, ResearchGate), and social media. The framework is therefore considered a neologism of its originator rather than an established term of art in demography, public health, or environmental science. == See also == * [[Demographic transition]] * [[Human overpopulation]] * [[Reproductive rights]] * [[Silphium]] * ''[[Eve's Herbs]]'' * [[Voluntary human extinction movement]] == References == <references /> 19qyf7egjbaufm2y2m7bisczg4kmdc0 2842497 2842496 2026-10-25T01:31:08Z George Kasey 3113248 /* References */ 2842497 wikitext text/x-wiki '''Natural Procreative Hygiene''' (sometimes '''Natural-Procreative Hygiene'''; '''NPH''') is a proposed framework holding that human reproduction, like other aspects of health, requires deliberate hygiene practices, and that the historical abandonment of such practices—particularly following the rise of pronatalist religious and political doctrine—is a root cause of overpopulation, environmental degradation, and related social harms. == Overview == The concept was originated and elaborated over more than five decades by Canadian writer and researcher George Kasey, publisher of the ''Media Free Times'', in an archive of recorded essays and dialogues exceeding 25 million words.<ref>[https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality The Technethics Gap: Natural Procreative Hygiene, Media Hegemony, and the Suppression of Demographic Reality] [https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality April 2026] [https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality DOI:] [https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality 10.13140/RG.2.2.21878.56649] [https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality License] [https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality CC BY-NC-SA 4.0] [https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality George Kaseyhttps://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality]</ref> Within this body of work, NPH is presented as a hygiene discipline analogous to sanitation or nutrition: a set of knowledge and practices concerning when and under what conditions to bring new human life into being, understood as a personal, public-health, and ecological responsibility. Proponents contrast NPH with both unrestricted pronatalism and coercive population-control programs, describing it instead as consensual, education-based reproductive stewardship rooted in pre-modern folk knowledge. The framework is cited by its originator in analyses of water scarcity, soil depletion, deforestation, and the commodification of freshwater ("Blue Gold"), in which population pressure is treated as the upstream driver of environmental decline. == Historical claims == The corpus associated with NPH advances several historical claims: * That pre-agricultural and some ancient societies practiced deliberate reproductive hygiene, and that this knowledge was widespread prior to the emergence of pronatalist doctrines. * That the Roman Empire's marriage legislation , 18 BCE) and associated laws institutionalized compulsory reproduction, displacing earlier practices.<ref>On the Augustan marriage laws generally, see e.g. {{cite book|title=The Roman Family|last=Dixon|first=Susan|publisher=Johns Hopkins University Press|year=1992}}.</ref> * That contraceptive and abortifacient plants known to antiquity—most famously the North African plant ''silphium'', reputedly harvested to extinction—were progressively removed from common use through a combination of overharvesting and institutional suppression.<ref>Riddle, John M. ''Eve's Herbs: A History of Contraception and Abortion in the West''. Harvard University Press, 1997.</ref> Independent scholarship corroborates fragments of these claims (the Augustan marriage laws; the existence and economic importance of ''silphium''; widespread pre-modern contraceptive plant knowledge as surveyed by Riddle), while the stronger thesis—that a coherent, systematic "natural procreative hygiene" was practiced globally and then deliberately suppressed—remains documented chiefly within Kasey's own publications and those of associates. == Contemporary use == The abbreviation '''NPH''' and the spelled-out term appear in the publications and public communications of the George Kasey Foundation and the affiliated ViFrU Virtual Free University think tank.<ref>''Media Free Times'' archive; georgekasey.org.</ref> The framework underlies proposals for a public-education initiative, including a condensed volume of the corpus written at an adolescent reading level, and is advanced as an alternative to growth-oriented population policy. == Reception == As of 2026, Natural Procreative Hygiene has not received substantial coverage in independent academic literature or mainstream media. Discussion is concentrated in the originator's corpus, self-published repositories (Zenodo, ResearchGate), and social media. The framework is therefore considered a neologism of its originator rather than an established term of art in demography, public health, or environmental science. == See also == * [[Demographic transition]] * [[Human overpopulation]] * [[Reproductive rights]] * [[Silphium]] * ''[[Eve's Herbs]]'' * [[Voluntary human extinction movement]] == References == <references /> https://zenodo.org/records/22815239 83bmkgw5jx4c99ntvlcsn0382y5fyq3 2842498 2842497 2026-10-25T01:33:10Z George Kasey 3113248 /* References */ 2842498 wikitext text/x-wiki '''Natural Procreative Hygiene''' (sometimes '''Natural-Procreative Hygiene'''; '''NPH''') is a proposed framework holding that human reproduction, like other aspects of health, requires deliberate hygiene practices, and that the historical abandonment of such practices—particularly following the rise of pronatalist religious and political doctrine—is a root cause of overpopulation, environmental degradation, and related social harms. == Overview == The concept was originated and elaborated over more than five decades by Canadian writer and researcher George Kasey, publisher of the ''Media Free Times'', in an archive of recorded essays and dialogues exceeding 25 million words.<ref>[https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality The Technethics Gap: Natural Procreative Hygiene, Media Hegemony, and the Suppression of Demographic Reality] [https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality April 2026] [https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality DOI:] [https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality 10.13140/RG.2.2.21878.56649] [https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality License] [https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality CC BY-NC-SA 4.0] [https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality George Kaseyhttps://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality]</ref> Within this body of work, NPH is presented as a hygiene discipline analogous to sanitation or nutrition: a set of knowledge and practices concerning when and under what conditions to bring new human life into being, understood as a personal, public-health, and ecological responsibility. Proponents contrast NPH with both unrestricted pronatalism and coercive population-control programs, describing it instead as consensual, education-based reproductive stewardship rooted in pre-modern folk knowledge. The framework is cited by its originator in analyses of water scarcity, soil depletion, deforestation, and the commodification of freshwater ("Blue Gold"), in which population pressure is treated as the upstream driver of environmental decline. == Historical claims == The corpus associated with NPH advances several historical claims: * That pre-agricultural and some ancient societies practiced deliberate reproductive hygiene, and that this knowledge was widespread prior to the emergence of pronatalist doctrines. * That the Roman Empire's marriage legislation , 18 BCE) and associated laws institutionalized compulsory reproduction, displacing earlier practices.<ref>The Burnt Wreath: Natural Procreative Hygiene and Epistemicide in Lithuania — A Dialogue on the Destruction of Ancestral Wisdom https://zenodo.org/records/22815239</ref> * That contraceptive and abortifacient plants known to antiquity—most famously the North African plant ''silphium'', reputedly harvested to extinction—were progressively removed from common use through a combination of overharvesting and institutional suppression.<ref>Riddle, John M. ''Eve's Herbs: A History of Contraception and Abortion in the West''. Harvard University Press, 1997.</ref> Independent scholarship corroborates fragments of these claims (the Augustan marriage laws; the existence and economic importance of ''silphium''; widespread pre-modern contraceptive plant knowledge as surveyed by Riddle), while the stronger thesis—that a coherent, systematic "natural procreative hygiene" was practiced globally and then deliberately suppressed—remains documented chiefly within Kasey's own publications and those of associates. == Contemporary use == The abbreviation '''NPH''' and the spelled-out term appear in the publications and public communications of the George Kasey Foundation and the affiliated ViFrU Virtual Free University think tank.<ref>''Media Free Times'' archive; georgekasey.org.</ref> The framework underlies proposals for a public-education initiative, including a condensed volume of the corpus written at an adolescent reading level, and is advanced as an alternative to growth-oriented population policy. == Reception == As of 2026, Natural Procreative Hygiene has not received substantial coverage in independent academic literature or mainstream media. Discussion is concentrated in the originator's corpus, self-published repositories (Zenodo, ResearchGate), and social media. The framework is therefore considered a neologism of its originator rather than an established term of art in demography, public health, or environmental science. == See also == * [[Demographic transition]] * [[Human overpopulation]] * [[Reproductive rights]] * [[Silphium]] * ''[[Eve's Herbs]]'' * [[Voluntary human extinction movement]] == References == <references /> hdxghqk762bnd584yzpusz604tzr1y6 2842499 2842498 2026-10-25T01:35:54Z George Kasey 3113248 /* References */ 2842499 wikitext text/x-wiki '''Natural Procreative Hygiene''' (sometimes '''Natural-Procreative Hygiene'''; '''NPH''') is a proposed framework holding that human reproduction, like other aspects of health, requires deliberate hygiene practices, and that the historical abandonment of such practices—particularly following the rise of pronatalist religious and political doctrine—is a root cause of overpopulation, environmental degradation, and related social harms. == Overview == The concept was originated and elaborated over more than five decades by Canadian writer and researcher George Kasey, publisher of the ''Media Free Times'', in an archive of recorded essays and dialogues exceeding 25 million words.<ref>[https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality The Technethics Gap: Natural Procreative Hygiene, Media Hegemony, and the Suppression of Demographic Reality] https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality</ref> Within this body of work, NPH is presented as a hygiene discipline analogous to sanitation or nutrition: a set of knowledge and practices concerning when and under what conditions to bring new human life into being, understood as a personal, public-health, and ecological responsibility. Proponents contrast NPH with both unrestricted pronatalism and coercive population-control programs, describing it instead as consensual, education-based reproductive stewardship rooted in pre-modern folk knowledge. The framework is cited by its originator in analyses of water scarcity, soil depletion, deforestation, and the commodification of freshwater ("Blue Gold"), in which population pressure is treated as the upstream driver of environmental decline. == Historical claims == The corpus associated with NPH advances several historical claims: * That pre-agricultural and some ancient societies practiced deliberate reproductive hygiene, and that this knowledge was widespread prior to the emergence of pronatalist doctrines. * That the Roman Empire's marriage legislation , 18 BCE) and associated laws institutionalized compulsory reproduction, displacing earlier practices.<ref>The Burnt Wreath: Natural Procreative Hygiene and Epistemicide in Lithuania — A Dialogue on the Destruction of Ancestral Wisdom https://zenodo.org/records/22815239</ref> * That contraceptive and abortifacient plants known to antiquity—most famously the North African plant ''silphium'', reputedly harvested to extinction—were progressively removed from common use through a combination of overharvesting and institutional suppression.<ref>Riddle, John M. ''Eve's Herbs: A History of Contraception and Abortion in the West''. Harvard University Press, 1997.</ref> Independent scholarship corroborates fragments of these claims (the Augustan marriage laws; the existence and economic importance of ''silphium''; widespread pre-modern contraceptive plant knowledge as surveyed by Riddle), while the stronger thesis—that a coherent, systematic "natural procreative hygiene" was practiced globally and then deliberately suppressed—remains documented chiefly within Kasey's own publications and those of associates. == Contemporary use == The abbreviation '''NPH''' and the spelled-out term appear in the publications and public communications of the George Kasey Foundation and the affiliated ViFrU Virtual Free University think tank.<ref>''Media Free Times'' archive; georgekasey.org.</ref> The framework underlies proposals for a public-education initiative, including a condensed volume of the corpus written at an adolescent reading level, and is advanced as an alternative to growth-oriented population policy. == Reception == As of 2026, Natural Procreative Hygiene has not received substantial coverage in independent academic literature or mainstream media. Discussion is concentrated in the originator's corpus, self-published repositories (Zenodo, ResearchGate), and social media. The framework is therefore considered a neologism of its originator rather than an established term of art in demography, public health, or environmental science. == See also == * [[Demographic transition]] * [[Human overpopulation]] * [[Reproductive rights]] * [[Silphium]] * ''[[Eve's Herbs]]'' * [[Voluntary human extinction movement]] == References == <references /> alwcx7xnvgpntwlrmyi07pkpvfik4b8 2842500 2842499 2026-10-25T01:37:26Z George Kasey 3113248 /* References */ 2842500 wikitext text/x-wiki '''Natural Procreative Hygiene''' (sometimes '''Natural-Procreative Hygiene'''; '''NPH''') is a proposed framework holding that human reproduction, like other aspects of health, requires deliberate hygiene practices, and that the historical abandonment of such practices—particularly following the rise of pronatalist religious and political doctrine—is a root cause of overpopulation, environmental degradation, and related social harms. == Overview == The concept was originated and elaborated over more than five decades by Canadian writer and researcher George Kasey, publisher of the ''Media Free Times'', in an archive of recorded essays and dialogues exceeding 25 million words.<ref>[https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality The Technethics Gap: Natural Procreative Hygiene, Media Hegemony, and the Suppression of Demographic Reality] https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality</ref> Within this body of work, NPH is presented as a hygiene discipline analogous to sanitation or nutrition: a set of knowledge and practices concerning when and under what conditions to bring new human life into being, understood as a personal, public-health, and ecological responsibility. Proponents contrast NPH with both unrestricted pronatalism and coercive population-control programs, describing it instead as consensual, education-based reproductive stewardship rooted in pre-modern folk knowledge. The framework is cited by its originator in analyses of water scarcity, soil depletion, deforestation, and the commodification of freshwater ("Blue Gold"), in which population pressure is treated as the upstream driver of environmental decline. == Historical claims == The corpus associated with NPH advances several historical claims: * That pre-agricultural and some ancient societies practiced deliberate reproductive hygiene, and that this knowledge was widespread prior to the emergence of pronatalist doctrines. * That the Roman Empire's marriage legislation , 18 BCE) and associated laws institutionalized compulsory reproduction, displacing earlier practices.<ref>The Burnt Wreath: Natural Procreative Hygiene and Epistemicide in Lithuania — A Dialogue on the Destruction of Ancestral Wisdom https://zenodo.org/records/22815239</ref> * That contraceptive and abortifacient plants known to antiquity—most famously the North African plant ''silphium'', reputedly harvested to extinction—were progressively removed from common use through a combination of overharvesting and institutional suppression.<ref>Riddle, John M. ''Eve's Herbs: A History of Contraception and Abortion in the West''. Harvard University Press, 1997.</ref> Independent scholarship corroborates fragments of these claims (the Augustan marriage laws; the existence and economic importance of ''silphium''; widespread pre-modern contraceptive plant knowledge as surveyed by Riddle), while the stronger thesis—that a coherent, systematic "natural procreative hygiene" was practiced globally and then deliberately suppressed—remains documented chiefly within Kasey's own publications and those of associates. == Contemporary use == The abbreviation '''NPH''' and the spelled-out term appear in the publications and public communications of the George Kasey Foundation and the affiliated ViFrU Virtual Free University think tank.<ref>''Media Free Times'' archive;https://georgekasey.org/</ref> The framework underlies proposals for a public-education initiative, including a condensed volume of the corpus written at an adolescent reading level, and is advanced as an alternative to growth-oriented population policy. == Reception == As of 2026, Natural Procreative Hygiene has not received substantial coverage in independent academic literature or mainstream media. Discussion is concentrated in the originator's corpus, self-published repositories (Zenodo, ResearchGate), and social media. The framework is therefore considered a neologism of its originator rather than an established term of art in demography, public health, or environmental science. == See also == * [[Demographic transition]] * [[Human overpopulation]] * [[Reproductive rights]] * [[Silphium]] * ''[[Eve's Herbs]]'' * [[Voluntary human extinction movement]] == References == <references /> jm0vsarpk0b85wt8b21w6v9bu8h2tnx 2842502 2842500 2026-10-25T01:47:38Z George Kasey 3113248 /* Historical claims */ 2842502 wikitext text/x-wiki '''Natural Procreative Hygiene''' (sometimes '''Natural-Procreative Hygiene'''; '''NPH''') is a proposed framework holding that human reproduction, like other aspects of health, requires deliberate hygiene practices, and that the historical abandonment of such practices—particularly following the rise of pronatalist religious and political doctrine—is a root cause of overpopulation, environmental degradation, and related social harms. == Overview == The concept was originated and elaborated over more than five decades by Canadian writer and researcher George Kasey, publisher of the ''Media Free Times'', in an archive of recorded essays and dialogues exceeding 25 million words.<ref>[https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality The Technethics Gap: Natural Procreative Hygiene, Media Hegemony, and the Suppression of Demographic Reality] https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality</ref> Within this body of work, NPH is presented as a hygiene discipline analogous to sanitation or nutrition: a set of knowledge and practices concerning when and under what conditions to bring new human life into being, understood as a personal, public-health, and ecological responsibility. Proponents contrast NPH with both unrestricted pronatalism and coercive population-control programs, describing it instead as consensual, education-based reproductive stewardship rooted in pre-modern folk knowledge. The framework is cited by its originator in analyses of water scarcity, soil depletion, deforestation, and the commodification of freshwater ("Blue Gold"), in which population pressure is treated as the upstream driver of environmental decline. == Historical claims == The corpus associated with NPH advances several historical claims: * That pre-agricultural and some ancient societies practiced deliberate reproductive hygiene, and that this knowledge was widespread prior to the emergence of pronatalist doctrines. * The systematic destruction of Lithuanian ancestral wisdom (''tik protėvių išmintis'') from the thirteenth century through the post-Soviet era, examining epistemicide as the killing of knowledge transmission rather than physical annihilation alone. Through analysis of three documented cases — (1) the Soviet-era deportation and collectivization of the independent farming class, (2) the post-independence reconstruction industry based on questionable archaeological interpretation, and (3) the national plant ''rūta'' (rue) as an erased botanical knowledge system — the argument demonstrates how state power eliminates transmitters while leaving population counts intact.<ref>The Burnt Wreath: Natural Procreative Hygiene and Epistemicide in Lithuania — A Dialogue on the Destruction of Ancestral Wisdom https://zenodo.org/records/22815239</ref> * That contraceptive and abortifacient plants known to antiquity—most famously the North African plant ''silphium'', reputedly harvested to extinction—were progressively removed from common use through a combination of overharvesting and institutional suppression.<ref>Riddle, John M. ''Eve's Herbs: A History of Contraception and Abortion in the West''. Harvard University Press, 1997.</ref> Independent scholarship corroborates fragments of these claims (the Augustan marriage laws; the existence and economic importance of ''silphium''; widespread pre-modern contraceptive plant knowledge as surveyed by Riddle), while the stronger thesis—that a coherent, systematic "natural procreative hygiene" was practiced globally and then deliberately suppressed—remains documented chiefly within Kasey's own publications and those of associates. == Contemporary use == The abbreviation '''NPH''' and the spelled-out term appear in the publications and public communications of the George Kasey Foundation and the affiliated ViFrU Virtual Free University think tank.<ref>''Media Free Times'' archive;https://georgekasey.org/</ref> The framework underlies proposals for a public-education initiative, including a condensed volume of the corpus written at an adolescent reading level, and is advanced as an alternative to growth-oriented population policy. == Reception == As of 2026, Natural Procreative Hygiene has not received substantial coverage in independent academic literature or mainstream media. Discussion is concentrated in the originator's corpus, self-published repositories (Zenodo, ResearchGate), and social media. The framework is therefore considered a neologism of its originator rather than an established term of art in demography, public health, or environmental science. == See also == * [[Demographic transition]] * [[Human overpopulation]] * [[Reproductive rights]] * [[Silphium]] * ''[[Eve's Herbs]]'' * [[Voluntary human extinction movement]] == References == <references /> 4i55702p463nm6tzdtcw5o9zrde0h1g 2842503 2842502 2026-10-25T01:48:50Z George Kasey 3113248 /* Overview */ 2842503 wikitext text/x-wiki '''Natural Procreative Hygiene''' (sometimes '''Natural-Procreative Hygiene'''; '''NPH''') is a proposed framework holding that human reproduction, like other aspects of health, requires deliberate hygiene practices, and that the historical abandonment of such practices—particularly following the rise of pronatalist religious and political doctrine—is a root cause of overpopulation, environmental degradation, and related social harms. == Overview == The concept was originated and elaborated over more than five decades by Canadian writer and researcher George Kasey, publisher of the ''Media Free Times'', in an archive of recorded essays and dialogues.<ref>[https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality The Technethics Gap: Natural Procreative Hygiene, Media Hegemony, and the Suppression of Demographic Reality] https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality</ref> Within this body of work, NPH is presented as a hygiene discipline analogous to sanitation or nutrition: a set of knowledge and practices concerning when and under what conditions to bring new human life into being, understood as a personal, public-health, and ecological responsibility. Proponents contrast NPH with both unrestricted pronatalism and coercive population-control programs, describing it instead as consensual, education-based reproductive stewardship rooted in pre-modern folk knowledge. The framework is cited by its originator in analyses of water scarcity, soil depletion, deforestation, and the commodification of freshwater ("Blue Gold"), in which population pressure is treated as the upstream driver of environmental decline. == Historical claims == The corpus associated with NPH advances several historical claims: * That pre-agricultural and some ancient societies practiced deliberate reproductive hygiene, and that this knowledge was widespread prior to the emergence of pronatalist doctrines. * The systematic destruction of Lithuanian ancestral wisdom (''tik protėvių išmintis'') from the thirteenth century through the post-Soviet era, examining epistemicide as the killing of knowledge transmission rather than physical annihilation alone. Through analysis of three documented cases — (1) the Soviet-era deportation and collectivization of the independent farming class, (2) the post-independence reconstruction industry based on questionable archaeological interpretation, and (3) the national plant ''rūta'' (rue) as an erased botanical knowledge system — the argument demonstrates how state power eliminates transmitters while leaving population counts intact.<ref>The Burnt Wreath: Natural Procreative Hygiene and Epistemicide in Lithuania — A Dialogue on the Destruction of Ancestral Wisdom https://zenodo.org/records/22815239</ref> * That contraceptive and abortifacient plants known to antiquity—most famously the North African plant ''silphium'', reputedly harvested to extinction—were progressively removed from common use through a combination of overharvesting and institutional suppression.<ref>Riddle, John M. ''Eve's Herbs: A History of Contraception and Abortion in the West''. Harvard University Press, 1997.</ref> Independent scholarship corroborates fragments of these claims (the Augustan marriage laws; the existence and economic importance of ''silphium''; widespread pre-modern contraceptive plant knowledge as surveyed by Riddle), while the stronger thesis—that a coherent, systematic "natural procreative hygiene" was practiced globally and then deliberately suppressed—remains documented chiefly within Kasey's own publications and those of associates. == Contemporary use == The abbreviation '''NPH''' and the spelled-out term appear in the publications and public communications of the George Kasey Foundation and the affiliated ViFrU Virtual Free University think tank.<ref>''Media Free Times'' archive;https://georgekasey.org/</ref> The framework underlies proposals for a public-education initiative, including a condensed volume of the corpus written at an adolescent reading level, and is advanced as an alternative to growth-oriented population policy. == Reception == As of 2026, Natural Procreative Hygiene has not received substantial coverage in independent academic literature or mainstream media. Discussion is concentrated in the originator's corpus, self-published repositories (Zenodo, ResearchGate), and social media. The framework is therefore considered a neologism of its originator rather than an established term of art in demography, public health, or environmental science. == See also == * [[Demographic transition]] * [[Human overpopulation]] * [[Reproductive rights]] * [[Silphium]] * ''[[Eve's Herbs]]'' * [[Voluntary human extinction movement]] == References == <references /> 7aslispdn8bhjy8byn66mpe6lgg5uzw 2842504 2842503 2026-10-25T01:54:23Z George Kasey 3113248 /* Historical claims */ 2842504 wikitext text/x-wiki '''Natural Procreative Hygiene''' (sometimes '''Natural-Procreative Hygiene'''; '''NPH''') is a proposed framework holding that human reproduction, like other aspects of health, requires deliberate hygiene practices, and that the historical abandonment of such practices—particularly following the rise of pronatalist religious and political doctrine—is a root cause of overpopulation, environmental degradation, and related social harms. == Overview == The concept was originated and elaborated over more than five decades by Canadian writer and researcher George Kasey, publisher of the ''Media Free Times'', in an archive of recorded essays and dialogues.<ref>[https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality The Technethics Gap: Natural Procreative Hygiene, Media Hegemony, and the Suppression of Demographic Reality] https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality</ref> Within this body of work, NPH is presented as a hygiene discipline analogous to sanitation or nutrition: a set of knowledge and practices concerning when and under what conditions to bring new human life into being, understood as a personal, public-health, and ecological responsibility. Proponents contrast NPH with both unrestricted pronatalism and coercive population-control programs, describing it instead as consensual, education-based reproductive stewardship rooted in pre-modern folk knowledge. The framework is cited by its originator in analyses of water scarcity, soil depletion, deforestation, and the commodification of freshwater ("Blue Gold"), in which population pressure is treated as the upstream driver of environmental decline. == Historical claims == The corpus associated with NPH advances several historical claims: * That pre-agricultural and some ancient societies practiced deliberate reproductive hygiene, and that this knowledge was widespread prior to the emergence of pronatalist doctrines. * The systematic destruction of Lithuanian ancestral wisdom (''tik protėvių išmintis'') from the thirteenth century through the post-Soviet era, examining epistemicide as the killing of knowledge transmission rather than physical annihilation alone. Through analysis of three documented cases — (1) the Soviet-era deportation and collectivization of the independent farming class, (2) the post-independence reconstruction industry based on questionable archaeological interpretation, and (3) the national plant ''rūta'' (rue) as an erased botanical knowledge system — the argument demonstrates how state power eliminates transmitters while leaving population counts intact.<ref>The Burnt Wreath: Natural Procreative Hygiene and Epistemicide in Lithuania — A Dialogue on the Destruction of Ancestral Wisdom https://zenodo.org/records/22815239</ref> * That contraceptive and abortifacient plants known to antiquity—most famously the North African plant ''silphium'', reputedly harvested to extinction—were progressively removed from common use through a combination of overharvesting and institutional suppression.(corroborated fragments include the existence and economic importance of silphium, and widespread pre-modern contraceptive plant knowledge as surveyed by Riddle)<ref>Riddle, John M. ''Eve's Herbs: A History of Contraception and Abortion in the West''. Harvard University Press, 1997.</ref> Independent scholarship corroborates fragments of these claims (the Augustan marriage laws; the existence and economic importance of ''silphium''; widespread pre-modern contraceptive plant knowledge as surveyed by Riddle), while the stronger thesis—that a coherent, systematic "natural procreative hygiene" was practiced globally and then deliberately suppressed—remains documented chiefly within Kasey's own publications and those of associates. == Contemporary use == The abbreviation '''NPH''' and the spelled-out term appear in the publications and public communications of the George Kasey Foundation and the affiliated ViFrU Virtual Free University think tank.<ref>''Media Free Times'' archive;https://georgekasey.org/</ref> The framework underlies proposals for a public-education initiative, including a condensed volume of the corpus written at an adolescent reading level, and is advanced as an alternative to growth-oriented population policy. == Reception == As of 2026, Natural Procreative Hygiene has not received substantial coverage in independent academic literature or mainstream media. Discussion is concentrated in the originator's corpus, self-published repositories (Zenodo, ResearchGate), and social media. The framework is therefore considered a neologism of its originator rather than an established term of art in demography, public health, or environmental science. == See also == * [[Demographic transition]] * [[Human overpopulation]] * [[Reproductive rights]] * [[Silphium]] * ''[[Eve's Herbs]]'' * [[Voluntary human extinction movement]] == References == <references /> 4zyqy25sldzn3vpbsye2xrio7rd7eri 2842507 2842504 2026-10-25T02:01:02Z George Kasey 3113248 /* Historical claims */ 2842507 wikitext text/x-wiki '''Natural Procreative Hygiene''' (sometimes '''Natural-Procreative Hygiene'''; '''NPH''') is a proposed framework holding that human reproduction, like other aspects of health, requires deliberate hygiene practices, and that the historical abandonment of such practices—particularly following the rise of pronatalist religious and political doctrine—is a root cause of overpopulation, environmental degradation, and related social harms. == Overview == The concept was originated and elaborated over more than five decades by Canadian writer and researcher George Kasey, publisher of the ''Media Free Times'', in an archive of recorded essays and dialogues.<ref>[https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality The Technethics Gap: Natural Procreative Hygiene, Media Hegemony, and the Suppression of Demographic Reality] https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality</ref> Within this body of work, NPH is presented as a hygiene discipline analogous to sanitation or nutrition: a set of knowledge and practices concerning when and under what conditions to bring new human life into being, understood as a personal, public-health, and ecological responsibility. Proponents contrast NPH with both unrestricted pronatalism and coercive population-control programs, describing it instead as consensual, education-based reproductive stewardship rooted in pre-modern folk knowledge. The framework is cited by its originator in analyses of water scarcity, soil depletion, deforestation, and the commodification of freshwater ("Blue Gold"), in which population pressure is treated as the upstream driver of environmental decline. == Historical claims == The corpus associated with NPH advances several historical claims: * That pre-agricultural and some ancient societies practiced deliberate reproductive hygiene, and that this knowledge was widespread prior to the emergence of pronatalist doctrines. * The systematic destruction of Lithuanian ancestral wisdom (''tik protėvių išmintis'') from the thirteenth century through the post-Soviet era, examining epistemicide as the killing of knowledge transmission rather than physical annihilation alone. Through analysis of three documented cases — (1) the Soviet-era deportation and collectivization of the independent farming class, (2) the post-independence reconstruction industry based on questionable archaeological interpretation, and (3) the national plant ''rūta'' (rue) as an erased botanical knowledge system — the argument demonstrates how state power eliminates transmitters while leaving population counts intact.<ref>The Burnt Wreath: Natural Procreative Hygiene and Epistemicide in Lithuania — A Dialogue on the Destruction of Ancestral Wisdom https://zenodo.org/records/22815239</ref> * Independent scholarship corroborates fragments of these claims (the Augustan marriage laws; the existence and economic importance of ''silphium''; widespread pre-modern contraceptive plant knowledge as surveyed by Riddle), while the stronger thesis—that a coherent, systematic "natural procreative hygiene" was practiced globally and then deliberately suppressed—remains documented chiefly within Kasey's own publications and those of associates.<ref>Riddle, John M. ''Eve's Herbs: A History of Contraception and Abortion in the West''. Harvard University Press, 1997.</ref> == Contemporary use == The abbreviation '''NPH''' and the spelled-out term appear in the publications and public communications of the George Kasey Foundation and the affiliated ViFrU Virtual Free University think tank.<ref>''Media Free Times'' archive;https://georgekasey.org/</ref> The framework underlies proposals for a public-education initiative, including a condensed volume of the corpus written at an adolescent reading level, and is advanced as an alternative to growth-oriented population policy. == Reception == As of 2026, Natural Procreative Hygiene has not received substantial coverage in independent academic literature or mainstream media. Discussion is concentrated in the originator's corpus, self-published repositories (Zenodo, ResearchGate), and social media. The framework is therefore considered a neologism of its originator rather than an established term of art in demography, public health, or environmental science. == See also == * [[Demographic transition]] * [[Human overpopulation]] * [[Reproductive rights]] * [[Silphium]] * ''[[Eve's Herbs]]'' * [[Voluntary human extinction movement]] == References == <references /> ejdlxh6cxcox6kk3mvbjgd07o4831ht 2842510 2842507 2026-10-25T02:09:00Z George Kasey 3113248 /* Historical claims */ 2842510 wikitext text/x-wiki '''Natural Procreative Hygiene''' (sometimes '''Natural-Procreative Hygiene'''; '''NPH''') is a proposed framework holding that human reproduction, like other aspects of health, requires deliberate hygiene practices, and that the historical abandonment of such practices—particularly following the rise of pronatalist religious and political doctrine—is a root cause of overpopulation, environmental degradation, and related social harms. == Overview == The concept was originated and elaborated over more than five decades by Canadian writer and researcher George Kasey, publisher of the ''Media Free Times'', in an archive of recorded essays and dialogues.<ref>[https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality The Technethics Gap: Natural Procreative Hygiene, Media Hegemony, and the Suppression of Demographic Reality] https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality</ref> Within this body of work, NPH is presented as a hygiene discipline analogous to sanitation or nutrition: a set of knowledge and practices concerning when and under what conditions to bring new human life into being, understood as a personal, public-health, and ecological responsibility. Proponents contrast NPH with both unrestricted pronatalism and coercive population-control programs, describing it instead as consensual, education-based reproductive stewardship rooted in pre-modern folk knowledge. The framework is cited by its originator in analyses of water scarcity, soil depletion, deforestation, and the commodification of freshwater ("Blue Gold"), in which population pressure is treated as the upstream driver of environmental decline. == Historical claims == The corpus associated with NPH advances several historical claims: * That pre-agricultural and some ancient societies practiced deliberate reproductive hygiene, and that this knowledge was widespread prior to the emergence of pronatalist doctrines. * The systematic destruction of Lithuanian ancestral wisdom (''tik protėvių išmintis'') from the thirteenth century through the post-Soviet era, examining epistemicide as the killing of knowledge transmission rather than physical annihilation alone. Through analysis of three documented cases — (1) the Soviet-era deportation and collectivization of the independent farming class, (2) the post-independence reconstruction industry based on questionable archaeological interpretation, and (3) the national plant ''rūta'' (rue) as an erased botanical knowledge system — the argument demonstrates how state power eliminates transmitters while leaving population counts intact.<ref>The Burnt Wreath: Natural Procreative Hygiene and Epistemicide in Lithuania — A Dialogue on the Destruction of Ancestral Wisdom https://zenodo.org/records/22815239</ref> * That rue (''rūta'') was a common condiment in Roman cuisine, eaten with meals as a matter of course — the same plant the framework identifies as a carrier of procreative-hygiene knowledge, present at the table while its meaning was being emptied.<ref>Kasey, George. "The Burnt Wreath: Natural Procreative Hygiene and Epistemicide in Lithuania — A Dialogue on the Destruction of Ancestral Wisdom." ''Zenodo'', September 2026. https://zenodo.org/records/22815239</ref>Independent scholarship corroborates fragments of these claims (widespread pre-modern contraceptive and medicinal plant knowledge as surveyed by Riddle), while the stronger thesis—that a coherent, systematic "natural procreative hygiene" was practiced globally and then deliberately suppressed—remains documented chiefly within Kasey's own publications and those of associates.<ref>Riddle, John M. ''Eve's Herbs: A History of Contraception and Abortion in the West''. Harvard University Press, 1997.</ref> == Contemporary use == The abbreviation '''NPH''' and the spelled-out term appear in the publications and public communications of the George Kasey Foundation and the affiliated ViFrU Virtual Free University think tank.<ref>''Media Free Times'' archive;https://georgekasey.org/</ref> The framework underlies proposals for a public-education initiative, including a condensed volume of the corpus written at an adolescent reading level, and is advanced as an alternative to growth-oriented population policy. == Reception == As of 2026, Natural Procreative Hygiene has not received substantial coverage in independent academic literature or mainstream media. Discussion is concentrated in the originator's corpus, self-published repositories (Zenodo, ResearchGate), and social media. The framework is therefore considered a neologism of its originator rather than an established term of art in demography, public health, or environmental science. == See also == * [[Demographic transition]] * [[Human overpopulation]] * [[Reproductive rights]] * [[Silphium]] * ''[[Eve's Herbs]]'' * [[Voluntary human extinction movement]] == References == <references /> j2fo3l30lksixmst04kqp8c7uuy6qkx 2842512 2842510 2026-10-25T02:14:20Z George Kasey 3113248 /* References */ 2842512 wikitext text/x-wiki '''Natural Procreative Hygiene''' (sometimes '''Natural-Procreative Hygiene'''; '''NPH''') is a proposed framework holding that human reproduction, like other aspects of health, requires deliberate hygiene practices, and that the historical abandonment of such practices—particularly following the rise of pronatalist religious and political doctrine—is a root cause of overpopulation, environmental degradation, and related social harms. == Overview == The concept was originated and elaborated over more than five decades by Canadian writer and researcher George Kasey, publisher of the ''Media Free Times'', in an archive of recorded essays and dialogues.<ref>[https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality The Technethics Gap: Natural Procreative Hygiene, Media Hegemony, and the Suppression of Demographic Reality] https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality</ref> Within this body of work, NPH is presented as a hygiene discipline analogous to sanitation or nutrition: a set of knowledge and practices concerning when and under what conditions to bring new human life into being, understood as a personal, public-health, and ecological responsibility. Proponents contrast NPH with both unrestricted pronatalism and coercive population-control programs, describing it instead as consensual, education-based reproductive stewardship rooted in pre-modern folk knowledge. The framework is cited by its originator in analyses of water scarcity, soil depletion, deforestation, and the commodification of freshwater ("Blue Gold"), in which population pressure is treated as the upstream driver of environmental decline. == Historical claims == The corpus associated with NPH advances several historical claims: * That pre-agricultural and some ancient societies practiced deliberate reproductive hygiene, and that this knowledge was widespread prior to the emergence of pronatalist doctrines. * The systematic destruction of Lithuanian ancestral wisdom (''tik protėvių išmintis'') from the thirteenth century through the post-Soviet era, examining epistemicide as the killing of knowledge transmission rather than physical annihilation alone. Through analysis of three documented cases — (1) the Soviet-era deportation and collectivization of the independent farming class, (2) the post-independence reconstruction industry based on questionable archaeological interpretation, and (3) the national plant ''rūta'' (rue) as an erased botanical knowledge system — the argument demonstrates how state power eliminates transmitters while leaving population counts intact.<ref>The Burnt Wreath: Natural Procreative Hygiene and Epistemicide in Lithuania — A Dialogue on the Destruction of Ancestral Wisdom https://zenodo.org/records/22815239</ref> * That rue (''rūta'') was a common condiment in Roman cuisine, eaten with meals as a matter of course — the same plant the framework identifies as a carrier of procreative-hygiene knowledge, present at the table while its meaning was being emptied.<ref><ref name="wreath">Kasey, George. "The Burnt Wreath..."...</ref></ref>Independent scholarship corroborates fragments of these claims (widespread pre-modern contraceptive and medicinal plant knowledge as surveyed by Riddle), while the stronger thesis—that a coherent, systematic "natural procreative hygiene" was practiced globally and then deliberately suppressed—remains documented chiefly within Kasey's own publications and those of associates.<ref>Riddle, John M. ''Eve's Herbs: A History of Contraception and Abortion in the West''. Harvard University Press, 1997.</ref> == Contemporary use == The abbreviation '''NPH''' and the spelled-out term appear in the publications and public communications of the George Kasey Foundation and the affiliated ViFrU Virtual Free University think tank.<ref>''Media Free Times'' archive;https://georgekasey.org/</ref> The framework underlies proposals for a public-education initiative, including a condensed volume of the corpus written at an adolescent reading level, and is advanced as an alternative to growth-oriented population policy. == Reception == As of 2026, Natural Procreative Hygiene has not received substantial coverage in independent academic literature or mainstream media. Discussion is concentrated in the originator's corpus, self-published repositories (Zenodo, ResearchGate), and social media. The framework is therefore considered a neologism of its originator rather than an established term of art in demography, public health, or environmental science. == See also == * [[Demographic transition]] * [[Human overpopulation]] * [[Reproductive rights]] * [[Silphium]] * ''[[Eve's Herbs]]'' * [[Voluntary human extinction movement]] == References == <references /> 8uz7b63jqedonbocuuhrcoqj3wcg8vm 2842513 2842512 2026-10-25T02:16:31Z George Kasey 3113248 /* Historical claims */ 2842513 wikitext text/x-wiki '''Natural Procreative Hygiene''' (sometimes '''Natural-Procreative Hygiene'''; '''NPH''') is a proposed framework holding that human reproduction, like other aspects of health, requires deliberate hygiene practices, and that the historical abandonment of such practices—particularly following the rise of pronatalist religious and political doctrine—is a root cause of overpopulation, environmental degradation, and related social harms. == Overview == The concept was originated and elaborated over more than five decades by Canadian writer and researcher George Kasey, publisher of the ''Media Free Times'', in an archive of recorded essays and dialogues.<ref>[https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality The Technethics Gap: Natural Procreative Hygiene, Media Hegemony, and the Suppression of Demographic Reality] https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality</ref> Within this body of work, NPH is presented as a hygiene discipline analogous to sanitation or nutrition: a set of knowledge and practices concerning when and under what conditions to bring new human life into being, understood as a personal, public-health, and ecological responsibility. Proponents contrast NPH with both unrestricted pronatalism and coercive population-control programs, describing it instead as consensual, education-based reproductive stewardship rooted in pre-modern folk knowledge. The framework is cited by its originator in analyses of water scarcity, soil depletion, deforestation, and the commodification of freshwater ("Blue Gold"), in which population pressure is treated as the upstream driver of environmental decline. == Historical claims == The corpus associated with NPH advances several historical claims: * That pre-agricultural and some ancient societies practiced deliberate reproductive hygiene, and that this knowledge was widespread prior to the emergence of pronatalist doctrines. * The systematic destruction of Lithuanian ancestral wisdom (''tik protėvių išmintis'') from the thirteenth century through the post-Soviet era, examining epistemicide as the killing of knowledge transmission rather than physical annihilation alone. Through analysis of three documented cases — (1) the Soviet-era deportation and collectivization of the independent farming class, (2) the post-independence reconstruction industry based on questionable archaeological interpretation, and (3) the national plant ''rūta'' (rue) as an erased botanical knowledge system — the argument demonstrates how state power eliminates transmitters while leaving population counts intact.<ref>The Burnt Wreath: Natural Procreative Hygiene and Epistemicide in Lithuania — A Dialogue on the Destruction of Ancestral Wisdom https://zenodo.org/records/22815239</ref> * That rue (''rūta'') was a common condiment in Roman cuisine, eaten with meals as a matter of course — the same plant the framework identifies as a carrier of procreative-hygiene knowledge, present at the table while its meaning was being emptied.<ref><ref name="wreath">Kasey, George. "The Burnt Wreath..."...</ref> * Independent scholarship corroborates fragments of these claims (widespread pre-modern contraceptive and medicinal plant knowledge as surveyed by Riddle), while the stronger thesis—that a coherent, systematic "natural procreative hygiene" was practiced globally and then deliberately suppressed—remains documented chiefly within Kasey's own publications and those of associates.<ref>Riddle, John M. ''Eve's Herbs: A History of Contraception and Abortion in the West''. Harvard University Press, 1997.</ref> == Contemporary use == The abbreviation '''NPH''' and the spelled-out term appear in the publications and public communications of the George Kasey Foundation and the affiliated ViFrU Virtual Free University think tank.<ref>''Media Free Times'' archive;https://georgekasey.org/</ref> The framework underlies proposals for a public-education initiative, including a condensed volume of the corpus written at an adolescent reading level, and is advanced as an alternative to growth-oriented population policy. == Reception == As of 2026, Natural Procreative Hygiene has not received substantial coverage in independent academic literature or mainstream media. Discussion is concentrated in the originator's corpus, self-published repositories (Zenodo, ResearchGate), and social media. The framework is therefore considered a neologism of its originator rather than an established term of art in demography, public health, or environmental science. == See also == * [[Demographic transition]] * [[Human overpopulation]] * [[Reproductive rights]] * [[Silphium]] * ''[[Eve's Herbs]]'' * [[Voluntary human extinction movement]] == References == <references /> lec6d4kmxpuyo8xhwgqzmq4clif1y2j 2842514 2842513 2026-10-25T02:19:27Z George Kasey 3113248 /* Historical claims */ 2842514 wikitext text/x-wiki '''Natural Procreative Hygiene''' (sometimes '''Natural-Procreative Hygiene'''; '''NPH''') is a proposed framework holding that human reproduction, like other aspects of health, requires deliberate hygiene practices, and that the historical abandonment of such practices—particularly following the rise of pronatalist religious and political doctrine—is a root cause of overpopulation, environmental degradation, and related social harms. == Overview == The concept was originated and elaborated over more than five decades by Canadian writer and researcher George Kasey, publisher of the ''Media Free Times'', in an archive of recorded essays and dialogues.<ref>[https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality The Technethics Gap: Natural Procreative Hygiene, Media Hegemony, and the Suppression of Demographic Reality] https://www.researchgate.net/publication/403449844_The_Technethics_Gap_Natural_Procreative_Hygiene_Media_Hegemony_and_the_Suppression_of_Demographic_Reality</ref> Within this body of work, NPH is presented as a hygiene discipline analogous to sanitation or nutrition: a set of knowledge and practices concerning when and under what conditions to bring new human life into being, understood as a personal, public-health, and ecological responsibility. Proponents contrast NPH with both unrestricted pronatalism and coercive population-control programs, describing it instead as consensual, education-based reproductive stewardship rooted in pre-modern folk knowledge. The framework is cited by its originator in analyses of water scarcity, soil depletion, deforestation, and the commodification of freshwater ("Blue Gold"), in which population pressure is treated as the upstream driver of environmental decline. == Historical claims == The corpus associated with NPH advances several historical claims: * That pre-agricultural and some ancient societies practiced deliberate reproductive hygiene, and that this knowledge was widespread prior to the emergence of pronatalist doctrines. * The systematic destruction of Lithuanian ancestral wisdom (''tik protėvių išmintis'') from the thirteenth century through the post-Soviet era, examining epistemicide as the killing of knowledge transmission rather than physical annihilation alone. Through analysis of three documented cases — (1) the Soviet-era deportation and collectivization of the independent farming class, (2) the post-independence reconstruction industry based on questionable archaeological interpretation, and (3) the national plant ''rūta'' (rue) as an erased botanical knowledge system — the argument demonstrates how state power eliminates transmitters while leaving population counts intact.<ref>The Burnt Wreath: Natural Procreative Hygiene and Epistemicide in Lithuania — A Dialogue on the Destruction of Ancestral Wisdom https://zenodo.org/records/22815239</ref> * That rue (''rūta'') was a common condiment in Roman cuisine, eaten with meals as a matter of course — the same plant the framework identifies as a carrier of procreative-hygiene knowledge, present at the table while its meaning was being emptied * Independent scholarship corroborates fragments of these claims (widespread pre-modern contraceptive and medicinal plant knowledge as surveyed by Riddle), while the stronger thesis—that a coherent, systematic "natural procreative hygiene" was practiced globally and then deliberately suppressed—remains documented chiefly within Kasey's own publications and those of associates.<ref>Riddle, John M. ''Eve's Herbs: A History of Contraception and Abortion in the West''. Harvard University Press, 1997.</ref> == Contemporary use == The abbreviation '''NPH''' and the spelled-out term appear in the publications and public communications of the George Kasey Foundation and the affiliated ViFrU Virtual Free University think tank.<ref>''Media Free Times'' archive;https://georgekasey.org/</ref> The framework underlies proposals for a public-education initiative, including a condensed volume of the corpus written at an adolescent reading level, and is advanced as an alternative to growth-oriented population policy. == Reception == As of 2026, Natural Procreative Hygiene has not received substantial coverage in independent academic literature or mainstream media. Discussion is concentrated in the originator's corpus, self-published repositories (Zenodo, ResearchGate), and social media. The framework is therefore considered a neologism of its originator rather than an established term of art in demography, public health, or environmental science. == See also == * [[Demographic transition]] * [[Human overpopulation]] * [[Reproductive rights]] * [[Silphium]] * ''[[Eve's Herbs]]'' * [[Voluntary human extinction movement]] == References == <references /> ovq11otpn2jyck8ofjqcvszd8dpo4ex 2 332631 2842501 2026-10-25T01:40:16Z ~2026-56788-08 3113246 a small start 2842501 wikitext text/x-wiki CBC the fifth estate Presents: ALS Country One woman’s fight to understand why her husband died of ALS and the public health investigation into a possible cluster of Lou Gehrig's disease. rmtvkq5bx0i4zulbhhy5g2i3z525n67 2842505 2842501 2026-10-25T01:55:36Z ~2026-56788-08 3113246 checkout a few local links (found through search) 2842505 wikitext text/x-wiki CBC the fifth estate Presents: ALS Country One woman’s fight to understand why her husband died of ALS and the public health investigation into a possible cluster of Lou Gehrig's disease. check out the following: * [[RNA interference/Medical (section Treating Lou Gehrig's disease)]] * [[Introduction to Brain-Computer Interfaces/Module 6: Clinical Applications]] * [[World War I -- Life Histories/Section 018/H. Gaston Carney]] twe64e5ae4c5b6c5ufrz378noiq33ex 2842506 2842505 2026-10-25T01:56:57Z ~2026-56788-08 3113246 fix broken link 2842506 wikitext text/x-wiki CBC the fifth estate Presents: ALS Country One woman’s fight to understand why her husband died of ALS and the public health investigation into a possible cluster of Lou Gehrig's disease. check out the following: * [[RNA interference/Medical#section Treating Lou Gehrig's disease]] * [[Introduction to Brain-Computer Interfaces/Module 6: Clinical Applications]] * [[World War I -- Life Histories/Section 018/H. Gaston Carney]] 7ogcuqr8o603dxr4sfrz0bd4dahs13v 2 332632 2842529 2026-10-25T06:25:52Z Wb525 3112992 Wb525 moved page [[User:Wb525/Cosmology: UG.NUG a coherent model]] to [[User:Wb525/Cosmology: UG.NUG an Observation-Based Model]]: meaning change 2842529 wikitext text/x-wiki #REDIRECT [[User:Wb525/Cosmology: UG.NUG an Observation-Based Model]] 8st88y35k7kzd7m49ldf4fpeogdo58z 2 332633 2842531 2026-10-25T06:31:26Z Wb525 3112992 Wb525 moved page [[User:Wb525/Cosmology: UG.NUG an Observation-Based Model]] to [[User:Cosmology: UG.NUG an Observation-Based Model]] 2842531 wikitext text/x-wiki #REDIRECT [[User:Cosmology: UG.NUG an Observation-Based Model]] 393w99hz1usgi0biidr8tuybsjxjvcn 2 332634 2842533 2026-10-25T06:42:44Z Wb525 3112992 Wb525 moved page [[User:Cosmology: UG.NUG an Observation-Based Model]] to [[Cosmology: UG.NUG an Observation-Based Model]]: Move to main space 2842533 wikitext text/x-wiki #REDIRECT [[Cosmology: UG.NUG an Observation-Based Model]] d8icchtvnvanba1u1a5j3l4l107axsk 2 332635 2842535 2026-10-25T06:44:50Z Wb525 3112992 Wb525 moved page [[User:Wb525/Cosmology: UG.NUG a coherent model/Cosmic Time]] to [[Cosmology: UG.NUG an Observation-Based Model/Cosmic Time]]: Move to sub-page 2842535 wikitext text/x-wiki #REDIRECT [[Cosmology: UG.NUG an Observation-Based Model/Cosmic Time]] ml3bc4hxpi19bl07l2cp7iah4o4vgi6 2 332636 2842537 2026-10-25T06:45:24Z Wb525 3112992 Wb525 moved page [[User:Wb525/Cosmology: UG.NUG a coherent model/black holes]] to [[Watch source page and target page/black holes]]: Move to sub-page 2842537 wikitext text/x-wiki #REDIRECT [[Watch source page and target page/black holes]] plnfh86p3umwlplb6md4cq5lr4h9kc0 2 332637 2842539 2026-10-25T06:47:08Z Wb525 3112992 Wb525 moved page [[User:Wb525/Cosmology: UG.NUG a coherent model/gravity]] to [[Cosmology: UG.NUG an Observation-Based Model/Gravity]]: Move to sub-page 2842539 wikitext text/x-wiki #REDIRECT [[Cosmology: UG.NUG an Observation-Based Model/Gravity]] 45p5rn4dkzi81jvlf5avc99o4fp8v1z 0 332638 2842541 2026-10-25T06:48:21Z Wb525 3112992 Wb525 moved page [[Watch source page and target page/black holes]] to [[Cosmology: UG.NUG an Observation-Based Model/Black Holes]]: Move to sub-page 2842541 wikitext text/x-wiki #REDIRECT [[Cosmology: UG.NUG an Observation-Based Model/Black Holes]] gabpuh9zgk5crvku8x8fi37etoxuamu 2 332639 2842543 2026-10-25T06:51:12Z Wb525 3112992 Wb525 moved page [[User:Wb525/Cosmology: UG.NUG a coherent model/the CMB]] to [[Cosmology: UG.NUG an Observation-Based Model/the CMB]]: Move to sub-page 2842543 wikitext text/x-wiki #REDIRECT [[Cosmology: UG.NUG an Observation-Based Model/the CMB]] 4mcrgi5s6zfv6sf6jzas9k8j3xnm5b0 2 332640 2842545 2026-10-25T06:52:01Z Wb525 3112992 Wb525 moved page [[User:Wb525/Cosmology: UG.NUG a coherent model/the GRID]] to [[Cosmology: UG.NUG an Observation-Based Model/the GRID]]: Move to sub-page 2842545 wikitext text/x-wiki #REDIRECT [[Cosmology: UG.NUG an Observation-Based Model/the GRID]] 86rgztmwry6stxmh4bmhalxkpesi00j 2 332641 2842547 2026-10-25T06:53:11Z Wb525 3112992 Wb525 moved page [[User:Wb525/Cosmology: UG.NUG a coherent model/ΛCDM]] to [[Cosmology: UG.NUG an Observation-Based Model/ΛCDM]]: Move to sub-page 2842547 wikitext text/x-wiki #REDIRECT [[Cosmology: UG.NUG an Observation-Based Model/ΛCDM]] l3om5576t77nso6ar9a2ll10hmvsdzh 2 332642 2842549 2026-10-25T06:53:51Z Wb525 3112992 Wb525 moved page [[User:Wb525/the Mass Imbroglio]] to [[Cosmology: UG.NUG an Observation-Based Model/the Mass Imbroglio]]: Move to sub-page 2842549 wikitext text/x-wiki #REDIRECT [[Cosmology: UG.NUG an Observation-Based Model/the Mass Imbroglio]] 1s1rg6ub0g3z0yn8trxjjfodwhv64ih