Wikiversity enwikiversity https://en.wikiversity.org/wiki/Wikiversity:Main_Page MediaWiki 1.48.0-wmf.3 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 2844503 2843373 2026-11-04T18:24:39Z ~2026-56788-08 3113246 /* Blocked users on wikimedia projects */ new section 2844503 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) ::Yes, that was my intent. I agree that a nom should request a duration with a "corresponding plan of work." [[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 04:17, 28 October 2026 (UTC) * {{support}} sensible proposal. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 15:20, 29 October 2026 (UTC) === Temporary rights extention === I propose that we clarify that temporary rights can be extended. If the community approves of an extension, it can be granted more than once. The temporary rights can repeatedly be extended, for new temporary durations. --[[User:Mu301|mikeu]] ^{[[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) == 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|2026-10-28|en}}'''. The switch will start at '''[https://zonestamp.toolforge.org/{{#time:U|2026-10-28T15:00|en}} {{#time:H:i e|2026-10-28T15:00}}]'''. Unfortunately, because of some limitations in [[mw:Special:MyLanguage/Manual:What is MediaWiki?|MediaWiki]], all editing must stop while the switch is made. We apologize for this disruption, and we are working to minimize it in the future. A banner will be displayed on all wikis 30 minutes before this operation happens. This banner will remain visible until the end of the operation. 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|2026-10-28|en}}. * If you try to edit or save during these times, you will see an error message. We hope that no edits will be lost during these minutes, but we can't guarantee it. If you see the error message, then please wait until everything is back to normal. Then you should be able to save your edit. But, we recommend that you make a copy of your changes first, just in case. ''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"/> [[user:Trizek_(WMF)|Trizek_(WMF)]] 14:00, 26 October 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 --> == Happy birthday to us == I just realized that en-wv was opened August 15, 2006. We've reached the 20 year mark! See [[Wikiversity:History of Wikiversity|History of Wikiversity]]. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 19:01, 28 October 2026 (UTC) == Blocked users on wikimedia projects == During my almost 20 years "editing" on various wikimedia wikis I have seen many users being blocked. Some are blocked for short periods, other indefinitely (or infinitely). In most cases a blocking admin provides a rationale for the blocking such as spamming, paid editing, etc. Some of the reasons for blocking are well documented on the English Wikipedia, b ut some are pretty obscure. I have not been contributing to this project on a regular basis', so am not sure if this is a topic that should be discussed at all at the English Wikiversity. I hope that there are others who share my interest. I believe that not allowing a discussion of this topic will end up on another website, and will draw attention away from wikimedia which is already suffering the loss of many contributions, human and financial. Any thoughts? [[Special:Contributions/~2026-56788-08|~2026-56788-08]] ([[User talk:~2026-56788-08|talk]]) 18:24, 4 November 2026 (UTC) 66owtzjqujs6t4dcmtg7jixukx9oep2 2844564 2844503 2026-11-05T02:15:45Z ~2026-56788-08 3113246 /* Blocked users on wikimedia projects */ clarification? 2844564 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) ::Yes, that was my intent. I agree that a nom should request a duration with a "corresponding plan of work." [[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 04:17, 28 October 2026 (UTC) * {{support}} sensible proposal. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 15:20, 29 October 2026 (UTC) === Temporary rights extention === I propose that we clarify that temporary rights can be extended. If the community approves of an extension, it can be granted more than once. The temporary rights can repeatedly be extended, for new temporary durations. --[[User:Mu301|mikeu]] ^{[[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) == 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|2026-10-28|en}}'''. The switch will start at '''[https://zonestamp.toolforge.org/{{#time:U|2026-10-28T15:00|en}} {{#time:H:i e|2026-10-28T15:00}}]'''. Unfortunately, because of some limitations in [[mw:Special:MyLanguage/Manual:What is MediaWiki?|MediaWiki]], all editing must stop while the switch is made. We apologize for this disruption, and we are working to minimize it in the future. A banner will be displayed on all wikis 30 minutes before this operation happens. This banner will remain visible until the end of the operation. 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|2026-10-28|en}}. * If you try to edit or save during these times, you will see an error message. We hope that no edits will be lost during these minutes, but we can't guarantee it. If you see the error message, then please wait until everything is back to normal. Then you should be able to save your edit. But, we recommend that you make a copy of your changes first, just in case. ''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"/> [[user:Trizek_(WMF)|Trizek_(WMF)]] 14:00, 26 October 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 --> == Happy birthday to us == I just realized that en-wv was opened August 15, 2006. We've reached the 20 year mark! See [[Wikiversity:History of Wikiversity|History of Wikiversity]]. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 19:01, 28 October 2026 (UTC) == Blocked users on wikimedia projects == During my almost 20 years "editing" on various wikimedia wikis I have seen many users being blocked. Some are blocked for short periods, others indefinitely (or infinitely). In most cases a blocking admin provides a rationale for the blocking such as vandalizing, spamming, paid editing, etc. Some of the reasons for blocking are well documented on the English Wikipedia, but some reasons are obscure. I have not been contributing to this project on a regular basis', so am not sure if this is a topic that should be discussed at all at the English Wikiversity. I hope that there are others who share my interest. I believe that not allowing a discussion of this topic will end up on another website, and will draw attention away from wikimedia which is already suffering the loss of many contributions, human and financial. Any thoughts? [[Special:Contributions/~2026-56788-08|~2026-56788-08]] ([[User talk:~2026-56788-08|talk]]) 18:24, 4 November 2026 (UTC) some minor changes [[Special:Contributions/~2026-56788-08|~2026-56788-08]] ([[User talk:~2026-56788-08|talk]]) 02:15, 5 November 2026 (UTC) oyjdtfhkawton72badjrviodz2pouub 2844603 2844564 2026-11-05T11:24:27Z Theologiministry89 3112580 /* Mathematical theology */ new section 2844603 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) ::Yes, that was my intent. I agree that a nom should request a duration with a "corresponding plan of work." [[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 04:17, 28 October 2026 (UTC) * {{support}} sensible proposal. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 15:20, 29 October 2026 (UTC) === Temporary rights extention === I propose that we clarify that temporary rights can be extended. If the community approves of an extension, it can be granted more than once. The temporary rights can repeatedly be extended, for new temporary durations. --[[User:Mu301|mikeu]] ^{[[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) == 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|2026-10-28|en}}'''. The switch will start at '''[https://zonestamp.toolforge.org/{{#time:U|2026-10-28T15:00|en}} {{#time:H:i e|2026-10-28T15:00}}]'''. Unfortunately, because of some limitations in [[mw:Special:MyLanguage/Manual:What is MediaWiki?|MediaWiki]], all editing must stop while the switch is made. We apologize for this disruption, and we are working to minimize it in the future. A banner will be displayed on all wikis 30 minutes before this operation happens. This banner will remain visible until the end of the operation. 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|2026-10-28|en}}. * If you try to edit or save during these times, you will see an error message. We hope that no edits will be lost during these minutes, but we can't guarantee it. If you see the error message, then please wait until everything is back to normal. Then you should be able to save your edit. But, we recommend that you make a copy of your changes first, just in case. ''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"/> [[user:Trizek_(WMF)|Trizek_(WMF)]] 14:00, 26 October 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 --> == Happy birthday to us == I just realized that en-wv was opened August 15, 2006. We've reached the 20 year mark! See [[Wikiversity:History of Wikiversity|History of Wikiversity]]. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 19:01, 28 October 2026 (UTC) == Blocked users on wikimedia projects == During my almost 20 years "editing" on various wikimedia wikis I have seen many users being blocked. Some are blocked for short periods, others indefinitely (or infinitely). In most cases a blocking admin provides a rationale for the blocking such as vandalizing, spamming, paid editing, etc. Some of the reasons for blocking are well documented on the English Wikipedia, but some reasons are obscure. I have not been contributing to this project on a regular basis', so am not sure if this is a topic that should be discussed at all at the English Wikiversity. I hope that there are others who share my interest. I believe that not allowing a discussion of this topic will end up on another website, and will draw attention away from wikimedia which is already suffering the loss of many contributions, human and financial. Any thoughts? [[Special:Contributions/~2026-56788-08|~2026-56788-08]] ([[User talk:~2026-56788-08|talk]]) 18:24, 4 November 2026 (UTC) some minor changes [[Special:Contributions/~2026-56788-08|~2026-56788-08]] ([[User talk:~2026-56788-08|talk]]) 02:15, 5 November 2026 (UTC) == Mathematical theology == <ref></ref> space -time universe, mathematical structures, mathematical objects, mathematical models, time as t1,t2,t3 , space as x,y,z and matter as solid, liquid and gas as 3 states of matter means 3 dimensions of matter or else (x,y,z,t) as the 4 parameters for experiment of the nuclear reactions within the neutron stars and the half life and radioactive decay constats by numerical computation {{Building structures}}{{Subpages}}[[[http://@theologiaministry.bsky.social|frameht |frame]]]–[[User:Theologiministry89|Theologiministry89]] ([[User talk:Theologiministry89|discuss]] • [[Special:Contributions/Theologiministry89|contribs]]) 11:24, 5 November 2026 (UTC) 5am3m5sw5xuew8z254jd5u7pn3exdxr 2844604 2844603 2026-11-05T11:26:17Z Theologiministry89 3112580 /* Mathematical theology */ new section 2844604 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) ::Yes, that was my intent. I agree that a nom should request a duration with a "corresponding plan of work." [[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 04:17, 28 October 2026 (UTC) * {{support}} sensible proposal. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 15:20, 29 October 2026 (UTC) === Temporary rights extention === I propose that we clarify that temporary rights can be extended. If the community approves of an extension, it can be granted more than once. The temporary rights can repeatedly be extended, for new temporary durations. --[[User:Mu301|mikeu]] ^{[[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) == 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|2026-10-28|en}}'''. The switch will start at '''[https://zonestamp.toolforge.org/{{#time:U|2026-10-28T15:00|en}} {{#time:H:i e|2026-10-28T15:00}}]'''. Unfortunately, because of some limitations in [[mw:Special:MyLanguage/Manual:What is MediaWiki?|MediaWiki]], all editing must stop while the switch is made. We apologize for this disruption, and we are working to minimize it in the future. A banner will be displayed on all wikis 30 minutes before this operation happens. This banner will remain visible until the end of the operation. 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|2026-10-28|en}}. * If you try to edit or save during these times, you will see an error message. We hope that no edits will be lost during these minutes, but we can't guarantee it. If you see the error message, then please wait until everything is back to normal. Then you should be able to save your edit. But, we recommend that you make a copy of your changes first, just in case. ''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"/> [[user:Trizek_(WMF)|Trizek_(WMF)]] 14:00, 26 October 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 --> == Happy birthday to us == I just realized that en-wv was opened August 15, 2006. We've reached the 20 year mark! See [[Wikiversity:History of Wikiversity|History of Wikiversity]]. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 19:01, 28 October 2026 (UTC) == Blocked users on wikimedia projects == During my almost 20 years "editing" on various wikimedia wikis I have seen many users being blocked. Some are blocked for short periods, others indefinitely (or infinitely). In most cases a blocking admin provides a rationale for the blocking such as vandalizing, spamming, paid editing, etc. Some of the reasons for blocking are well documented on the English Wikipedia, but some reasons are obscure. I have not been contributing to this project on a regular basis', so am not sure if this is a topic that should be discussed at all at the English Wikiversity. I hope that there are others who share my interest. I believe that not allowing a discussion of this topic will end up on another website, and will draw attention away from wikimedia which is already suffering the loss of many contributions, human and financial. Any thoughts? [[Special:Contributions/~2026-56788-08|~2026-56788-08]] ([[User talk:~2026-56788-08|talk]]) 18:24, 4 November 2026 (UTC) some minor changes [[Special:Contributions/~2026-56788-08|~2026-56788-08]] ([[User talk:~2026-56788-08|talk]]) 02:15, 5 November 2026 (UTC) == Mathematical theology == <ref></ref> space -time universe, mathematical structures, mathematical objects, mathematical models, time as t1,t2,t3 , space as x,y,z and matter as solid, liquid and gas as 3 states of matter means 3 dimensions of matter or else (x,y,z,t) as the 4 parameters for experiment of the nuclear reactions within the neutron stars and the half life and radioactive decay constats by numerical computation {{Building structures}}{{Subpages}}[[[http://@theologiaministry.bsky.social|frameht |frame]]]–[[User:Theologiministry89|Theologiministry89]] ([[User talk:Theologiministry89|discuss]] • [[Special:Contributions/Theologiministry89|contribs]]) 11:24, 5 November 2026 (UTC) == Mathematical theology == origin of the universe by scientific theology, cosmology, mathematical theology [[User:Theologiministry89|Theologiministry89]] ([[User talk:Theologiministry89|discuss]] • [[Special:Contributions/Theologiministry89|contribs]]) 11:26, 5 November 2026 (UTC) 8qvmeay3ojafc38fnchaol1ued3t2pv 4 1558 2844455 2844444 2026-11-04T12:35:25Z Atcovi 276019 Reverted edit by [[Special:Contributions/TechBenchmarkHQ|TechBenchmarkHQ]] ([[User_talk:TechBenchmarkHQ|talk]]) to last version by [[User:MathXplore|MathXplore]] using [[Wikiversity:Rollback|rollback]] 2843617 wikitext text/x-wiki {{Please leave this line alone (sandbox heading)}} What <quiz> 1111111-000000=1111111 90+10=100 </quiz> final work <quiz> 10-2=8 212+8+220 scotland forever A A G AG A KAGA KG:AS LAG SGSA"ASG G GSA AGSA GS AG GA GS GSAG AS <nowiki><quiz></nowiki> 22-1=21 23+21(11)=354 <======= What does this mean??? i don't know you tell me EDUCATION = GOOD who tells what to tell who Sandbox practice edit by Milan Horvat 9: learning how editing works before creating a course page. 631lp4lxbwdz2enzpzlp2bpx5thr6hn 102 1808 2844506 2802370 2026-11-04T19:25:29Z ~2026-58700-96 3113829 Made the spelling more accurate to modern ideals 2844506 wikitext text/x-wiki 2844522 2844506 2026-11-04T22:31:08Z MathXplore 2888076 Reverted edit by [[Special:Contributions/~2026-58700-96|~2026-58700-96]] ([[User_talk:~2026-58700-96|talk]]) to last version by [[User:PieWriter|PieWriter]] using [[Wikiversity:Rollback|rollback]] 2802370 wikitext text/x-wiki {{proposed deletion|since a large portion of Wikiversity are learning materials, it is not clear what a portal is too achieve here}} <!-- This portal was created using subst:box portal skeleton --> <div style="float:right; width:100%"> {{box-transclude|Intro|<big>The {{PAGENAME}} Portal</big>}} </div> <div style="float:left; width:60%;"> <!-- This width added to the the margin below to equal 99%--> {{box-transclude|Featured learning resource}} {{box-transclude|Selected picture}} {{box-transclude|Selected biography|Selected research}} </div> <div style="float:right; width:39%"> <!-- This margin should be right of the above --> {{box-transclude|Related portals}} {{box-transclude|WikiProjects|Content development projects}} {{box-transclude|Categories}} {{box-transclude|Quotes}} {{box-transclude|Did you know|Did you know...}} </div> <div style="float:right; width:100%"> {{box-transclude|{{PAGENAME}} topics|{{PAGENAME}} learning projects}} </div> <div style="float:left; width:60%;"> <!-- This width added to the the margin below to equal 99%--> {{box-transclude|Things you can do}} </div> <div style="float:right; width:39%"> <!-- This margin should be right of the above --> {{box-transclude|{{PAGENAME}} news}} </div> <div style="float:right; width:100%"> {{gateways}} </div> __NOTOC__ __NOEDITSECTION__ [[Category:Resources|*{{PAGENAME}}]] [[Category:Wikiversity portals|{{PAGENAME}}]] 040ycot7k6ygf8rkrrkizz6e22d5p36 2844577 2844522 2026-11-05T03:51:18Z ~2026-58700-96 3113829 2844577 wikitext text/x-wiki 2844579 2844577 2026-11-05T03:53:39Z ~2026-58700-96 3113829 2844579 wikitext text/x-wiki 2844584 2844579 2026-11-05T04:34:44Z Morkoz 3096691 Reverted edits by [[Special:Contributions/~2026-58700-96|~2026-58700-96]] ([[User_talk:~2026-58700-96|talk]]) to last version by [[User:MathXplore|MathXplore]] using [[Wikiversity:Rollback|rollback]] 2802370 wikitext text/x-wiki {{proposed deletion|since a large portion of Wikiversity are learning materials, it is not clear what a portal is too achieve here}} <!-- This portal was created using subst:box portal skeleton --> <div style="float:right; width:100%"> {{box-transclude|Intro|<big>The {{PAGENAME}} Portal</big>}} </div> <div style="float:left; width:60%;"> <!-- This width added to the the margin below to equal 99%--> {{box-transclude|Featured learning resource}} {{box-transclude|Selected picture}} {{box-transclude|Selected biography|Selected research}} </div> <div style="float:right; width:39%"> <!-- This margin should be right of the above --> {{box-transclude|Related portals}} {{box-transclude|WikiProjects|Content development projects}} {{box-transclude|Categories}} {{box-transclude|Quotes}} {{box-transclude|Did you know|Did you know...}} </div> <div style="float:right; width:100%"> {{box-transclude|{{PAGENAME}} topics|{{PAGENAME}} learning projects}} </div> <div style="float:left; width:60%;"> <!-- This width added to the the margin below to equal 99%--> {{box-transclude|Things you can do}} </div> <div style="float:right; width:39%"> <!-- This margin should be right of the above --> {{box-transclude|{{PAGENAME}} news}} </div> <div style="float:right; width:100%"> {{gateways}} </div> __NOTOC__ __NOEDITSECTION__ [[Category:Resources|*{{PAGENAME}}]] [[Category:Wikiversity portals|{{PAGENAME}}]] 040ycot7k6ygf8rkrrkizz6e22d5p36 100 1969 2844581 2804874 2026-11-05T03:59:51Z ~2026-58713-84 3113842 Expansion of resource information for Canadian aviation. 2844581 wikitext text/x-wiki <div style="float:right; width:100%"> {{/box-header|<big>The {{PAGENAME}} School</big>|{{FULLPAGENAME}}/Intro|}} The [[Wikiversity:Introduction|Wikiversity]] School of Aviation is hoping to start with courses of Practical Aviation content. Please feel free to contribute with as much or as little as you wish. Don't fear, there is no harm you can do that can't be undone. A school is a large organizational structure which can contain various departments and divisions. The departments and divisions should be listed in the departments and divisions section. The school should not contain any learning resources. The school can contain [[Portal:Learning Projects|projects]] for developing [[Portal:Learning Materials|learning resources]]. '''Please add anything you can to any of the existing areas. Feel free to edit, rewrite or start a whole new topic. We really need some more organization to this school.''' {{box-footer}} == What Aviation Is == </div> {{RightTOC}} [[File:RWBA Flughafen(R).svg|thumb|Pursuit of a dream or practical transportation? How much of each is in your desire to fly?]] Aviation deals with flight and all factors involved in flight. It covers not only pilots, but also all the fields necessary to support flight. * Pilots, Flight Instructors, Ground Instructors * Crewmembers Other Than Pilots * Aircraft and Maintenance * Airports * Air Traffic Control * Air Carriers and For Compensation Operations * Space Transportation * Rotorcraft operations Related studies include [[Aerospace engineering|Aerospace Engineering]]. ==Pilots== For [[Aircraft piloting/Pilot basic information|basic information]] you need to know before beginning training to become a pilot, read the [[Aircraft piloting/Pilot basic information|Pilot's Basic Information]] page. ==Divisions and Departments== Divisions and Departments of the School exist on pages in "topic" namespace. Start the name of departments with the "" prefix; departments reside in the [[Wikiversity:Namespaces| namespace]]. Departments and divisions link to learning materials and learning projects. Divisions can link subdivisions or to departments. For more information on schools, divisions and departments look at the [[Wikiversity:Naming_conventions|Naming Conventions]]. There are three basic Genres in Aviation: * [[General aviation|General Aviation]] (SPL, RPL and PPL) [[w:general_aviation|GA]] * [[Commercial aviation|Commercial and Air Transport Aviation]] (CPL and ATPL) [[w:airline|Airline]] * [[Military aviation|Military Aviation]] [[w:military_aviation|MA]] ===Practical aviation=== Includes Theory Courses required of students (cadets) who are pursuing a license or a rating such as: * [[Sport License|Sport Licenses]] (SPL) * [[Recreational License|Recreational Licenses]] (RPL) * [[Private Pilot License|Private Pilot Licenses]] [[w:private_pilot_license|PPL]] * [[Instrument rating|Instrument Ratings]] [[w:instrument_rating|IR]] * [[Multiengine Rating|Multiengine Ratings]] (MR) * [[Commercial Pilot License|Commercial Pilot Licenses]] [[w:commercial_pilot_license|CPL]] * [[Certificated Flight Instructor|Certificated Flight Instructor Licenses]] [[w:certificated_flight_instructor_license|CFI]] * [[Air Transport Pilot License|Air Transport Pilot Licenses]] [[w:ATPL|ATPL]] * [[Airframe and Powerplant|Airframe & Powerplant Mechanic(A&P) and related]] [[w:Aircraft Maintenance Technician]] Etc. Each of these levels require some theory study plus actual and simulated flying hours. They also include some maintenance and other courses but mostly in a practical theme. Although this study is mainly static from year to year and some pilots have had twenty years of experience and found out that their kids are studying the same courses and information they did, changes in regulations and changes in technology require continual updating. Many Aviation schools started by providing only the basics, then moved up to Aviation Science after adding resources and the staff. We urge you to take the placement test just to know where you belong ==Courses== Courses we will be working on are: # [[Aircrafts and aerodynamics|Aircrafts and Aerodynamics]] # [[Aircraft Instruments, Engines and Systems]] # [[Airports, Air Traffic Control, and Airspace]] # [[Federal Aviation Regulations]] # [[Aircraft Performance and Weight and Balance]] # [[Aeromedical Factors and Aeronautical Decision Making]] # [[Aviation Weather]] # [[Aviation Weather Services]] # [[Navigation]] # [[Navigation Systems]] # [[Cross Country Flight Planning]] # [[Rotorcraft operations]] # [[Glider operations]] # [[Lighter-than-air operations]] Before we begin, I'd like to state that you are beginning on a journey that may take you to places you've yet to imagine. You'll never regret pursuing your pilot's certificate. Maybe your goal is to become a professional pilot. GOOD FOR YOU. Pilots love going to work. Most pilots pay their dues by Flight instructing, giving scenic rides, towing banners & gliders, The pay is low, but the experience is priceless. I wish you all the best of luck. ===Practical aviation course contents=== *Chapter 1 [[Aircraft components|Aircraft Components]] *Chapter 2 [[Aerodynamics|Aerodynamics]] *Chapter 3 [[Engine|Engines]] *Chapter 4 [[Electrical Systems]] *Chapter 5 [[Flight Instruments|Flight Instruments]] *Chapter 6 [[Federal Aviation Regulations]] *Chapter 7 [[Airport operations|Airport Operations]] *Chapter 8 [[Radio Operations]] *Chapter 9 [[Airspace|Airspace]] *Chapter 10 [[Aviation Maps]] *Chapter 11 [[Radio navigation|Radio Navigation]] *Chapter 12 [[Understanding Weather]] *Chapter 13 [[Weather Charts & Briefings]] *Chapter 14 [[Flight planning]] *Chapter 15 [[Airplane Performance Charts]] *Chapter 16 [[Weight & Balance]] *Chapter 17 [[Aeronautical Information]] *Chapter 18 [[Checklists and Aircraft Specific Information]] *Chapter 19 [[Helicopters and rotorcraft|Helicopters and Rotorcraft]] ==Aviation Science== This is the more scientific branch of study and includes some engineering courses and sometimes even safety courses. Please feel free to start new categories or new courses ==Active participants== The histories of Wikiversity pages indicate who the active participants are. If you are an active participant in this school, you can list your name here (this can help small schools grow and the participants communicate better; for large schools it is not needed). * Since 10 March 2012‎, working on [[Draft:Repellor vehicles|Repellor vehicles]] as an EAA project! --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 04:01, 13 April 2018 (UTC) ===Inactive participants=== * [[User:Aerodrome]] * [[User:nbx909]] * [[User:hunt1200]] * [[User:AuntPeggy]] * [[User:Swatjester]] * [[User:Joeldcox]] * [[User:Ikluft]] * [[User:RandyRostie]] ==Things you can do!== * Clean up [[Draft:Applications of power electronics]] and move it to become a subpage of a supporting main page learning project. * Clean up [[Draft:Electrochemical capacitors]] and move it to become a subpage of a supporting main page learning project. * Clean up [[Draft:Geographic coordinates]] and move it to become a subpage of a supporting main page learning project. * Clean up [[Draft:Lofting technology]] and move it to become a subpage of a supporting main page learning project. * Clean up [[Draft:Mems capacitors]] and move it to become a subpage of a supporting main page learning project. * Clean up [[Draft:Repellor vehicles]] and move it to become a subpage of a supporting main page learning project. * Clean up [[Draft:Shielding]] and move it to become a subpage of a supporting main page learning project. * Clean up [[Draft:Small solar vehicle]] and move it to become a subpage of a supporting main page learning project. * Clean up [[Draft:Spaceflights]] and move it to become a subpage of a supporting main page learning project. ==School news== * '''August 18, 2006''' - School founded! * '''March 1, 2007''' - Much more work needed! * '''September 2, 2008''' - Resources section reorganized, new documentation links {| cellpadding="10" cellspacing="5" style="width: 100%; background-color: inherit; color:inherit; margin-left: auto; margin-right: auto" | style="width: 60%; background-color: cornsilk;color:var(--color-base-fixed,#202122); border: 1px solid #777777; vertical-align: top; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" rowspan="1" | [[Image:Crystal Clear action info.png|right|128px]] ==Resources== These are online resources to assist with personal research. ===United States=== The following online resources are from the [[w:Federal Aviation Administration|US Federal Aviation Administration]]. While the scope of any regulatory material only applies to the United States, anything about how aircraft work is globally applicable. There is a lot of information here. * [http://ecfr.gpoaccess.gov/cgi/t/text/text-idx?&c=ecfr&tpl=/ecfrbrowse/Title14/14tab_02.tpl Federal Aviation Regulations (FARs)] - aviation regulations in the United States * [http://www.faa.gov/airports_airtraffic/air_traffic/publications/atpubs/aim/ FAA Aeronautical Information Manual (AIM)] - procedures and regulatory overview * [http://www.faa.gov/library/manuals/aviation/pilot_handbook/ Pilot's Handbook of Aeronautical Knowledge] - details on piloting for all categories of aircraft * [http://www.faa.gov/library/manuals/aviation/media/faa-h-8083-27a.pdf Student Pilot Guide] (PDF) - how to learn to fly * [http://www.faa.gov/library/manuals/aviation/media/faa-h-8083-19.pdf Plane Sense - General Aviation Information] (PDF) - aircraft ownership * Aircraft flying handbooks ** [http://www.faa.gov/library/manuals/aircraft/airplane_handbook/ Airplane Flying Handbook] ** [http://www.faa.gov/library/manuals/aircraft/glider_handbook/ Glider Flying Handbook] ** [http://www.faa.gov/library/manuals/aircraft/media/faa-h-8083-21.pdf Rotorcraft Flying Handbook] (PDF) ** [http://www.faa.gov/library/manuals/aircraft/media/Balloon%20Flying%20Handbook.pdf Balloon Flying Handbook] (PDF) ** [http://www.faa.gov/library/manuals/aircraft/seaplane_handbook/ Seaplane, Skiplane, and Float/Ski Equipped Helicopter Operations Handbook] ** [http://www.faa.gov/library/manuals/aircraft/media/FAA-H-8083-29.pdf Powered Parachute Flying Handbook] (PDF) * Instrument flight ** [http://www.faa.gov/library/manuals/aviation/instrument_flying_handbook/ Instrument Flying Handbook] ** [http://www.faa.gov/library/manuals/aviation/instrument_procedures_handbook/ Instrument Procedures Handbook] * Safety brochures ** [http://www.faa.gov/pilots/safety/pilotsafetybrochures/media/alcohol.pdf Alcohol and Flying: A Deadly Combination] (PDF) ** [http://www.faa.gov/pilots/safety/pilotsafetybrochures/media/DCS.pdf Altitude-Induced Decompression Sickness] (PDF) ** [http://www.faa.gov/pilots/safety/pilotsafetybrochures/media/CObroforweb.pdf Carbon Monoxide: A Deadly Menace] (PDF) ** [http://www.faa.gov/pilots/safety/pilotsafetybrochures/media/DVT_07182005.pdf Deep Vein Thrombosis & Travel] (PDF) ** [http://www.faa.gov/pilots/safety/pilotsafetybrochures/media/Fatigue_Aviation.pdf Fatigue in Aviation] (PDF) ** [http://www.faa.gov/pilots/safety/pilotsafetybrochures/media/hearing_brochure.pdf Hearing and Noise in Aviation] (PDF) ** [http://www.faa.gov/pilots/safety/pilotsafetybrochures/media/hypoxia.pdf Hypoxia: The Higher You Fly...The Less Air In The Sky] (PDF) ** [http://www.faa.gov/pilots/safety/pilotsafetybrochures/media/LaserEye_II.pdf Information for Pilots Considering Laser Eye Surgery] (PDF) ** [http://www.faa.gov/pilots/safety/pilotsafetybrochures/media/Checklist.pdf Medical Certification Questions and Answers] (PDF) ** [http://www.faa.gov/pilots/safety/pilotsafetybrochures/media/Meds_flying_web.pdf Medications and Flying] (PDF) ** [http://www.faa.gov/pilots/safety/pilotsafetybrochures/media/meds_flying_pstr.pdf Medications and Flying Poster] (PDF) ** [http://www.faa.gov/pilots/safety/pilotsafetybrochures/media/PT_06272005_web.pdf Physiological Training Courses for Civil Aviation Pilots] (PDF) ** [http://www.faa.gov/pilots/safety/pilotsafetybrochures/media/MedCertforWeb.pdf Pilot Medical Certification Information for the Aviation Community] (PDF) ** [http://www.faa.gov/pilots/safety/pilotsafetybrochures/media/Pilot_Vision.pdf Pilot Vision] (PDF) ** [http://www.faa.gov/pilots/safety/pilotsafetybrochures/media/seatbelt_web2.pdf Seat Belts & Shoulder Harnesses: Smart Protection in Small Airplanes] (PDF) ** [http://www.faa.gov/pilots/safety/pilotsafetybrochures/media/Smoke_Web.pdf Smoke] (PDF) ** [http://www.faa.gov/pilots/safety/pilotsafetybrochures/media/SpatialD_Seeing.pdf Spatial Disorientation: Visual Illusions] (PDF) ** [http://www.faa.gov/pilots/safety/pilotsafetybrochures/media/SpatialD.pdf Spatial Disorientation: Why You Shouldn't Fly By the Seat of Your Pants] (PDF) ** [http://www.faa.gov/pilots/safety/pilotsafetybrochures/media/sunglasses.pdf Sunglasses for Pilots: Beyond the Image] (PDF) * Misc handbooks ** [http://www.faa.gov/airports_airtraffic/air_traffic/publications/ifim/ International Flight Information Manual] ** [http://www.faa.gov/library/manuals/aviation/media/FAA-H-8083-9.pdf Aviation Instructor's Handbook] (PDF) ** [http://www.faa.gov/library/manuals/aircraft/media/FAA-H-8083-1A.pdf Aircraft Weight and Balance Handbook] (PDF) ** [http://www.faa.gov/library/manuals/aircraft/media/ac90-89a.pdf Amateur-built Aircraft & Ultralight Flight Testing Handbook] (PDF) ** [http://www.faa.gov/library/manuals/aircraft/media/FAA-H-8083-17.pdf Parachute Rigger Handbook] (PDF) ** [http://www.faa.gov/library/manuals/aviation/media/faa-fs.pdf Banner Tow Operations] (PDF) * Examiner handbooks ** [http://rgl.faa.gov/Regulatory_and_Guidance_Library/rgOrders.nsf/0/98F29EF75138870886257163005B63A1?OpenDocument Aviation Mechanic Examiner Handbook] ** [http://rgl.faa.gov/Regulatory_and_Guidance_Library/rgOrders.nsf/0/F5678F339C9FFF38862572BC006D5A5A?OpenDocument Designated Pilot and Flight Engineer Examiner's Handbook] ** [http://rgl.faa.gov/Regulatory_and_Guidance_Library/rgOrders.nsf/0/A42EA72503111140862572DB00595F5A?OpenDocument Designee Management Handbook] ** [http://www.faa.gov/about/office_org/headquarters_offices/avs/offices/aam/ame/guide/ Guide for Aviation Medical Examiners] ** [http://rgl.faa.gov/Regulatory_and_Guidance_Library/rgOrders.nsf/0/F818FFFD191664F586257163005B625F?OpenDocument Parachute Rigger Examiner Handbook] ** [http://rgl.faa.gov/Regulatory_and_Guidance_Library/rgOrders.nsf/0/C5136D3DD30B463186256F4D0056D295?OpenDocument Sport Pilot Examiner's Handbook] ===New Zealand=== The [[New Zealand Written Examinations Manuals]] is trying to build up a manual based on the syllabus requirements from the NZCAA Advisory Circulars ===Canada=== Canadian resources largely follow or copy [[w:Federal Aviation Administration|US Federal Aviation Administration]] resources, Though they are not interchangeable. When searching for official Canadian resources the [https://publications.gc.ca/site/eng/home.html Government of Canada Publications site] should be used. Resources can also be found at the [https://tc.canada.ca/en/aviation Transport Canada Aviation site] for documents, licensing, certificates, ratings, aviation medical examiners and more. *[http://www.tc.gc.ca/eng/civilaviation/regserv/cars/menu.htm Aviation regulations in Canada] *[https://tc.canada.ca/en/aviation Transport Canada Aviation site] *[https://publications.gc.ca/site/eng/9.649390/publication.html Flight training manual] *Aeronautical Information Manual (TC AIM) - Updated and released every 6 months ===What if your country isn't listed here?=== This is a volunteer-maintained site. You can add the information for your country. Please take care to ensure accuracy. When possible, please include links that others can use to confirm its accuracy as well - but don't hesitate to get it started. |} == See also == * [[w: Quiet Supersonic Transport]] == External links == [[Image:Cessna 172 3.jpg|right|thumb|260px]] * [http://www.faa.gov/pilots/training/ US Federal Aviation Administration, Pilot training resources] * [http://www.aopa.org/learntofly/ Aircraft Owners and Pilots Association] * [http://www.faa-ground-school.com Online Ground School - Free Multimedia Lessons] * [http://www.pilotsalmanac.com The Pilots Almanac (wiki)] * [http://www.flightcentral.net/overview FlightCentral.net (Wx, Airport Info, Flight Planning)] * [http://cafefoundation.org/v2/pav_home.php Personal Air Vehicle Page at Cafe Foundation (in affiliation with NASA)] * [http://seattlepi.nwsource.com/business/130398_electplane11.html Boeing's electric plane using fuelcells.] * [http://www.groundschoolacademy.com Ground School] * [https://www.mzeroa.com/ Flight Training - Online Ground School] ===Open source software=== * [http://sourceforge.net/projects/fplan/ fPlan] - Flight plan creator * [http://sourceforge.net/projects/csp/ The Combat Simulator Project] * [http://www.flightgear.org/ FlightGear] - open-source, multi-platform, highly customizable flight simulator. == Related news == * April 2008 [http://www.cleanedge.com/news/story.php?nID=5264 Boeing Successfully Flies Fuel Cell-Powered Airplane] * March 2007 [http://www.sciencedaily.com/releases/2007/03/070319175853.htm Researchers investigate the use of bio-jet propellant.] [[Category:Wikiversity schools]] kb0typ6nyjqo8ewwtq2izu8lu08h2bs 2844583 2844581 2026-11-05T04:01:15Z ~2026-58713-84 3113842 small edit to use less authoritative language. 2844583 wikitext text/x-wiki <div style="float:right; width:100%"> {{/box-header|<big>The {{PAGENAME}} School</big>|{{FULLPAGENAME}}/Intro|}} The [[Wikiversity:Introduction|Wikiversity]] School of Aviation is hoping to start with courses of Practical Aviation content. Please feel free to contribute with as much or as little as you wish. Don't fear, there is no harm you can do that can't be undone. A school is a large organizational structure which can contain various departments and divisions. The departments and divisions should be listed in the departments and divisions section. The school should not contain any learning resources. The school can contain [[Portal:Learning Projects|projects]] for developing [[Portal:Learning Materials|learning resources]]. '''Please add anything you can to any of the existing areas. Feel free to edit, rewrite or start a whole new topic. We really need some more organization to this school.''' {{box-footer}} == What Aviation Is == </div> {{RightTOC}} [[File:RWBA Flughafen(R).svg|thumb|Pursuit of a dream or practical transportation? How much of each is in your desire to fly?]] Aviation deals with flight and all factors involved in flight. It covers not only pilots, but also all the fields necessary to support flight. * Pilots, Flight Instructors, Ground Instructors * Crewmembers Other Than Pilots * Aircraft and Maintenance * Airports * Air Traffic Control * Air Carriers and For Compensation Operations * Space Transportation * Rotorcraft operations Related studies include [[Aerospace engineering|Aerospace Engineering]]. ==Pilots== For [[Aircraft piloting/Pilot basic information|basic information]] you need to know before beginning training to become a pilot, read the [[Aircraft piloting/Pilot basic information|Pilot's Basic Information]] page. ==Divisions and Departments== Divisions and Departments of the School exist on pages in "topic" namespace. Start the name of departments with the "" prefix; departments reside in the [[Wikiversity:Namespaces| namespace]]. Departments and divisions link to learning materials and learning projects. Divisions can link subdivisions or to departments. For more information on schools, divisions and departments look at the [[Wikiversity:Naming_conventions|Naming Conventions]]. There are three basic Genres in Aviation: * [[General aviation|General Aviation]] (SPL, RPL and PPL) [[w:general_aviation|GA]] * [[Commercial aviation|Commercial and Air Transport Aviation]] (CPL and ATPL) [[w:airline|Airline]] * [[Military aviation|Military Aviation]] [[w:military_aviation|MA]] ===Practical aviation=== Includes Theory Courses required of students (cadets) who are pursuing a license or a rating such as: * [[Sport License|Sport Licenses]] (SPL) * [[Recreational License|Recreational Licenses]] (RPL) * [[Private Pilot License|Private Pilot Licenses]] [[w:private_pilot_license|PPL]] * [[Instrument rating|Instrument Ratings]] [[w:instrument_rating|IR]] * [[Multiengine Rating|Multiengine Ratings]] (MR) * [[Commercial Pilot License|Commercial Pilot Licenses]] [[w:commercial_pilot_license|CPL]] * [[Certificated Flight Instructor|Certificated Flight Instructor Licenses]] [[w:certificated_flight_instructor_license|CFI]] * [[Air Transport Pilot License|Air Transport Pilot Licenses]] [[w:ATPL|ATPL]] * [[Airframe and Powerplant|Airframe & Powerplant Mechanic(A&P) and related]] [[w:Aircraft Maintenance Technician]] Etc. Each of these levels require some theory study plus actual and simulated flying hours. They also include some maintenance and other courses but mostly in a practical theme. Although this study is mainly static from year to year and some pilots have had twenty years of experience and found out that their kids are studying the same courses and information they did, changes in regulations and changes in technology require continual updating. Many Aviation schools started by providing only the basics, then moved up to Aviation Science after adding resources and the staff. We urge you to take the placement test just to know where you belong ==Courses== Courses we will be working on are: # [[Aircrafts and aerodynamics|Aircrafts and Aerodynamics]] # [[Aircraft Instruments, Engines and Systems]] # [[Airports, Air Traffic Control, and Airspace]] # [[Federal Aviation Regulations]] # [[Aircraft Performance and Weight and Balance]] # [[Aeromedical Factors and Aeronautical Decision Making]] # [[Aviation Weather]] # [[Aviation Weather Services]] # [[Navigation]] # [[Navigation Systems]] # [[Cross Country Flight Planning]] # [[Rotorcraft operations]] # [[Glider operations]] # [[Lighter-than-air operations]] Before we begin, I'd like to state that you are beginning on a journey that may take you to places you've yet to imagine. You'll never regret pursuing your pilot's certificate. Maybe your goal is to become a professional pilot. GOOD FOR YOU. Pilots love going to work. Most pilots pay their dues by Flight instructing, giving scenic rides, towing banners & gliders, The pay is low, but the experience is priceless. I wish you all the best of luck. ===Practical aviation course contents=== *Chapter 1 [[Aircraft components|Aircraft Components]] *Chapter 2 [[Aerodynamics|Aerodynamics]] *Chapter 3 [[Engine|Engines]] *Chapter 4 [[Electrical Systems]] *Chapter 5 [[Flight Instruments|Flight Instruments]] *Chapter 6 [[Federal Aviation Regulations]] *Chapter 7 [[Airport operations|Airport Operations]] *Chapter 8 [[Radio Operations]] *Chapter 9 [[Airspace|Airspace]] *Chapter 10 [[Aviation Maps]] *Chapter 11 [[Radio navigation|Radio Navigation]] *Chapter 12 [[Understanding Weather]] *Chapter 13 [[Weather Charts & Briefings]] *Chapter 14 [[Flight planning]] *Chapter 15 [[Airplane Performance Charts]] *Chapter 16 [[Weight & Balance]] *Chapter 17 [[Aeronautical Information]] *Chapter 18 [[Checklists and Aircraft Specific Information]] *Chapter 19 [[Helicopters and rotorcraft|Helicopters and Rotorcraft]] ==Aviation Science== This is the more scientific branch of study and includes some engineering courses and sometimes even safety courses. Please feel free to start new categories or new courses ==Active participants== The histories of Wikiversity pages indicate who the active participants are. If you are an active participant in this school, you can list your name here (this can help small schools grow and the participants communicate better; for large schools it is not needed). * Since 10 March 2012‎, working on [[Draft:Repellor vehicles|Repellor vehicles]] as an EAA project! --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 04:01, 13 April 2018 (UTC) ===Inactive participants=== * [[User:Aerodrome]] * [[User:nbx909]] * [[User:hunt1200]] * [[User:AuntPeggy]] * [[User:Swatjester]] * [[User:Joeldcox]] * [[User:Ikluft]] * [[User:RandyRostie]] ==Things you can do!== * Clean up [[Draft:Applications of power electronics]] and move it to become a subpage of a supporting main page learning project. * Clean up [[Draft:Electrochemical capacitors]] and move it to become a subpage of a supporting main page learning project. * Clean up [[Draft:Geographic coordinates]] and move it to become a subpage of a supporting main page learning project. * Clean up [[Draft:Lofting technology]] and move it to become a subpage of a supporting main page learning project. * Clean up [[Draft:Mems capacitors]] and move it to become a subpage of a supporting main page learning project. * Clean up [[Draft:Repellor vehicles]] and move it to become a subpage of a supporting main page learning project. * Clean up [[Draft:Shielding]] and move it to become a subpage of a supporting main page learning project. * Clean up [[Draft:Small solar vehicle]] and move it to become a subpage of a supporting main page learning project. * Clean up [[Draft:Spaceflights]] and move it to become a subpage of a supporting main page learning project. ==School news== * '''August 18, 2006''' - School founded! * '''March 1, 2007''' - Much more work needed! * '''September 2, 2008''' - Resources section reorganized, new documentation links {| cellpadding="10" cellspacing="5" style="width: 100%; background-color: inherit; color:inherit; margin-left: auto; margin-right: auto" | style="width: 60%; background-color: cornsilk;color:var(--color-base-fixed,#202122); border: 1px solid #777777; vertical-align: top; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;" rowspan="1" | [[Image:Crystal Clear action info.png|right|128px]] ==Resources== These are online resources to assist with personal research. ===United States=== The following online resources are from the [[w:Federal Aviation Administration|US Federal Aviation Administration]]. While the scope of any regulatory material only applies to the United States, anything about how aircraft work is globally applicable. There is a lot of information here. * [http://ecfr.gpoaccess.gov/cgi/t/text/text-idx?&c=ecfr&tpl=/ecfrbrowse/Title14/14tab_02.tpl Federal Aviation Regulations (FARs)] - aviation regulations in the United States * [http://www.faa.gov/airports_airtraffic/air_traffic/publications/atpubs/aim/ FAA Aeronautical Information Manual (AIM)] - procedures and regulatory overview * [http://www.faa.gov/library/manuals/aviation/pilot_handbook/ Pilot's Handbook of Aeronautical Knowledge] - details on piloting for all categories of aircraft * [http://www.faa.gov/library/manuals/aviation/media/faa-h-8083-27a.pdf Student Pilot Guide] (PDF) - how to learn to fly * [http://www.faa.gov/library/manuals/aviation/media/faa-h-8083-19.pdf Plane Sense - General Aviation Information] (PDF) - aircraft ownership * Aircraft flying handbooks ** [http://www.faa.gov/library/manuals/aircraft/airplane_handbook/ Airplane Flying Handbook] ** [http://www.faa.gov/library/manuals/aircraft/glider_handbook/ Glider Flying Handbook] ** [http://www.faa.gov/library/manuals/aircraft/media/faa-h-8083-21.pdf Rotorcraft Flying Handbook] (PDF) ** [http://www.faa.gov/library/manuals/aircraft/media/Balloon%20Flying%20Handbook.pdf Balloon Flying Handbook] (PDF) ** [http://www.faa.gov/library/manuals/aircraft/seaplane_handbook/ Seaplane, Skiplane, and Float/Ski Equipped Helicopter Operations Handbook] ** [http://www.faa.gov/library/manuals/aircraft/media/FAA-H-8083-29.pdf Powered Parachute Flying Handbook] (PDF) * Instrument flight ** [http://www.faa.gov/library/manuals/aviation/instrument_flying_handbook/ Instrument Flying Handbook] ** [http://www.faa.gov/library/manuals/aviation/instrument_procedures_handbook/ Instrument Procedures Handbook] * Safety brochures ** [http://www.faa.gov/pilots/safety/pilotsafetybrochures/media/alcohol.pdf Alcohol and Flying: A Deadly Combination] (PDF) ** [http://www.faa.gov/pilots/safety/pilotsafetybrochures/media/DCS.pdf Altitude-Induced Decompression Sickness] (PDF) ** [http://www.faa.gov/pilots/safety/pilotsafetybrochures/media/CObroforweb.pdf Carbon Monoxide: A Deadly Menace] (PDF) ** [http://www.faa.gov/pilots/safety/pilotsafetybrochures/media/DVT_07182005.pdf Deep Vein Thrombosis & Travel] (PDF) ** [http://www.faa.gov/pilots/safety/pilotsafetybrochures/media/Fatigue_Aviation.pdf Fatigue in Aviation] (PDF) ** [http://www.faa.gov/pilots/safety/pilotsafetybrochures/media/hearing_brochure.pdf Hearing and Noise in Aviation] (PDF) ** [http://www.faa.gov/pilots/safety/pilotsafetybrochures/media/hypoxia.pdf Hypoxia: The Higher You Fly...The Less Air In The Sky] (PDF) ** [http://www.faa.gov/pilots/safety/pilotsafetybrochures/media/LaserEye_II.pdf Information for Pilots Considering Laser Eye Surgery] (PDF) ** [http://www.faa.gov/pilots/safety/pilotsafetybrochures/media/Checklist.pdf Medical Certification Questions and Answers] (PDF) ** [http://www.faa.gov/pilots/safety/pilotsafetybrochures/media/Meds_flying_web.pdf Medications and Flying] (PDF) ** [http://www.faa.gov/pilots/safety/pilotsafetybrochures/media/meds_flying_pstr.pdf Medications and Flying Poster] (PDF) ** [http://www.faa.gov/pilots/safety/pilotsafetybrochures/media/PT_06272005_web.pdf Physiological Training Courses for Civil Aviation Pilots] (PDF) ** [http://www.faa.gov/pilots/safety/pilotsafetybrochures/media/MedCertforWeb.pdf Pilot Medical Certification Information for the Aviation Community] (PDF) ** [http://www.faa.gov/pilots/safety/pilotsafetybrochures/media/Pilot_Vision.pdf Pilot Vision] (PDF) ** [http://www.faa.gov/pilots/safety/pilotsafetybrochures/media/seatbelt_web2.pdf Seat Belts & Shoulder Harnesses: Smart Protection in Small Airplanes] (PDF) ** [http://www.faa.gov/pilots/safety/pilotsafetybrochures/media/Smoke_Web.pdf Smoke] (PDF) ** [http://www.faa.gov/pilots/safety/pilotsafetybrochures/media/SpatialD_Seeing.pdf Spatial Disorientation: Visual Illusions] (PDF) ** [http://www.faa.gov/pilots/safety/pilotsafetybrochures/media/SpatialD.pdf Spatial Disorientation: Why You Shouldn't Fly By the Seat of Your Pants] (PDF) ** [http://www.faa.gov/pilots/safety/pilotsafetybrochures/media/sunglasses.pdf Sunglasses for Pilots: Beyond the Image] (PDF) * Misc handbooks ** [http://www.faa.gov/airports_airtraffic/air_traffic/publications/ifim/ International Flight Information Manual] ** [http://www.faa.gov/library/manuals/aviation/media/FAA-H-8083-9.pdf Aviation Instructor's Handbook] (PDF) ** [http://www.faa.gov/library/manuals/aircraft/media/FAA-H-8083-1A.pdf Aircraft Weight and Balance Handbook] (PDF) ** [http://www.faa.gov/library/manuals/aircraft/media/ac90-89a.pdf Amateur-built Aircraft & Ultralight Flight Testing Handbook] (PDF) ** [http://www.faa.gov/library/manuals/aircraft/media/FAA-H-8083-17.pdf Parachute Rigger Handbook] (PDF) ** [http://www.faa.gov/library/manuals/aviation/media/faa-fs.pdf Banner Tow Operations] (PDF) * Examiner handbooks ** [http://rgl.faa.gov/Regulatory_and_Guidance_Library/rgOrders.nsf/0/98F29EF75138870886257163005B63A1?OpenDocument Aviation Mechanic Examiner Handbook] ** [http://rgl.faa.gov/Regulatory_and_Guidance_Library/rgOrders.nsf/0/F5678F339C9FFF38862572BC006D5A5A?OpenDocument Designated Pilot and Flight Engineer Examiner's Handbook] ** [http://rgl.faa.gov/Regulatory_and_Guidance_Library/rgOrders.nsf/0/A42EA72503111140862572DB00595F5A?OpenDocument Designee Management Handbook] ** [http://www.faa.gov/about/office_org/headquarters_offices/avs/offices/aam/ame/guide/ Guide for Aviation Medical Examiners] ** [http://rgl.faa.gov/Regulatory_and_Guidance_Library/rgOrders.nsf/0/F818FFFD191664F586257163005B625F?OpenDocument Parachute Rigger Examiner Handbook] ** [http://rgl.faa.gov/Regulatory_and_Guidance_Library/rgOrders.nsf/0/C5136D3DD30B463186256F4D0056D295?OpenDocument Sport Pilot Examiner's Handbook] ===New Zealand=== The [[New Zealand Written Examinations Manuals]] is trying to build up a manual based on the syllabus requirements from the NZCAA Advisory Circulars ===Canada=== Canadian resources largely follow or copy [[w:Federal Aviation Administration|US Federal Aviation Administration]] resources, Though they are not interchangeable. When searching for official Canadian resources the [https://publications.gc.ca/site/eng/home.html Government of Canada Publications site] is recommended. Resources can also be found at the [https://tc.canada.ca/en/aviation Transport Canada Aviation site] for documents, licensing, certificates, ratings, aviation medical examiners and more. *[http://www.tc.gc.ca/eng/civilaviation/regserv/cars/menu.htm Aviation regulations in Canada] *[https://tc.canada.ca/en/aviation Transport Canada Aviation site] *[https://publications.gc.ca/site/eng/9.649390/publication.html Flight training manual] *Aeronautical Information Manual (TC AIM) - Updated and released every 6 months ===What if your country isn't listed here?=== This is a volunteer-maintained site. You can add the information for your country. Please take care to ensure accuracy. When possible, please include links that others can use to confirm its accuracy as well - but don't hesitate to get it started. |} == See also == * [[w: Quiet Supersonic Transport]] == External links == [[Image:Cessna 172 3.jpg|right|thumb|260px]] * [http://www.faa.gov/pilots/training/ US Federal Aviation Administration, Pilot training resources] * [http://www.aopa.org/learntofly/ Aircraft Owners and Pilots Association] * [http://www.faa-ground-school.com Online Ground School - Free Multimedia Lessons] * [http://www.pilotsalmanac.com The Pilots Almanac (wiki)] * [http://www.flightcentral.net/overview FlightCentral.net (Wx, Airport Info, Flight Planning)] * [http://cafefoundation.org/v2/pav_home.php Personal Air Vehicle Page at Cafe Foundation (in affiliation with NASA)] * [http://seattlepi.nwsource.com/business/130398_electplane11.html Boeing's electric plane using fuelcells.] * [http://www.groundschoolacademy.com Ground School] * [https://www.mzeroa.com/ Flight Training - Online Ground School] ===Open source software=== * [http://sourceforge.net/projects/fplan/ fPlan] - Flight plan creator * [http://sourceforge.net/projects/csp/ The Combat Simulator Project] * [http://www.flightgear.org/ FlightGear] - open-source, multi-platform, highly customizable flight simulator. == Related news == * April 2008 [http://www.cleanedge.com/news/story.php?nID=5264 Boeing Successfully Flies Fuel Cell-Powered Airplane] * March 2007 [http://www.sciencedaily.com/releases/2007/03/070319175853.htm Researchers investigate the use of bio-jet propellant.] [[Category:Wikiversity schools]] kwd7d9plse1dzcgyvj98oue5s8gedoz 103 25259 2844507 84573 2026-11-04T19:30:28Z ~2026-58700-96 3113829 /* Vandalism? */ new section 2844507 wikitext text/x-wiki If you want to experiment with a radical new version of [[Portal:Learning Materials]] you can work at [[Portal:Learning Materialsnew]]. If you update featured content on this portal, make sure that you update [[Portal:Learning Materials/Featured]]. ---- == Vandalism? == Someone is talking about the very respectable user @[[User:Materialscientist|Materialscientist]] in a very profane way! Someone please revert this! [[Special:Contributions/~2026-58700-96|~2026-58700-96]] ([[User talk:~2026-58700-96|talk]]) 19:30, 4 November 2026 (UTC) a10zmwua4347bp982u1w9o0slmiwgos 2844508 2844507 2026-11-04T19:31:02Z ~2026-58700-96 3113829 /* Vandalism? */ Reply 2844508 wikitext text/x-wiki 2844523 2844508 2026-11-04T22:31:27Z MathXplore 2888076 Reverted edits by [[Special:Contributions/~2026-58700-96|~2026-58700-96]] ([[User_talk:~2026-58700-96|talk]]) to last version by [[User:JWSchmidt|JWSchmidt]] using [[Wikiversity:Rollback|rollback]] 84573 wikitext text/x-wiki If you want to experiment with a radical new version of [[Portal:Learning Materials]] you can work at [[Portal:Learning Materialsnew]]. If you update featured content on this portal, make sure that you update [[Portal:Learning Materials/Featured]]. ---- 1t7auz7wwfq4z2rmvxpocnhmsanzsxn 2844578 2844523 2026-11-05T03:51:49Z ~2026-58700-96 3113829 /* mmmm yummy poop */ new section 2844578 wikitext text/x-wiki 2844585 2844578 2026-11-05T04:35:01Z Morkoz 3096691 Reverted edit by [[Special:Contributions/~2026-58700-96|~2026-58700-96]] ([[User_talk:~2026-58700-96|talk]]) to last version by [[User:MathXplore|MathXplore]] using [[Wikiversity:Rollback|rollback]] 84573 wikitext text/x-wiki If you want to experiment with a radical new version of [[Portal:Learning Materials]] you can work at [[Portal:Learning Materialsnew]]. If you update featured content on this portal, make sure that you update [[Portal:Learning Materials/Featured]]. ---- 1t7auz7wwfq4z2rmvxpocnhmsanzsxn 0 35613 2844505 2777857 2026-11-04T18:55:21Z ~2026-58700-96 3113829 Corrected spelling 2844505 wikitext text/x-wiki <noinclude> {{Other uses|Learning resources}} {{shortcut|WV:LR}} </noinclude> '''Learning Resources''' are text, media, and other digital assets that are useful for teaching, learning, and assessing as well as for research purposes.<ref>[[Wikipedia:Open educational resources]]</ref> Wikiversity has many [[Help:resource types#Introduction|types of learning resources]] not available at other Wikimedia Foundation projects (see: [[Wikiversity:What Wikiversity is not|What Wikiversity is not]]). However, Wikiversity doesn't automatically include all of the learning resources available in other projects, such as [[Wikipedia:|encyclopedia articles]], [[Wikibooks:|textbooks]], or [[Wikisource:|source texts]] (see: [[Wikimedia]]). In special cases these types of resources may be hoted by Wikiversity projects. == What learning resources does Wikiversiy include? == Wikiversity includes learning resources in all subject areas for all age groups. Learning resources can be roughly divided into: # Learning projects that support [[Portal:Education/Wikiversity model|active learning]], and # Static learning materials for outside of Wikiversity. ===Learning project=== [[Wikiversity:Learning projects|Learning projects]] guide active learners to exercises and collaborative learning projects. See: [[Portal:Learning Projects]]. ===Research project=== Research is an integral part of advanced education and profesional work in many fields. See [[Wikiversity:Research]] for information on Wikiversity research projects. ===Learning materials=== [[Wikiversity:Learning materials|Learning materials]] can be audio and video files, text files and are generally static learning objects that can be used either by Wikiversity participants or downloaded for use at bricks-and-mortar educational institutions. ==Finding Wikiversity Learning resources== '''Portals'''. Every Wikiversity learning resource may be linked to by at least one [[Wikiversity:Portals|portal page]]. You can browse the Wikiversity portals. See [[:Category:Portals]]. '''Categories'''. Every Wikiversity learning resource should be in at least one [[Help:Category|Category]]. You can browse Wikiversity by category. See: [[:Category:Categories]]. '''Search'''. Try using the search feature to find learning resoures. See: [[Wikiversity:Searching]]. ===Your turn=== Wikiversity had many learning resources that are on hard-to-find pages. You can help make sure that every learning resource has a [[link]] to it from at least one portal page. Particularly exciting learning resources should become [[Wikiversity:Featured|featured content]] on portals or the [[Main Page]]. When existing portals become too crowded you can [[Wikiversity:Portals#Starting a new portal|make new portals]]. ==Creating learning resources== See: [[Wikiversity:Content development]]. <noinclude> ==See also== * [[Wikiversity:Learning resources]] - more on Wikiversity's resource policy * [[Wikiversity:What is Wikiversity?|What is Wikiversity?]] * [[Portal:Learning Materials]] - cataloguing and integrating resources * [[Rich Learning Resources Platform]] * [[Other Free Learning Resources]] - find, list, and evaluate Intrnet learning resources == References == {{Reflist}} {{Using Wikiversity}} [[Category:Education]] [[Category:Introduction to Wikiversity]] [[Category:Introductory articles in need of repair]] </noinclude> mhhnk0cyan4azhieq8f9c863wij758b 2844588 2844505 2026-11-05T05:35:44Z Koavf 147 Reverted edit by [[Special:Contributions/~2026-58700-96|~2026-58700-96]] ([[User_talk:~2026-58700-96|talk]]) to last version by [[User:Atcovi|Atcovi]] using [[Wikiversity:Rollback|rollback]] 2686048 wikitext text/x-wiki <noinclude> {{Other uses|Learning resources}} {{shortcut|WV:LR}} </noinclude> '''Learning resources''' are text, media, and other digital assets that are useful for teaching, learning, and assessing as well as for research purposes.<ref>[[Wikipedia:Open educational resources]]</ref> Wikiversity has many [[Help:resource types#Introduction|types of learning resources]] not available at other Wikimedia Foundation projects (see: [[Wikiversity:What Wikiversity is not|What Wikiversity is not]]). However, Wikiversity doesn't automatically include all of the learning resources available in other projects, such as [[Wikipedia:|encyclopedia articles]], [[Wikibooks:|textbooks]], or [[Wikisource:|source texts]] (see: [[Wikimedia]]). In special cases these types of resources may be hosted by Wikiversity projects. == What learning resources does Wikiversity include? == Wikiversity includes learning resources in all subject areas for all age groups. Learning resources can be roughly divided into: # Learning projects that support [[Portal:Education/Wikiversity model|active learning]], and # Static learning materials for use outside of Wikiversity. ===Learning projects=== [[Wikiversity:Learning projects|Learning projects]] guide active learners to exercises and collaborative learning projects. See: [[Portal:Learning Projects]]. ===Research projects=== Research is an integral part of advanced education and professional work in many fields. See [[Wikiversity:Research]] for information on Wikiversity research projects. ===Learning materials=== [[Wikiversity:Learning materials|Learning materials]] can be audio and video files, text files and are generally static learning objects that can be used either by Wikiversity participants or downloaded for use at bricks-and-mortar educational institutions. ==Finding Wikiversity Learning resources== '''Portals'''. Every Wikiversity learning resource may be linked to by at least one [[Wikiversity:Portals|portal page]]. You can browse the Wikiversity portals. See [[:Category:Portals]]. '''Categories'''. Every Wikiversity learning resource should be in at least one [[Help:Category|Category]]. You can browse Wikiversity by category. See: [[:Category:Categories]]. '''Search'''. Try using the search feature to find learning resources. See: [[Wikiversity:Searching]]. ===Your turn=== Wikiversity had many learning resources that are on hard-to-find pages. You can help make sure that every learning resource has a [[link]] to it from at least one portal page. Particularly exciting learning resources should become [[Wikiversity:Featured|featured content]] on portals or the [[Main Page]]. When existing portals become too crowded you can [[Wikiversity:Portals#Starting a new portal|make new portals]]. ==Creating learning resources== See: [[Wikiversity:Content development]]. <noinclude> ==See also== * [[Wikiversity:Learning resources]] - more on Wikiversity's resource policy * [[Wikiversity:What is Wikiversity?|What is Wikiversity?]] * [[Portal:Learning Materials]] - cataloguing and integrating resources * [[Rich Learning Resources Platform]] * [[Other Free Learning Resources]] - find, list, and evaluate Internet learning resources == References == {{Reflist}} {{Using Wikiversity}} [[Category:Education]] [[Category:Introduction to Wikiversity]] [[Category:Introductory articles in need of repair]] </noinclude> 6by3y8vf8gipxosbt6gwwh56curaerp 4 55201 2844516 2839914 2026-11-04T22:06:36Z Codename Noreste 2969951 We now require 5 edits. 2844516 wikitext text/x-wiki [[File:Wikiversity Autoconfirmed.svg|150px|right]] A user is autoconfirmed when the user account has existed for 4 days ''and'' includes a minimum number of 5 edits. An anonymous user is not autoconfirmed. Users who are not autoconfirmed have restricted editing rights, including not being able to: * [[Special:Upload|upload a file]] * move a page * edit semi-protected pages * create new pages via an IP address without logging in If you are not autoconfirmed and an action you want to perform is thereby blocked, you have the following options: * Create an account at [[commons:Main Page|Commons]] and upload files at Commons. You can use the files here at Wikiversity by referencing their file names. * Contact [[Wikiversity:Support staff]] (also by [[Wikiversity:Chat]]) * Write a message in the [[Wikiversity:Colloquium]]. [[Category:Wikiversity administration]] 8xlwuy2tqftjrtw293aeohdnbex16g4 2844517 2844516 2026-11-04T22:06:54Z Codename Noreste 2969951 2844517 wikitext text/x-wiki [[File:Wikiversity Autoconfirmed.svg|150px|right]] A user is autoconfirmed when the user account has existed for 4 days ''and'' includes a minimum of 5 edits. An anonymous user is not autoconfirmed. Users who are not autoconfirmed have restricted editing rights, including not being able to: * [[Special:Upload|upload a file]] * move a page * edit semi-protected pages * create new pages via an IP address without logging in If you are not autoconfirmed and an action you want to perform is thereby blocked, you have the following options: * Create an account at [[commons:Main Page|Commons]] and upload files at Commons. You can use the files here at Wikiversity by referencing their file names. * Contact [[Wikiversity:Support staff]] (also by [[Wikiversity:Chat]]) * Write a message in the [[Wikiversity:Colloquium]]. [[Category:Wikiversity administration]] ezorb44gwv74nq69zsekril5w4cmmng 4 75745 2844587 2843934 2026-11-05T04:37:35Z Morkoz 3096691 + 2844587 wikitext text/x-wiki {{/Header}} == Unable to Create Page (?) == Hi, I'm trying to create a page: [[Magnetoencephalography (MEG)]]. </br> When I attempt to create it however, it does not allow me, spitting the following message: </br> "This action has been automatically identified as potentially harmful, and therefore disallowed. If you believe your action was constructive, please [[Wikiversity:Request custodian action|inform an administrator]] of what you were trying to do. A brief description of the abuse rule which your action matched is: New user exceeded new page limit" Any help on this would be greatly appreciated. Thanks, [[User:SamLaxarox|SamLaxarox]] ([[User talk:SamLaxarox|discuss]] • [[Special:Contributions/SamLaxarox|contribs]]) 23:34, 12 October 2026 (UTC) :There are restrictions on new accounts. We could do two things: I personally recommend that you create it in your userspace and build it up to be ready for primetime and then it can be moved or I could make a blank page and you can edit it from there. I'm willing to do that, but think it's not the best method. ―[[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> 02:09, 13 October 2026 (UTC) ::No worries, I'll just put it in my userspace; Thanks for the help! [[User:SamLaxarox|SamLaxarox]] ([[User talk:SamLaxarox|discuss]] • [[Special:Contributions/SamLaxarox|contribs]]) 19:08, 13 October 2026 (UTC) ::@[[User:Koavf|Koavf]], on 23:39, 15 March 2020, I posted a similar request <nowiki>'''</nowiki>[[Wikiversity:Request custodian action#trying to create a page in my user space here|<bdi>trying to create a page in my user space here</bdi>]]''<nowiki>'''</nowiki>.'' That was my first successful "edit" to this project. It took awhile to get my page established, and I believe but have not checked recently that it is still here. However, unlike my experience with another wiki project I never felt I could actually contribute here and as a result my contributions have been far and few between. ::I hope @ [[User:SamLaxarox|SamLaxarox]] will have better luck, but I sometimes wonder if all these bureaucratic restrictions , that create a lot of work for admins, are actually beneficial for this wiki as a whole. Just my $.02 [[Special:Contributions/~2026-40514-81|~2026-40514-81]] ([[User talk:~2026-40514-81|talk]]) 16:26, 14 October 2026 (UTC) :::Temporary account users are welcome and we appreciate you wanting to add to the educational resources, but as you can imagine, we need to have some restrictions on accounts. It's hard to have perfect settings for these. ―[[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> 17:41, 14 October 2026 (UTC) ::::Thanks for replying @Koavf. I guess what I was trying to probe is why Wikiversity makes it so difficult for new users to start contributing here. I have not been "editing" at the English Wikipedia for years, but I believe one can create a new page without having to post a request at the Village Pump or at the admin noticeboard? [[Special:Contributions/~2026-40514-81|~2026-40514-81]] ([[User talk:~2026-40514-81|talk]]) 14:55, 15 October 2026 (UTC) :::::Per [[w:Wikipedia:User groups]]: "Several actions on the English Wikipedia (such as article creation) are restricted to user accounts that have made at least ''10 edits'' to the English Wikipedia '''and''' after ''4 days'' have passed since the time of their first edit." Wikiversity is not much different from Wikipedia in this regard. [[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 18:37, 15 October 2026 (UTC) ::::::@[[User:Mu301|Mu301]]: ::::::I admit I did not check the small nuances of difference on enwp and enwv between new users who can and those who cannot create new pages. My excuse, which many here attribute to laziness, is that I do a lot of Wikimedia-project-hopping and cannot remember all the rules for contributing. ::::::In any event I was just wondering out loud if there is a better way of communicating with new users to let them know how they can contribute without requiring a phd on the topic first. [[Special:Contributions/~2026-56788-08|~2026-56788-08]] ([[User talk:~2026-56788-08|talk]]) 20:15, 29 October 2026 (UTC) ::::::: Once a user creates an account, they usually receive a Welcome message on their talk page with introductory information. Anonymous users can also access e.g., [[Wikiversity:Welcome]] which is linked from the en.wv homepage. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 22:09, 29 October 2026 (UTC) == Report concerning [[special:contribs/Smallestredboy2|Smallestredboy2]] == '''Reason''': Impersonation of [[user:Smallestredboy|Smallestredboy]] + vandalism. --[[User:Morkoz|Morkoz]] ([[User talk:Morkoz|discuss]] • [[Special:Contributions/Morkoz|contribs]]) 20:40, 28 October 2026 (UTC) :{{done}} ―[[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> 23:16, 28 October 2026 (UTC) == [[:File:Pelevin 01 body.jpg]] == The image [[:File:Pelevin 01 body.jpg]] was uploaded in 2012 under claim of "own work". However, the same image was actually published in [https://bombmagazine.org/articles/2002/04/01/victor-pelevin/ this 2002 article] credited to Viking Books. [[User:Howardcorn33|Howardcorn33]] ([[User talk:Howardcorn33|discuss]] • [[Special:Contributions/Howardcorn33|contribs]]) 10:38, 29 October 2026 (UTC) :{{done}} ―[[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> 14:58, 29 October 2026 (UTC) == Report concerning [[special:contribs/BoiaBoiaMans|BoiaBoiaMans]] == '''Reason''': Long-term abuse, see also [[special:contribs/Smallestredboy2|Smallestredboy2]]. --[[User:Morkoz|Morkoz]] ([[User talk:Morkoz|discuss]] • [[Special:Contributions/Morkoz|contribs]]) 19:30, 1 November 2026 (UTC) :{{done}} ―[[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> 19:30, 1 November 2026 (UTC) == I was not able to publish my page == Every time I try to publish my page, it says, "This action has been automatically identified as potentially harmful and therefore disallowed". All I was trying to do was an assignment I have for class. It's my first time posting on this website, and I had around 6 external links that could have triggered the reason for this. [[User:MM0665|MM0665]] ([[User talk:MM0665|discuss]] • [[Special:Contributions/MM0665|contribs]]) 04:49, 2 November 2026 (UTC) :I know this is frustrating and can be confusing. It looks like you were trying to create [[Baseball in Technology]]. I made this as a blank page and you should be able to edit it now. ―[[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> 05:28, 2 November 2026 (UTC) == Report concerning [[special:contribs/~2026-58700-96|~2026-58700-96]] == '''Reason''': [[WV:Vandalism|Vandalism]]. --[[User:Morkoz|Morkoz]] ([[User talk:Morkoz|discuss]] • [[Special:Contributions/Morkoz|contribs]]) 04:37, 5 November 2026 (UTC) n24liqbpmdcylo428fp9aeks6uh08ps 2844589 2844587 2026-11-05T05:36:24Z Koavf 147 /* Report concerning ~2026-58700-96 */ Reply 2844589 wikitext text/x-wiki {{/Header}} == Unable to Create Page (?) == Hi, I'm trying to create a page: [[Magnetoencephalography (MEG)]]. </br> When I attempt to create it however, it does not allow me, spitting the following message: </br> "This action has been automatically identified as potentially harmful, and therefore disallowed. If you believe your action was constructive, please [[Wikiversity:Request custodian action|inform an administrator]] of what you were trying to do. A brief description of the abuse rule which your action matched is: New user exceeded new page limit" Any help on this would be greatly appreciated. Thanks, [[User:SamLaxarox|SamLaxarox]] ([[User talk:SamLaxarox|discuss]] • [[Special:Contributions/SamLaxarox|contribs]]) 23:34, 12 October 2026 (UTC) :There are restrictions on new accounts. We could do two things: I personally recommend that you create it in your userspace and build it up to be ready for primetime and then it can be moved or I could make a blank page and you can edit it from there. I'm willing to do that, but think it's not the best method. ―[[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> 02:09, 13 October 2026 (UTC) ::No worries, I'll just put it in my userspace; Thanks for the help! [[User:SamLaxarox|SamLaxarox]] ([[User talk:SamLaxarox|discuss]] • [[Special:Contributions/SamLaxarox|contribs]]) 19:08, 13 October 2026 (UTC) ::@[[User:Koavf|Koavf]], on 23:39, 15 March 2020, I posted a similar request <nowiki>'''</nowiki>[[Wikiversity:Request custodian action#trying to create a page in my user space here|<bdi>trying to create a page in my user space here</bdi>]]''<nowiki>'''</nowiki>.'' That was my first successful "edit" to this project. It took awhile to get my page established, and I believe but have not checked recently that it is still here. However, unlike my experience with another wiki project I never felt I could actually contribute here and as a result my contributions have been far and few between. ::I hope @ [[User:SamLaxarox|SamLaxarox]] will have better luck, but I sometimes wonder if all these bureaucratic restrictions , that create a lot of work for admins, are actually beneficial for this wiki as a whole. Just my $.02 [[Special:Contributions/~2026-40514-81|~2026-40514-81]] ([[User talk:~2026-40514-81|talk]]) 16:26, 14 October 2026 (UTC) :::Temporary account users are welcome and we appreciate you wanting to add to the educational resources, but as you can imagine, we need to have some restrictions on accounts. It's hard to have perfect settings for these. ―[[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> 17:41, 14 October 2026 (UTC) ::::Thanks for replying @Koavf. I guess what I was trying to probe is why Wikiversity makes it so difficult for new users to start contributing here. I have not been "editing" at the English Wikipedia for years, but I believe one can create a new page without having to post a request at the Village Pump or at the admin noticeboard? [[Special:Contributions/~2026-40514-81|~2026-40514-81]] ([[User talk:~2026-40514-81|talk]]) 14:55, 15 October 2026 (UTC) :::::Per [[w:Wikipedia:User groups]]: "Several actions on the English Wikipedia (such as article creation) are restricted to user accounts that have made at least ''10 edits'' to the English Wikipedia '''and''' after ''4 days'' have passed since the time of their first edit." Wikiversity is not much different from Wikipedia in this regard. [[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 18:37, 15 October 2026 (UTC) ::::::@[[User:Mu301|Mu301]]: ::::::I admit I did not check the small nuances of difference on enwp and enwv between new users who can and those who cannot create new pages. My excuse, which many here attribute to laziness, is that I do a lot of Wikimedia-project-hopping and cannot remember all the rules for contributing. ::::::In any event I was just wondering out loud if there is a better way of communicating with new users to let them know how they can contribute without requiring a phd on the topic first. [[Special:Contributions/~2026-56788-08|~2026-56788-08]] ([[User talk:~2026-56788-08|talk]]) 20:15, 29 October 2026 (UTC) ::::::: Once a user creates an account, they usually receive a Welcome message on their talk page with introductory information. Anonymous users can also access e.g., [[Wikiversity:Welcome]] which is linked from the en.wv homepage. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 22:09, 29 October 2026 (UTC) == Report concerning [[special:contribs/Smallestredboy2|Smallestredboy2]] == '''Reason''': Impersonation of [[user:Smallestredboy|Smallestredboy]] + vandalism. --[[User:Morkoz|Morkoz]] ([[User talk:Morkoz|discuss]] • [[Special:Contributions/Morkoz|contribs]]) 20:40, 28 October 2026 (UTC) :{{done}} ―[[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> 23:16, 28 October 2026 (UTC) == [[:File:Pelevin 01 body.jpg]] == The image [[:File:Pelevin 01 body.jpg]] was uploaded in 2012 under claim of "own work". However, the same image was actually published in [https://bombmagazine.org/articles/2002/04/01/victor-pelevin/ this 2002 article] credited to Viking Books. [[User:Howardcorn33|Howardcorn33]] ([[User talk:Howardcorn33|discuss]] • [[Special:Contributions/Howardcorn33|contribs]]) 10:38, 29 October 2026 (UTC) :{{done}} ―[[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> 14:58, 29 October 2026 (UTC) == Report concerning [[special:contribs/BoiaBoiaMans|BoiaBoiaMans]] == '''Reason''': Long-term abuse, see also [[special:contribs/Smallestredboy2|Smallestredboy2]]. --[[User:Morkoz|Morkoz]] ([[User talk:Morkoz|discuss]] • [[Special:Contributions/Morkoz|contribs]]) 19:30, 1 November 2026 (UTC) :{{done}} ―[[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> 19:30, 1 November 2026 (UTC) == I was not able to publish my page == Every time I try to publish my page, it says, "This action has been automatically identified as potentially harmful and therefore disallowed". All I was trying to do was an assignment I have for class. It's my first time posting on this website, and I had around 6 external links that could have triggered the reason for this. [[User:MM0665|MM0665]] ([[User talk:MM0665|discuss]] • [[Special:Contributions/MM0665|contribs]]) 04:49, 2 November 2026 (UTC) :I know this is frustrating and can be confusing. It looks like you were trying to create [[Baseball in Technology]]. I made this as a blank page and you should be able to edit it now. ―[[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> 05:28, 2 November 2026 (UTC) == Report concerning [[special:contribs/~2026-58700-96|~2026-58700-96]] == '''Reason''': [[WV:Vandalism|Vandalism]]. --[[User:Morkoz|Morkoz]] ([[User talk:Morkoz|discuss]] • [[Special:Contributions/Morkoz|contribs]]) 04:37, 5 November 2026 (UTC) :{{done}} ―[[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> 05:36, 5 November 2026 (UTC) op4jn8da9o0p4crsw87s95xuvejfpm3 0 78429 2844453 2712376 2026-11-04T12:31:09Z ~2026-58509-18 3113812 Undid revision [[Special:Diff/2399809|2399809]] by [[Special:Contributions/41.116.131.193|41.116.131.193]] ([[User talk:41.116.131.193|talk]]) 2844453 wikitext text/x-wiki '''Fashion''' is a style that is accepted and used by the majority of a group at any one time for a substantial period. They tend to grow slowly, remain popular for a while, then decline slowly. There are certain seasons for fashion like the weather. Fashion is an important part of everyday life and has to do with everyone since everyone wears clothes. Its patterns change with the changing times, the designs are linked to modernity and the method of embroidery changes and develops. == High fashion == It refers to those styles or designs accepted by a limited group of fashion leaders-elite class-who are first to accept fashion change. These articles are sold in low quantity but at outrageous prices to socialites, entertainers and fashion innovators. == Mass fashion/volume fashion == Mass produced fashion, fast fashion or retail fashion are those apparel items which conform to fashion trends introduced by the high fashion industry. Mass produced fashion, although emulating colour and/or silhouette of high fashion, is usually produced using less expensive fabric and less labour intensive techniques in order to be accessible to the retail or mass consumer market. Furthermore, depending on the regional consumer base, local designers might adjust the style in order to satisfy mores of modesty, practicality or particular climate conditions of the local consumer base. These adjustments might be related to fabrication (materials and trims) or style (necklines, hemlines, fit). [[Category:Fashion| ]] a897invfq54vjx7da45f5lwbcrfdbtq 0 104030 2844512 2844253 2026-11-04T21:31:20Z CommonsDelinker 9184 Removing [[:c:File:Pseudotribos_BW_(flipped).jpg|Pseudotribos_BW_(flipped).jpg]], it has been deleted from Commons by [[:c:User:Túrelio|Túrelio]] because: This file is ONLY published under a license that does not allow unrestricted commercial use. Under [ 2844512 wikitext text/x-wiki <div class="thumb tleft" style="border: 1px solid #CCCCCC; margin:0.5em;"><table border=0><tr><td width=10><p> </p><td> {| class="wikitable" |- | colspan="5" align="center" style="color: black; background: #ccccff" | '''Orders of Mammals''' |- ! align="center" style="color: black; background: #ccccff" |Infraclass ! align="center" style="color: black; background: #ccccff" |Superorder ! align="center" style="color: black; background: #ccccff" |Order ! align="center" style="color: black; background: #ccccff" |Common Representatives |- ||Eutheria||Afrotheria|| Afrosoricida || golden moles |- ||Eutheria||Afrotheria|| Hyracoidea || hyraxes |- ||Eutheria||Afrotheria|| Macroscelidea || elephant shrews |- ||Eutheria||Afrotheria|| Sirenia || manatees |- ||Eutheria||Afrotheria|| Proboscidea || elephants |- ||Eutheria||Afrotheria|| Tubulidentata || aardvarks |- |||||||| |- ||Eutheria||Euarchontoglires|| Dermoptera || flying lemurs |- ||Eutheria||Euarchontoglires|| Lagomorpha || rabbits |- ||Eutheria||Euarchontoglires|| Primates || chimps |- ||Eutheria||Euarchontoglires|| [[/Rodentia/]] || mice |- ||Eutheria||Euarchontoglires|| Scandentia || tree shrews |- |||||||| |- ||Eutheria||Laurasiatheria|| Artiodactyla || pigs, goats, sheep, cows |- ||Eutheria||Laurasiatheria|| Carnivora || cats and dogs |- ||Eutheria||Laurasiatheria|| Cetacea || whales and dolphins |- ||Eutheria||Laurasiatheria|| Chiroptera || bats |- ||Eutheria||Laurasiatheria|| Erinaceomorpha || hedgehogs |- ||Eutheria||Laurasiatheria|| Perissodactyla || horses |- ||Eutheria||Laurasiatheria|| Pholidota || scaly anteaters |- ||Eutheria||Laurasiatheria|| Soricomorpha || white-toothed shrews |- |||||||| |- ||Eutheria||Xenarthra|| Cingulata || armadillos |- ||Eutheria||Xenarthra|| Pilosa || giant anteaters |- |||||||| |- |||||||| |- ||Marsupial||Ameridelphia|| Didelphimorphia || Virginia opossums |- ||Marsupial||Ameridelphia|| Paucituberculata || shrew opossums |- |||||||| |- ||Marsupial||Australidelphia|| Dasyuromorphia || Tasmanian devils |- ||Marsupial||Australidelphia|| Diprotodontia || kangaroos |- ||Marsupial||Australidelphia|| Microbiotheria || monito del monte |- ||Marsupial||Australidelphia|| Notoryctemorphia || marsupial moles |- ||Marsupial||Australidelphia|| Peramelemorphia || marsupial omnivores |- |}</td></tr></table></div> Mammals are a class of [[tetrapods]] that include [[camels]], chimpanzees, [[Felidae|cats]] and rats and elephants, goats and sheep, horses and cows. <ref> http://species.wikimedia.org/wiki/Mammalia </ref> The class of mammals can be divided into two main infraclasses: eutheria and the [[marsupials]]. ===EUTHERIA = the crown placental mammals=== Eutheria (Greek: "true beasts") is a group of mammals consisting of the first placental mammal and all of her decendants. The earliest known fossil eutherian, Eomaia, was found in Asia, and is dated to the Early Cretaceous period, about 125 million years ago. <ref> http://en.wikipedia.org/wiki/Eutheria </ref> Most placental mammals belong to one of four main superorders: Afrotheria, Euarchontoglires, Laurasiatheria, and Xenarthra. <div class="thumb tright" style="border: 1px solid #CCCCCC; margin:0.5em;"> {|border="0" cellpadding="2" cellspacing="0" |{{clade| style=font-size:90%;line-height:75% | label1=Eutheria <br> <br> | 1={{clade | label1=Boreoeutheria <br> <br> | 1={{clade | 1=Euarchontoglires [[File:Eastern_Cottontail.JPG|60px]] | 2=Laurasiatheria [[File:Asiatische-Goldkatze-Catopuma-temminckii-tier-katze-0001_2.JPG|60px]]}} | label2=Atlantogenata <br> <br> | 2={{clade | 1=Xenarthra [[File:Bicho-preguiça_3.jpg|60px]] | 2=Afrotheria [[File:African_Elephant_(Loxodonta_africana)_male_(17289351322).jpg|55x60px]]}} }} }} |} </div> Euarchontoglires + Laurasiatheria = Boreoeutheria and Xenarthra + Afrotheria = Atlantogenata. Xenarthra diverged from Afrotheria relatively rapidly after Atlantogenata diverged from Boreoeutheria. There was considerable gene flow among the nascent clades prior to their divergence. This made it difficult to resolve the above phylogenic tree until relatively recently. <ref> http://mbe.oxfordjournals.org/content/24/9/2059.short </ref> '''MARSUPIALS''' The seven living orders of marsupials are divided into two clades: Ameridelphia and Australidelphia. <ref> http://www.ncbi.nlm.nih.gov/pubmed/12878458 </ref> '''MONOTREMATA''' <div class="thumb tright" style="border: 1px solid #CCCCCC; margin:0.5em;"> {|border="0" cellpadding="2" cellspacing="0" |{{clade| style=font-size:90%;line-height:75% | label1= mammals <br> <br> | 1={{clade | label1= Theria <br> <br> | 1={{clade | 1= Eutheria [[File:Elaphurus_davidianus_001.jpg|60px]] | 2=Marsupials [[File:Koala_climbing_tree.jpg|60px]]}} | 2={{clade | 1= Monotremata [[File:Duck-billed_platypus_(Ornithorhynchus_anatinus)_Scottsdale.jpg|60px]] }} }} }} |} </div> Order [[monotremata]] = Subclass prototheria Eutheria + Marsupials = Subclass theria ==Phylogeny== {{Clade | style=font-size:90%;line-height:70% |label1='''Mammalia''' |1={{Clade |label1=Yinotheria |1={{clade |1=†Shuotheriidae |label2=Australosphenida |2={{clade |1=†Ausktribosphenidae |2=†Bishopidae |3=†Henosferidae |label4=Monotremata |4={{clade |1=''Ornithorhynchus'' [[File:Genera mammalium Ornithorhynchus anatinus.jpg|50 px|''Ornithorhynchus anatinus'']] |label2=Tachyglossidae |sublabel2=(Echidna) |2={{clade |1=''Tachyglossus'' [[File:Tachyglossus aculeatus side on.jpg|60px|''Tachyglossus aculeatus'']] |label2=''Zaglossus'' |2={{clade |1=''Zaglossus attenboroughi'' [[File:Zaglossus attenboroughi Flannery.jpg|60px| ''Zaglossus attenboroughi'']] |2=''Zaglossus bruijnii'' [[File:Long-beakedEchidna.jpg|60px|''Zaglossus bruijnii'']] |3=''Zaglossus bartoni'' [[File:Zaglossus bartoni - MUSE.JPG|60px|''Zaglossus bartoni'']]}}}} }}}} }} |label2=Theria |2={{Clade |label1=Marsupialia |1={{Clade |1=Paucituberculata [[File:Phylogenetic tree of marsupials derived from retroposon data (Paucituberculata).png|50 px]] |2={{Clade |1=Didelphimorphia [[File:A hand-book to the marsupialia and monotremata (Plate XXXII) (white background).jpg|50 px]] |label2=Australidelphia |2={{Clade |1=Microbiotheria |label2=Eomarsupialia |2={{Clade |label1=Agreodontia |1={{Clade |1=Notoryctemorphia [[File:Phylogenetic tree of marsupials derived from retroposon data (Notoryctemorphia).png|50 px]] |2={{Clade |1=Peramelemorphia [[File:Phylogenetic tree of marsupials derived from retroposon data (Paramelemorphia).png|50 px]] |2=Dasyuromorphia [[File:Phylogenetic tree of marsupials derived from retroposon data (Dasyuromorphia).png|50 px]] }} }} |2=Diprotodontia [[File:A monograph of the Macropodidæ, or family of kangaroos (9398404841) white background.jpg|50 px|Macropodidae]] }} }} }} }} |label2=Placentalia |2={{Clade |label1=Atlantogenata |1={{Clade |label1=Xenarthra |1={{Clade |1=Cingulata [[File:Nine-banded-Armadillo white background.jpg|50 px|''Dasypus novemcinctus'']] |label2=Pilosa |2={{clade |1=Folivora [[File:Bradypus.jpg|50x50px]] |2=Vermilingua [[File:Natural history of the animal kingdom for the use of young people (Plate XV) (Myrmecophaga tridactyla).jpg|50 px|''Myrmecophaga tridactyla'']]}} }} |label2=Afrotheria |2={{Clade |label1=Paenungulata |1={{Clade |1=Hyracoidea [[File:DendrohyraxEminiSmit white background.jpg|50 px]] |label2=Tethytheria |2={{Clade |1=Sirenia [[File:Manatee white background.jpg|50 px|''Trichechus'']] |2=Proboscidea [[File:Indian elephant white background.jpg|50 px|''Elephas maximus'']] }} }} |label2=Afroinsectiphilia |2={{Clade |1=Tubulidentata [[File:Aardvark2 (PSF) colourised.png|50 px]] |label2=Afroinsectivora |2={{Clade |1=Macroscelidea [[File:Rhynchocyon chrysopygus-J Smit white background.jpg|50 px]] |2=Afrosoricida [[File:Potamogale velox illustration.jpg|50 px]] }} }} }} }} |label2=Boreoeutheria |2={{Clade |label1=Laurasiatheria |1={{Clade |1=Eulipotyphla [[File:Mole white background.jpg|50 px|Talpidae]] |label2=Scrotifera |2={{Clade |1=Chiroptera [[File:Vampire bat white background.jpg|50 px|Desmodontinae]] |label2=Ferungulata |2={{Clade |label1=Ferae |1={{clade |1=Pholidota [[File:FMIB 46859 Pangolin a grosse queue white background.jpeg|50 px|Manidae]] |2=Carnivora [[File:Cynailurus guttata - 1818-1842 - Print - Iconographia Zoologica - Special Collections University of Amsterdam - (white background).jpg|50px|''Acinonyx jubatus'']]}} |label2=Euungulata |2={{Clade |1=Perissodactyla [[File:Equus quagga (white background).jpg|50 px|''Equus quagga'']] |2=Artiodactyla [[File:Walia ibex illustration white background.png|50 px|''Capra walie'']] }} }} }} }} |label2=Euarchontoglires |2={{Clade |1={{Clade |label1=Glires |1={{Clade |1=Lagomorpha [[File:Bruno Liljefors - Hare studies 1885 white background.jpg|50 px|''Lepus'']] |2=Rodentia [[File:Ruskea rotta.png|50 px|''Rattus'']] }} }} |label2=Euarchonta |2={{Clade |1=Scandentia [[File:Die Säugthiere in Abbildungen nach der Natur, mit Beschreibungen (Plate 34) (white background).jpg|50 px]] |label2=Primatomorpha |2={{Clade |1=Dermoptera [[File:Cynocephalus volans Brehm1883 (white background).jpg|50 px]] |2=Primates [[File:Die Säugthiere in Abbildungen nach der Natur, mit Beschreibungen (Plate 8) (white background).jpg|50 px|''Cebus olivaceus'']] }} }} }} }} }} }} }} }} ==References== <references/> [[Category:Eukaryote phylogeny]] [[Category:Animals]] [[Category:Zoology]] [[Category:Biology]] [[Category:Taxonomy]] 2q6r37dqskewaldqflvjwwkl20lhvsq 0 139384 2844466 2843961 2026-11-04T13:32:06Z Young1lim 21186 /* Adder */ 2844466 wikitext text/x-wiki == Adder == * Binary Adder Architecture Exploration ( [[Media:Adder.20131113.pdf|pdf]] ) {| class="wikitable" |- ! Adder type !! Overview !! Analysis !! VHDL Level Design !! CMOS Level Design |- | '''1. Ripple Carry Adder''' || [[Media:VLSI.Arith.1A.RCA.20250522.pdf|A]]|| || [[Media:Adder.rca.20140313.pdf|pdf]] || [[Media:VLSI.Arith.1D.RCA.CMOS.20211108.pdf|pdf]] |- | '''2. Carry Lookahead Adder''' || [[Media:VLSI.Arith.2A.CLA.20260722.pdf|A]], [[Media:VLSI.Arith.2B.CLA.20261103.pdf|B]], [[Media:VLSI.Arith.2C.CLA.20261103.pdf|C]], [[Media:VLSI.Arith.2D.CLA.20260720.pdf|D]] || || [[Media:Adder.cla.20140313.pdf|pdf]]|| |- | '''3. Carry Save Adder''' || [[Media:VLSI.Arith.1.A.CSave.20151209.pdf|A]]|| || || |- || '''4. Carry Select Adder''' || [[Media:VLSI.Arith.1.A.CSelA.20191002.pdf|A]]|| || || |- || '''5. Carry Skip Adder''' || [[Media:VLSI.Arith.5A.CSkip.20250405.pdf|A]]|| || || [[Media:VLSI.Arith.5D.CSkip.CMOS.20211108.pdf|pdf]] |- || '''6. Carry Chain Adder''' || [[Media:VLSI.Arith.6A.CCA.20211109.pdf|A]]|| || [[Media:VLSI.Arith.6C.CCA.VHDL.20211109.pdf|pdf]], [[Media:Adder.cca.20140313.pdf|pdf]] || [[Media:VLSI.Arith.6D.CCA.CMOS.20211109.pdf|pdf]] |- || '''7. Kogge-Stone Adder''' || [[Media:VLSI.Arith.1.A.KSA.20140315.pdf|A]]|| || [[Media:Adder.ksa.20140409.pdf|pdf]]|| |- || '''8. Prefix Adder''' || [[Media:VLSI.Arith.1.A.PFA.20140314.pdf|A]]|| || || |- || '''9.1 Variable Block Adder''' || [[Media:VLSI.Arith.1A.VBA.20221110.pdf|A]], [[Media:VLSI.Arith.1B.VBA.20230911.pdf|B]], [[Media:VLSI.Arith.1C.VBA.20240622.pdf|C]], [[Media:VLSI.Arith.1C.VBA.20250218.pdf|D]]|| || || |- || '''9.2 Multi-Level Variable Block Adder''' || [[Media:VLSI.Arith.1.A.VBA-Multi.20221031.pdf|A]]|| || || |} </br> === Adder Architectures Suitable for FPGA === * FPGA Carry-Chain Adder ([[Media:VLSI.Arith.1.A.FPGA-CCA.20210421.pdf|pdf]]) * FPGA Carry Select Adder ([[Media:VLSI.Arith.1.B.FPGA-CarrySelect.20210522.pdf|pdf]]) * FPGA Variable Block Adder ([[Media:VLSI.Arith.1.C.FPGA-VariableBlock.20220125.pdf|pdf]]) * FPGA Carry Lookahead Adder ([[Media:VLSI.Arith.1.D.FPGA-CLookahead.20210304.pdf|pdf]]) * Carry-Skip Adder </br> == Barrel Shifter == * Barrel Shifter Architecture Exploration ([[Media:Bshift.20131105.pdf|bshfit.vhdl]], [[Media:Bshift.makefile.20131109.pdf|bshfit.makefile]]) </br> '''Mux Based Barrel Shifter''' * Analysis ([[Media:Arith.BShfiter.20151207.pdf|pdf]]) * Implementation </br> == Multiplier == === Array Multipliers === * Analysis ([[Media:VLSI.Arith.1.A.Mult.20151209.pdf|pdf]]) </br> === Tree Mulltipliers === * Lattice Multiplication ([[Media:VLSI.Arith.LatticeMult.20170204.pdf|pdf]]) * Wallace Tree ([[Media:VLSI.Arith.WallaceTree.20170204.pdf|pdf]]) * Dadda Tree ([[Media:VLSI.Arith.DaddaTree.20170701.pdf|pdf]]) </br> === Booth Multipliers === * [[Media:RNS4.BoothEncode.20161005.pdf|Booth Encoding Note]] * Booth Multiplier Note ([[Media:BoothMult.20160929.pdf|H1.pdf]]) </br> == Divider == * Binary Divider ([[Media:VLSI.Arith.1.A.Divider.20131217.pdf|pdf]])</br> </br> </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] [[Category:Digital Circuit Design]] [[Category:FPGA]] 7phx1gt5qvt62z69ipbc1tlbh3i6xdg 2844468 2844466 2026-11-04T13:33:29Z Young1lim 21186 /* Adder */ 2844468 wikitext text/x-wiki == Adder == * Binary Adder Architecture Exploration ( [[Media:Adder.20131113.pdf|pdf]] ) {| class="wikitable" |- ! Adder type !! Overview !! Analysis !! VHDL Level Design !! CMOS Level Design |- | '''1. Ripple Carry Adder''' || [[Media:VLSI.Arith.1A.RCA.20250522.pdf|A]]|| || [[Media:Adder.rca.20140313.pdf|pdf]] || [[Media:VLSI.Arith.1D.RCA.CMOS.20211108.pdf|pdf]] |- | '''2. Carry Lookahead Adder''' || [[Media:VLSI.Arith.2A.CLA.20260722.pdf|A]], [[Media:VLSI.Arith.2B.CLA.20261104.pdf|B]], [[Media:VLSI.Arith.2C.CLA.20261103.pdf|C]], [[Media:VLSI.Arith.2D.CLA.20260720.pdf|D]] || || [[Media:Adder.cla.20140313.pdf|pdf]]|| |- | '''3. Carry Save Adder''' || [[Media:VLSI.Arith.1.A.CSave.20151209.pdf|A]]|| || || |- || '''4. Carry Select Adder''' || [[Media:VLSI.Arith.1.A.CSelA.20191002.pdf|A]]|| || || |- || '''5. Carry Skip Adder''' || [[Media:VLSI.Arith.5A.CSkip.20250405.pdf|A]]|| || || [[Media:VLSI.Arith.5D.CSkip.CMOS.20211108.pdf|pdf]] |- || '''6. Carry Chain Adder''' || [[Media:VLSI.Arith.6A.CCA.20211109.pdf|A]]|| || [[Media:VLSI.Arith.6C.CCA.VHDL.20211109.pdf|pdf]], [[Media:Adder.cca.20140313.pdf|pdf]] || [[Media:VLSI.Arith.6D.CCA.CMOS.20211109.pdf|pdf]] |- || '''7. Kogge-Stone Adder''' || [[Media:VLSI.Arith.1.A.KSA.20140315.pdf|A]]|| || [[Media:Adder.ksa.20140409.pdf|pdf]]|| |- || '''8. Prefix Adder''' || [[Media:VLSI.Arith.1.A.PFA.20140314.pdf|A]]|| || || |- || '''9.1 Variable Block Adder''' || [[Media:VLSI.Arith.1A.VBA.20221110.pdf|A]], [[Media:VLSI.Arith.1B.VBA.20230911.pdf|B]], [[Media:VLSI.Arith.1C.VBA.20240622.pdf|C]], [[Media:VLSI.Arith.1C.VBA.20250218.pdf|D]]|| || || |- || '''9.2 Multi-Level Variable Block Adder''' || [[Media:VLSI.Arith.1.A.VBA-Multi.20221031.pdf|A]]|| || || |} </br> === Adder Architectures Suitable for FPGA === * FPGA Carry-Chain Adder ([[Media:VLSI.Arith.1.A.FPGA-CCA.20210421.pdf|pdf]]) * FPGA Carry Select Adder ([[Media:VLSI.Arith.1.B.FPGA-CarrySelect.20210522.pdf|pdf]]) * FPGA Variable Block Adder ([[Media:VLSI.Arith.1.C.FPGA-VariableBlock.20220125.pdf|pdf]]) * FPGA Carry Lookahead Adder ([[Media:VLSI.Arith.1.D.FPGA-CLookahead.20210304.pdf|pdf]]) * Carry-Skip Adder </br> == Barrel Shifter == * Barrel Shifter Architecture Exploration ([[Media:Bshift.20131105.pdf|bshfit.vhdl]], [[Media:Bshift.makefile.20131109.pdf|bshfit.makefile]]) </br> '''Mux Based Barrel Shifter''' * Analysis ([[Media:Arith.BShfiter.20151207.pdf|pdf]]) * Implementation </br> == Multiplier == === Array Multipliers === * Analysis ([[Media:VLSI.Arith.1.A.Mult.20151209.pdf|pdf]]) </br> === Tree Mulltipliers === * Lattice Multiplication ([[Media:VLSI.Arith.LatticeMult.20170204.pdf|pdf]]) * Wallace Tree ([[Media:VLSI.Arith.WallaceTree.20170204.pdf|pdf]]) * Dadda Tree ([[Media:VLSI.Arith.DaddaTree.20170701.pdf|pdf]]) </br> === Booth Multipliers === * [[Media:RNS4.BoothEncode.20161005.pdf|Booth Encoding Note]] * Booth Multiplier Note ([[Media:BoothMult.20160929.pdf|H1.pdf]]) </br> == Divider == * Binary Divider ([[Media:VLSI.Arith.1.A.Divider.20131217.pdf|pdf]])</br> </br> </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] [[Category:Digital Circuit Design]] [[Category:FPGA]] ln2bf91yr9j2f02azhslnk8mnjphn6b 2844471 2844468 2026-11-04T13:35:04Z Young1lim 21186 /* Adder */ 2844471 wikitext text/x-wiki == Adder == * Binary Adder Architecture Exploration ( [[Media:Adder.20131113.pdf|pdf]] ) {| class="wikitable" |- ! Adder type !! Overview !! Analysis !! VHDL Level Design !! CMOS Level Design |- | '''1. Ripple Carry Adder''' || [[Media:VLSI.Arith.1A.RCA.20250522.pdf|A]]|| || [[Media:Adder.rca.20140313.pdf|pdf]] || [[Media:VLSI.Arith.1D.RCA.CMOS.20211108.pdf|pdf]] |- | '''2. Carry Lookahead Adder''' || [[Media:VLSI.Arith.2A.CLA.20260722.pdf|A]], [[Media:VLSI.Arith.2B.CLA.20261104.pdf|B]], [[Media:VLSI.Arith.2C.CLA.20261104.pdf|C]], [[Media:VLSI.Arith.2D.CLA.20260720.pdf|D]] || || [[Media:Adder.cla.20140313.pdf|pdf]]|| |- | '''3. Carry Save Adder''' || [[Media:VLSI.Arith.1.A.CSave.20151209.pdf|A]]|| || || |- || '''4. Carry Select Adder''' || [[Media:VLSI.Arith.1.A.CSelA.20191002.pdf|A]]|| || || |- || '''5. Carry Skip Adder''' || [[Media:VLSI.Arith.5A.CSkip.20250405.pdf|A]]|| || || [[Media:VLSI.Arith.5D.CSkip.CMOS.20211108.pdf|pdf]] |- || '''6. Carry Chain Adder''' || [[Media:VLSI.Arith.6A.CCA.20211109.pdf|A]]|| || [[Media:VLSI.Arith.6C.CCA.VHDL.20211109.pdf|pdf]], [[Media:Adder.cca.20140313.pdf|pdf]] || [[Media:VLSI.Arith.6D.CCA.CMOS.20211109.pdf|pdf]] |- || '''7. Kogge-Stone Adder''' || [[Media:VLSI.Arith.1.A.KSA.20140315.pdf|A]]|| || [[Media:Adder.ksa.20140409.pdf|pdf]]|| |- || '''8. Prefix Adder''' || [[Media:VLSI.Arith.1.A.PFA.20140314.pdf|A]]|| || || |- || '''9.1 Variable Block Adder''' || [[Media:VLSI.Arith.1A.VBA.20221110.pdf|A]], [[Media:VLSI.Arith.1B.VBA.20230911.pdf|B]], [[Media:VLSI.Arith.1C.VBA.20240622.pdf|C]], [[Media:VLSI.Arith.1C.VBA.20250218.pdf|D]]|| || || |- || '''9.2 Multi-Level Variable Block Adder''' || [[Media:VLSI.Arith.1.A.VBA-Multi.20221031.pdf|A]]|| || || |} </br> === Adder Architectures Suitable for FPGA === * FPGA Carry-Chain Adder ([[Media:VLSI.Arith.1.A.FPGA-CCA.20210421.pdf|pdf]]) * FPGA Carry Select Adder ([[Media:VLSI.Arith.1.B.FPGA-CarrySelect.20210522.pdf|pdf]]) * FPGA Variable Block Adder ([[Media:VLSI.Arith.1.C.FPGA-VariableBlock.20220125.pdf|pdf]]) * FPGA Carry Lookahead Adder ([[Media:VLSI.Arith.1.D.FPGA-CLookahead.20210304.pdf|pdf]]) * Carry-Skip Adder </br> == Barrel Shifter == * Barrel Shifter Architecture Exploration ([[Media:Bshift.20131105.pdf|bshfit.vhdl]], [[Media:Bshift.makefile.20131109.pdf|bshfit.makefile]]) </br> '''Mux Based Barrel Shifter''' * Analysis ([[Media:Arith.BShfiter.20151207.pdf|pdf]]) * Implementation </br> == Multiplier == === Array Multipliers === * Analysis ([[Media:VLSI.Arith.1.A.Mult.20151209.pdf|pdf]]) </br> === Tree Mulltipliers === * Lattice Multiplication ([[Media:VLSI.Arith.LatticeMult.20170204.pdf|pdf]]) * Wallace Tree ([[Media:VLSI.Arith.WallaceTree.20170204.pdf|pdf]]) * Dadda Tree ([[Media:VLSI.Arith.DaddaTree.20170701.pdf|pdf]]) </br> === Booth Multipliers === * [[Media:RNS4.BoothEncode.20161005.pdf|Booth Encoding Note]] * Booth Multiplier Note ([[Media:BoothMult.20160929.pdf|H1.pdf]]) </br> == Divider == * Binary Divider ([[Media:VLSI.Arith.1.A.Divider.20131217.pdf|pdf]])</br> </br> </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] [[Category:Digital Circuit Design]] [[Category:FPGA]] 6rl7pvn3897ao2f8ge99ba4za4fggqf 102 206414 2844509 2817684 2026-11-04T19:32:15Z ~2026-58700-96 3113829 2844509 wikitext text/x-wiki 2844520 2844509 2026-11-04T22:26:00Z Atcovi 276019 Reverted edit by [[Special:Contributions/~2026-58700-96|~2026-58700-96]] ([[User_talk:~2026-58700-96|talk]]) to last version by [[User:MathXplore|MathXplore]] using [[Wikiversity:Rollback|rollback]] 2723014 wikitext text/x-wiki __NOTOC__ __NOEDITSECTION__ {{{{#titleparts:{{FULLPAGENAME}}|1}}/start tab}} {{box-transclude|Introduction|The {{BASEPAGENAME}} Portal}} <div style="float:left; width:66%; min-width:300px;"> {{{{FULLPAGENAME}}/box-header|Featured Resource|{{FULLPAGENAME}}/Featured Resource}} * [[WikiJournal of Medicine]] {{{{FULLPAGENAME}}/box-footer|}} {{{{FULLPAGENAME}}/box-header|Featured Picture|{{FULLPAGENAME}}/Featured Picture}} [[{{#invoke:Portals|randomlink|{{FULLPAGENAME}}/Featured Picture}}]] {{{{FULLPAGENAME}}/box-footer|}} </div> <div style="float:right; width:33%; min-width:150px;"> {{box-transclude|Categories}} {{box-transclude|Related Categories}} {{box-transclude|Related Portals}} </div> {{clear}} {{{{FULLPAGENAME}}/box-header|Major Wikiversity Portals|{{FULLPAGENAME}}}} {{Portals}} {{{{FULLPAGENAME}}/box-footer|}} {{{{FULLPAGENAME}}/box-header|Wikimedia|{{FULLPAGENAME}}}} {{Wikimedia for portals|page={{titleparts|1}}}} {{{{FULLPAGENAME}}/box-footer|}} {{Reflist}} {{end tab}} [[Category:{{PAGENAME}}| {{FULLPAGENAME}}]] [[Category:Subject portals|{{PAGENAME}}]] [[Category:Medicine portals]] 8ea4og7fc65ke7mze0b3yoyuw71rgbh 0 212733 2844499 2843574 2026-11-04T17:56:28Z Young1lim 21186 /* Using Libraries */ 2844499 wikitext text/x-wiki ==''' Part I '''== <!----------------------------------------------------------------------> === Introduction === * Overview * Memory * Number <!----------------------------------------------------------------------> === Python for C programmers === * Hello, World! ([[Media:CProg.Hello.1A.20230406.pdf |pdf]]) * Statement Level ([[Media:CProg.Statement.1A.20230509.pdf |pdf]]) * Output with print * Formatted output * File IO <!----------------------------------------------------------------------> === Using Libraries === * Scripts ([[Media:Python.Work2.Script.1A.20231129.pdf |pdf]]) * Modules ([[Media:Python.Work2.Module.1A.20261027.pdf |pdf]]) * Packages ([[Media:Python.Work2.Package.1A.20261027.pdf |pdf]]) * Libraries ([[Media:Python.Work2.Library.1A.20261102.pdf |A]], [[Media:Python.Work2.Library.1B.20260720.pdf |B]]) * Namespaces ([[Media:Python.Work2.Scope.1A.20231021.pdf |pdf]]) <!----------------------------------------------------------------------> === Handling Repetition === * Control ([[Media:Python.Repeat1.Control.1.A.20230314.pdf |pdf]]) * Loop ([[Media:Repeat2.Loop.1A.20230401.pdf |pdf]]) <!----------------------------------------------------------------------> === Handling a Big Work === * Functions ([[Media:Python.Work1.Function.1A.20230529.pdf |pdf]]) * Lambda ([[Media:Python.Work2.Lambda.1A.20230705.pdf |pdf]]) * Type Annotations ([[Media:Python.Work2.AtypeAnnot.1A.20230817.pdf |pdf]]) <!----------------------------------------------------------------------> === Handling Series of Data === * Arrays ([[Media:Python.Series1.Array.1A.pdf |pdf]]) * Tuples ([[Media:Python.Series2.Tuple.1A.pdf |pdf]]) * Lists ([[Media:Python.Series3.List.1A.pdf |pdf]]) * Tuples ([[Media:Python.Series4.Tuple.1A.pdf |pdf]]) * Sets ([[Media:Python.Series5.Set.1A.pdf |pdf]]) * Dictionary ([[Media:Python.Series6.Dictionary.1A.pdf |pdf]]) <!----------------------------------------------------------------------> === Handling Various Kinds of Data === * Types * Operators ([[Media:Python.Data3.Operators.1.A.pdf |pdf]]) * Files ([[Media:Python.Data4.File.1.A.pdf |pdf]]) <!----------------------------------------------------------------------> === Class and Objects === * Classes & Objects ([[Media:Python.Work2.Class.1A.20230906.pdf |pdf]]) * Inheritance <!----------------------------------------------------------------------> </br> == Python in Numerical Analysis == </br> </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] ==External links== * [http://www.southampton.ac.uk/~fangohr/training/python/pdfs/Python-for-Computational-Science-and-Engineering.pdf Python and Computational Science and Engineering] 2cbbhyxgkfxdfzccwoaxozfyh01iuxf 2844501 2844499 2026-11-04T17:58:19Z Young1lim 21186 /* Using Libraries */ 2844501 wikitext text/x-wiki ==''' Part I '''== <!----------------------------------------------------------------------> === Introduction === * Overview * Memory * Number <!----------------------------------------------------------------------> === Python for C programmers === * Hello, World! ([[Media:CProg.Hello.1A.20230406.pdf |pdf]]) * Statement Level ([[Media:CProg.Statement.1A.20230509.pdf |pdf]]) * Output with print * Formatted output * File IO <!----------------------------------------------------------------------> === Using Libraries === * Scripts ([[Media:Python.Work2.Script.1A.20231129.pdf |pdf]]) * Modules ([[Media:Python.Work2.Module.1A.20261027.pdf |pdf]]) * Packages ([[Media:Python.Work2.Package.1A.20261027.pdf |pdf]]) * Libraries ([[Media:Python.Work2.Library.1A.20261103.pdf |A]], [[Media:Python.Work2.Library.1B.20260720.pdf |B]]) * Namespaces ([[Media:Python.Work2.Scope.1A.20231021.pdf |pdf]]) <!----------------------------------------------------------------------> === Handling Repetition === * Control ([[Media:Python.Repeat1.Control.1.A.20230314.pdf |pdf]]) * Loop ([[Media:Repeat2.Loop.1A.20230401.pdf |pdf]]) <!----------------------------------------------------------------------> === Handling a Big Work === * Functions ([[Media:Python.Work1.Function.1A.20230529.pdf |pdf]]) * Lambda ([[Media:Python.Work2.Lambda.1A.20230705.pdf |pdf]]) * Type Annotations ([[Media:Python.Work2.AtypeAnnot.1A.20230817.pdf |pdf]]) <!----------------------------------------------------------------------> === Handling Series of Data === * Arrays ([[Media:Python.Series1.Array.1A.pdf |pdf]]) * Tuples ([[Media:Python.Series2.Tuple.1A.pdf |pdf]]) * Lists ([[Media:Python.Series3.List.1A.pdf |pdf]]) * Tuples ([[Media:Python.Series4.Tuple.1A.pdf |pdf]]) * Sets ([[Media:Python.Series5.Set.1A.pdf |pdf]]) * Dictionary ([[Media:Python.Series6.Dictionary.1A.pdf |pdf]]) <!----------------------------------------------------------------------> === Handling Various Kinds of Data === * Types * Operators ([[Media:Python.Data3.Operators.1.A.pdf |pdf]]) * Files ([[Media:Python.Data4.File.1.A.pdf |pdf]]) <!----------------------------------------------------------------------> === Class and Objects === * Classes & Objects ([[Media:Python.Work2.Class.1A.20230906.pdf |pdf]]) * Inheritance <!----------------------------------------------------------------------> </br> == Python in Numerical Analysis == </br> </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] ==External links== * [http://www.southampton.ac.uk/~fangohr/training/python/pdfs/Python-for-Computational-Science-and-Engineering.pdf Python and Computational Science and Engineering] 0spsm8qqq2jss29455di8gr8f5h1p21 0 241565 2844545 2844402 2026-11-05T01:35:23Z Jtneill 10242 /* Motivation */ Add tutorial comments 2844545 wikitext text/x-wiki {{/Banner}} ==2026== ===Motivation=== * All behaviour is motivated. * Self-determination theory ** Basic psychological needs: We need autonomy, competence and relatedness to experience intrinsic motivation. ** Intrinsic-extrinsic motivation (taxonomy of motivation) *** I think learning about concepts such as '''intrinsic and extrinsic motivation''' was interesting, because I was able to relate a concept to things we all do in our daily lives *** The emotional underpinning of motivation, specifically how extrinsic and intrinsic motivation are considered (completing intrinsically motivated tasks feels good). *** The internalisation continuum was my biggest aha moment. I recognised many of my “shoulds” as '''introjected regulation''': rules taken from outside but never really digested, held in place by shame, guilt and contingent self-worth. For example, “doing nothing means you’re lazy” and “there must be one correct answer”. ** Cognitive evaluation theory ===Emotion=== * Widened my base knowledge and expanded what I knew and I realised there is so much more to learn * Learning about all the '''different emotions''', and how they are separate etc '''basic, cognitively complex, and self conscious''' * How emotions influence behaviour and how there's '''no such thing as a bad emotion''' ===Other=== * My book chapter helped me dive so deeply in a topic I found interesting which was really insightful ==See also== * [[/Previous years/]] [[Category:Motivation and emotion]] tds9kp6fdl6mpdkm4sev5gkpv3217mf 2844546 2844545 2026-11-05T01:36:58Z Jtneill 10242 /* Emotion */ Add comments from tutorials 2844546 wikitext text/x-wiki {{/Banner}} ==2026== ===Motivation=== * All behaviour is motivated. * Self-determination theory ** Basic psychological needs: We need autonomy, competence and relatedness to experience intrinsic motivation. ** Intrinsic-extrinsic motivation (taxonomy of motivation) *** I think learning about concepts such as '''intrinsic and extrinsic motivation''' was interesting, because I was able to relate a concept to things we all do in our daily lives *** The emotional underpinning of motivation, specifically how extrinsic and intrinsic motivation are considered (completing intrinsically motivated tasks feels good). *** The internalisation continuum was my biggest aha moment. I recognised many of my “shoulds” as '''introjected regulation''': rules taken from outside but never really digested, held in place by shame, guilt and contingent self-worth. For example, “doing nothing means you’re lazy” and “there must be one correct answer”. ** Cognitive evaluation theory ===Emotion=== * Utility of emotions ** All emotions have some sort of utility, we shouldn’t necessarily try avoid them but can use them as important information about what our body and mind needs. ** How emotions influence behaviour and how there's '''no such thing as a bad emotion''' * Widened my base knowledge and expanded what I knew and I realised there is so much more to learn * Learning about all the '''different emotions''', and how they are separate etc '''basic, cognitively complex, and self conscious''' ===Other=== * My book chapter helped me dive so deeply in a topic I found interesting which was really insightful ==See also== * [[/Previous years/]] [[Category:Motivation and emotion]] e67zosm4hz1uhag05b9rmceeufso9q7 2844552 2844546 2026-11-05T01:58:00Z Jtneill 10242 2844552 wikitext text/x-wiki {{/Banner}} ==2026== ===Motivation=== * All behaviour is motivated. * Self-determination theory ** Basic psychological needs: We need autonomy, competence and relatedness to experience intrinsic motivation. ** Intrinsic-extrinsic motivation (taxonomy of motivation) *** I think learning about concepts such as '''intrinsic and extrinsic motivation''' was interesting, because I was able to relate a concept to things we all do in our daily lives *** The emotional underpinning of motivation, specifically how extrinsic and intrinsic motivation are considered (completing intrinsically motivated tasks feels good). *** The internalisation continuum was my biggest aha moment. I recognised many of my “shoulds” as '''introjected regulation''': rules taken from outside but never really digested, held in place by shame, guilt and contingent self-worth. For example, “doing nothing means you’re lazy” and “there must be one correct answer”. ** Cognitive evaluation theory ===Emotion=== * Utility of emotions ** All emotions have some sort of utility, we shouldn’t necessarily try avoid them but can use them as important information about what our body and mind needs. ** How emotions influence behaviour and how there's '''no such thing as a bad emotion''' * Learning about all the '''different emotions''', and how they are separate etc '''basic, cognitively complex, and self conscious''' * Feeling emotions in your body is important but when empathising with others it’s important to make a distinction between yourself and others - affective empathy is a high energy task that will quickly lead to burnout if left unregulated. Cognitive empathy is effective and a lower energy process, and often leads to motivation to support change. * Positive psychology ** Enhancing strengths is important, psychology shouldn’t just focus on solving psychopathology. (Positive psych) ** Happiness (a pleasant life) is not very malleable, fostering a good life (eudaimonic happiness) and a meaningful life is more important for wellbeing ===Other=== * Widened my base knowledge and expanded what I knew and I realised there is so much more to learn * My book chapter helped me dive so deeply in a topic I found interesting which was really insightful ==See also== * [[/Previous years/]] [[Category:Motivation and emotion]] j22e0ted6hjhwc7o0g1bxb3inniram9 2844553 2844552 2026-11-05T02:00:28Z Jtneill 10242 /* 2026 */ Add 2844553 wikitext text/x-wiki {{/Banner}} ==2026== ===Motivation=== * All behaviour is motivated. * The human needs that are essential: autonomy, self efficacy, and the human desire for relatedness. When you sitiate this in your own life, it resonates with what I would presume to be a general view, where people want agency or choice over their own life. * I like the positive and humanistic psychology thinking around motivation * Self-determination theory ** Basic psychological needs: We need autonomy, competence and relatedness to experience intrinsic motivation. ** Intrinsic-extrinsic motivation (taxonomy of motivation) *** I think learning about concepts such as '''intrinsic and extrinsic motivation''' was interesting, because I was able to relate a concept to things we all do in our daily lives *** The emotional underpinning of motivation, specifically how extrinsic and intrinsic motivation are considered (completing intrinsically motivated tasks feels good). *** The internalisation continuum was my biggest aha moment. I recognised many of my “shoulds” as '''introjected regulation''': rules taken from outside but never really digested, held in place by shame, guilt and contingent self-worth. For example, “doing nothing means you’re lazy” and “there must be one correct answer”. ** Cognitive evaluation theory ===Emotion=== * Utility of emotions ** All emotions have some sort of utility, we shouldn’t necessarily try avoid them but can use them as important information about what our body and mind needs. ** How emotions influence behaviour and how there's '''no such thing as a bad emotion''' * Learning about all the '''different emotions''', and how they are separate etc '''basic, cognitively complex, and self conscious''' * Feeling emotions in your body is important but when empathising with others it’s important to make a distinction between yourself and others - affective empathy is a high energy task that will quickly lead to burnout if left unregulated. Cognitive empathy is effective and a lower energy process, and often leads to motivation to support change. * Positive psychology ** Enhancing strengths is important, psychology shouldn’t just focus on solving psychopathology. (Positive psych) ** Happiness (a pleasant life) is not very malleable, fostering a good life (eudaimonic happiness) and a meaningful life is more important for wellbeing ===Other=== * Widened my base knowledge and expanded what I knew and I realised there is so much more to learn * My book chapter helped me dive so deeply in a topic I found interesting which was really insightful ==See also== * [[/Previous years/]] [[Category:Motivation and emotion]] je3ng0an72k1m9h956lsp9v7mlex2e2 0 264724 2844498 2844293 2026-11-04T17:10:40Z Scogdill 1331941 2844498 wikitext text/x-wiki {{Short description|Dress worn by Queen Victoria at her wedding to Prince Albert in 1840}} = Sandbox = Page to draft revisions for Wikipedia articles. For Gwladys Robinson, see Gwladys Lowther Robinson, [[Social Victorians/People/Ripon|Marchioness of Ripon]] and, earlier, [[Social Victorians/People/Lowther|Countess of Lonsdale]] ==References== {{reflist|2}} [[Category:1840 works]] [[Category:Royal wedding dresses|Victoria Queen]] [[Category:1840s fashion]] [[Category:British royal attire]] [[Category:Dresses in the Royal Collection of the United Kingdom|Victoria, Wedding]] [[Category:Diamond Jubilee of Queen Victoria]] = Victorian fashion = == Notes and Quotes == * Women followed Paris fashion in Europe and America, but London tailors led the way for men’s fashion. The Industrial Revolution actually impacted men’s fashion more than women’s. * Sewing machines in factories began mass producing cheap clothing for the largest group in the population, working class men. * Ready-made clothing affected the social structure, the economy, and gender differentiation. * Paper patterns, introduced in fashion magazines that targeted women, along with domestic sewing machines had more impact on women’s fashion than men’s. Haute couture was still bespoke (being individually constructed by tailors or dressmakers or fashion houses like Maison Worth). Sewing Machines and Clothes Before the industrial revolution wealthy people paid others to sew their clothes by hand and those who were of much lower economic status did all the manual labor necessary to feed and clothe the rich minority. The working class had to make all their own clothes. Sewing, however, came at the end of the process.  They began with raw wool cut from the backs of sheep or gathering flax. Only the wealthy could afford to buy fabric. They followed processes for cleaning and spinning the wool into thread and the processes of soaking and pulling out the fibrous strips to make into thread for linen fabric. After they had thread, it was tied to a loom so it could be woven into cloth. Then it had to be cut or torn into pieces. For many families scissors were a luxury. And the last step is sewing the pieces together to create a garment. Machines could not only sew, they could process the raw materials into thread and weave it into fabric much  quicker. Regarding clothing, working class men and their families benefitted greatly from ready-made clothes. Manufacturers marketed to them because they were the largest class. For example,<blockquote>–Men’s shirts, made by machine, take 3 hours and 10 minutes each to do; by hand, 15 hours and 20 minutes; a lady’s dress, by machine, takes 12 hours 4 minutes; by hand 20 hours 35 minutes; . . . a common frock-coat, by the machine occupies 17 hours 20 minutes; by the hand, 27 hours and 40 minutes: a linen vest, by machine, 3 hours 44 minutes; by hand 7 hours 18 minutes; a fine overcoat, by machine, 28 hours 13 minutes; by hand 31 hours 20 minutes; . . . <ref>Mitchell, Rebecca, ed. "22. 'Progress of the Sewing Machine,' ''Bow Bells'' (19 April 1865), p. 283." Section 6: Production and Industry. ''Fashioning the Victorians: A Critical Sourcebook''. Bloomsbury Visual Arts, 2018.</ref></blockquote> === Industrial Revolution and Technological Advancements === * the railroad * mass production of fabrics for working-class men's clothing ** Jacquard looms * sewing machines * aniline dyes, including black === Expanding Middle Class === * gender roles getting more clearly defined and rigid (referring back to page overview) === French Revolution === Sarah Gharmallah Alzahrani and Safia Abdelaziz Saroukh (https://www.researchgate.net/profile/Safia-Saroukh/publication/385099194_The_Semiotic_Dimension_of_Men's_Fashion_in_Modern_Eras/links/671686fbd796f96b8ec4f90e/The-Semiotic-Dimension-of-Mens-Fashion-in-Modern-Eras.pdf):<blockquote>The 19th century started with a fashion landscape that was changing dramatically and rapidly from the styles of a generation earlier. The French Revolution brought fashions that had been emerging since the 1780s to the forefront. Neoclassicism now defined fashion as both men and women taking inspiration from classical antiquity. For women, the high-waisted silhouette in lightweight muslin was the dominant style, while fashionable men looked to the tailors of Britain for a new, refined look [17]. [17]  Franklin, H. (Aug 18, 2020). Published on Jun 25, 2020, Retrieved: <nowiki>https://fashionhistory.fitnyc.edu/1800-1809/</nowiki> 11/11/2023. Edited. ...</blockquote>Brent Shannon:<blockquote>"Costume," wrote Max Beerbohm in 1896, "enables us to classify any 'professional man' at a glance, be he lawyer, leech or who not" (24–25). A man's profession and class were read by his jacket, his hat, what he rode in, and how he carried himself. "Perhaps there is a tendency among Englishmen to judge a man too much by the shape of his hat or the kind of collar he wears," conduct author John Wanamaker confessed; "But one must remember that in England if you ''wear'' the wrong thing, you will probably ''do'' the wrong thing, and generally ''be'' the wrong thing" (1).¹¹ Such assertions were predicated on the powerful Victorian conviction that outward appearance reflected inner qualities.<ref>{{Cite book|title=The Cut of His Coat: Men, Dress, and Consumer Culture in Britain, 1860–1914|last=Shannon|first=Brent Alan|publisher=Ohio University Press|year=2006|location=Athens, Ohio}}</ref> (148)</blockquote> ==Women's fashion== == Men's fashion == (Two intros: one to Victorian Fashion, giving background for both women's and men's fashions; two to the decades for men's fashion) Among the influences on men's styles: * differentiation of genders (domestic ideology, ideology of the spheres) * the "Great Male Renunciation," the French Revolution and English tailoring * industrial revolution: mass production of fabrics and articles of clothing, especially for working-class men * sports and country clothing * Albert Edward, Prince of Wales, later King Edward VII, influence went beyond England; tailors led by him led fashion for the upper classes * then later, cult of youth * military uniforms * comfort and convenience * evolving definitions of gender, both femininity and masculinity evolved as concepts. The changing definitions of masculinity affected men's clothing. For the growing and rising middle classes, gender roles became more and more rigid around the concept of the separate spheres, the public for men and the private sphere for women. Cultural sources for men's clothing and men's fashion: industrial revolution; expanding middle class; technological advancements; an association in the culture between outward appearance and inward nature; French Revolution; Albert Edward, Prince of Wales Citation for Victorian Hell: <ref name=":29">{{Cite web|url=https://victorianhell.substack.com/p/victorian-mens-fashion-history-and|title=Victorian Men’s Fashion History and Clothing Guide|last="It's Monty, Actually"|date=31 October 2025|website=Victorian Hell|access-date=7 September 2026}}</ref> === Waistcoats === *c. 1830, dark brown or black w faces, gold buttons, standing collar: File:Waistcoat MET 38.68 F.jpg * c. 1830, beige and white woven fabric with rose buds stamped or embroidered??: File:Waistcoat MET 19.181.4 F.jpg * 1830–1835, dotted and floral stamped fabric, brown and tan: File:Waistcoat MET 1976.254.1.jpg * * c. 1840, flowered fabric: File:Waistcoat MET 1979.346.45 F.jpg * c. 1840, red, jacquard-like, double-breasted: File:Waistcoat MET 1979.346.47 F.jpg * 1840s, white brocade: File:Waistcoat MET 1980.132.8 F.jpg * 1840s, dark blue, red, green? stamped?: File:Waistcoat MET 1976.254.1.jpg * Mid 1840s, brown, embroidered: File:Waistcoat MET 24.160.3 F.jpg * late 1840s, embossed white with hash thingies: File:Waistcoat MET 1978.88.9 F.jpg * 1845 waistcoat made from blue Chinese silk damask: File:1845 waistcoat made from blue Chinese silk damask.jpg * 1845–1855, white brocade wedding waistcoat, piping on the edges: File:Waistcoat MET 1986.373.3 F.jpg * '''1849–1860''', black w gold lining, ribbon: File:Waistcoat MET 1978.477.25 F.jpg * * 1850, print of St. James's Park: File:Print (BM 1880,1113.2351).jpg * 1850s, white and blue, w blue floral design stamped on white-and-grey stamped on white fabric: File:Waistcoat MET 1978.477.28 F.jpg *c. 1853, brown and tan plaid, with red squares: File:Waistcoat MET 1984.90.2 F.jpg *1854: Fashion plates, groups of ppl: File:Paris, New York & Philadelphia fashions for spring and summer 1854, published & sold by F. Mahan, no. 211 Chestnut Street, Philadelphia. LCCN2003655770.jpg *c. 1855, French fashion plate, frock coats, waistcoats, etc.: File:Album z ilustracjami do czasopism z moda francuska, niemiecka i angielska. ca 1855 (9109644).jpg * 1857, cream, printed flowers on check, watch pocket: File:Waistcoat MET 1978.88.8 F.jpg * * c. 1860, dark floral? fabric with machine embroidery?: File:Waistcoat MET 1983.479.13 F.jpg * * late 1870s–early 1880s, gold, woven: File:Waistcoat MET 1979.34.1 F.jpg === Frock coats === * Apostol Mănescu, politician, wearing a frock coat, butterfly tie, black waistcoat with button chain and light coloured pants, drainpipe type. Circa 1866. Photo by Franz Duschek.jpg (1866)[[File:Apostol Mănescu, politician, wearing a frock coat, butterfly tie, black waistcoat with button chain and light coloured pants, drainpipe type. Circa 1866. Photo by Franz Duschek.jpg|thumb|Romanian (?) Politician, c. 1866, ill-fitting but probably handmade suit|center]] Full-length portrait of two young men wearing frock coats with sashes and pistols at their waists LCCN2006689564.tif (right)[[File:Full-length portrait of two young men wearing frock coats with sashes and pistols at their waists LCCN2006689564.jpg|thumb|1860, American, young men with frock coats, performers??|center]] * George Augustus Henry Sala.jpg (Mathew Brady, 1855–1865) * [[File:George Augustus Henry Sala.jpg|center|thumb|George Augustus Sala, 1855–1865, LoC]] ==== Dress and Conduct Manuals, Histories, Victorian Theory, Catalogs ==== # [Baker, William Henry.]  ''Clothes Dictionary''. Chicago, Ill.: Cahn, Wampold, 1901. https://commons.wikimedia.org/wiki/File:Clothes_dictionary;_(IA_clothesdictionar00bake).pdf. # [[c:File:Hint's_about_men's_dress,_right_principles_economically_applied_(IA_hintsaboutmensdr00barr).pdf|Hint's about men's dress, right principles economically applied (IA hintsaboutmensdr00barr).pdf]] (book, conduct manual, American, 1888) #[https://upload.wikimedia.org/wikipedia/commons/6/69/Fashion-_the_power_that_influences_the_world_%28IA_fashionpowerthat00foxg%29.pdf?utm_source=commons.wikimedia.org&utm_campaign=index&utm_content=original Fashion- the power that influences the world (IA fashionpowerthat00foxg).pdf] (1871) (American, incl court dress, Victorian fashion theory, livery, "White, as it is the color of day, is expressive to us of the cheerfulness or gaiety which the return of day brings." [145], "Gentlemen's Furnishing and Goods," "Head and Foot Covering," Index) # [https://upload.wikimedia.org/wikipedia/commons/a/ae/How_to_behave_and_how_to_amuse._A_handy_manual_of_etiquette_and_parlor_games_%28IA_howtobehavehowto00sand%29.pdf?utm_source=commons.wikimedia.org&utm_campaign=index&utm_content=original How to behave and how to amuse. A handy manual of etiquette and parlor games (IA howtobehavehowto00sand).pdf] (1895; American; Men's Dress, p. 23; besides conduct and dress, it has parlor games) # [https://upload.wikimedia.org/wikipedia/commons/8/8e/The_etiquette_of_men%27s_dress_%28IA_etiquetteofmensd00cluc%29.pdf?utm_source=commons.wikimedia.org&utm_campaign=index&utm_content=original The etiquette of men's dress (IA etiquetteofmensd00cluc).pdf] (1888; American; some ads; anthology of individually written articles; lots of sports, outdoor events; search for "frock" — cutaway vs Prince Albert) #[https://upload.wikimedia.org/wikipedia/commons/e/e9/S.W._Silver_%26_Co.%27s_handbook_for_transatlantic_travellers_.._%28IA_swsilvercoshandb00silv%29.pdf?utm_source=commons.wikimedia.org&utm_campaign=index&utm_content=original S.W. Silver & Co.'s handbook for transatlantic travellers .. (IA swsilvercoshandb00silv).pdf] (1883; American; lots of ads from SW Silver, with prices; has blank pages where ppl can write "Introductions" and "Engagements," but this is not a conduct book, it's travel guide / catalog to London) #[https://upload.wikimedia.org/wikipedia/commons/f/f6/How_we_go_to_those_who_cannot_come_to_us.._%28IA_howwegotothosewh00roge%29.pdf?utm_source=commons.wikimedia.org&utm_campaign=index&utm_content=original How we go to those who cannot come to us.. (IA howwegotothosewh00roge).pdf] (Roger, Peet & Co.; American; catalog, spring 1891; pp. 11, clerical frock; 7 "Prince Alberts" double-breasted frocks; p. 23 shirts as underwear; has a thing on fitting and mail order; bathing suits, collars and cuffs; tennis shoes) == Victorian Men's Fashion (Intro to the Men's Fashion Section) == Museums and other historical sites have plenty of women’s clothes from the nineteenth century. There is great variety in both outer wear and underwear. “Men’s garments of the Victorian period have survived in far less quantity than women’s.” Buck p. 185 More waistcoats have survived than any other garment. The hardest garment to find are trousers. Very few survived the nineteenth century. Buck p. 185 By 1850 trousers dominated the fashion for men; the trouser straps under the shoe had disappeared; and a fly front had replaced the fall-front. Buck 185 It took most of the nineteenth century for men’s fashion to evolve into the coat, trousers and vest (waistcoat) that dominated the twentieth century and appears to be unchanging in the twenty-first century. “Men’s clothing, like women’s, was specific to each occasion and type of activity during the late nineteenth century.” Payne2, p. 518 For example, formal wear for social events was different from the casual, everyday wear for upper middle class and aristocratic men. Victorians also dressed in ensembles for hunting and riding (included knickerbockers–breeches that had a fuller cut). Casual and looser clothing was worn when participating in sports. Many different cuts of coats were available, like the Norfolk jacket, the morning coat, or the sack coat. “Tailored wool garments , originally inspired by the clothing of English country gentry, formed the backbone of men’s wardrobes in the early nineteenth century.” Men’s clothing began to change from styles that were colorful and flamboyant in the eighteenth century to styles that emphasized cut over color. By the end of the century men’s clothing had settled into a kind of uniform with coats and trousers usually made of the same fabric, often with a vest or waistcoat of the same fabric. The uniform included a shirt with a stiff collar, a necktie and very little jewelry. It took the entire nineteenth century to change to a rather static style that is still worn today. Ready-made clothing affected the social structure, the economy, and the gender differentiation. David Kutcha traces the origin of the 3-piece men's suit to King Charles II of England, who may be said to be wearing a coat, pants and a waistcoat, but otherwise is not wearing a "suit" we — or the Victorians — would recognize as such: <blockquote>... since 1666, male gentility has been associated with modesty and plainness in dress. Eschewing fashion as an increasingly feminized realm Charles II's vest inaugurated a new and essentially modern era of masculine aesthetics, one that reversed a long-held association between elaborate display and high social status. Manly thrift now displayed elite status.<ref>{{Cite book|title=The Three-Piece Suit and Modern Masculinity, England 1550–1850|last=Kutcha|first=David|publisher=University of California Press|year=2002|location=Berkeley and Los Angeles}}</ref> (2)</blockquote>So it's really Cromwell who should get credit?? The French Revolution caused both men and women to change stylistic direction radically. For one generation, stylish women wore simple white muslin dresses without foundation garments underneath. The changes in men's fashion was more lasting: men turned away from the ornamentation and ruffles of the ''ancien régime'', choosing instead understated and subtle tailoring to indicate good taste and wealth. Poole "comfort and convenience" — trousers, sack coats, collars and cuffs without a shirt, wool rather than silks and satins; knee breeches constantly riding up men as leaders of fashion, in spite of our sense that men didn't do fashion in the way that women did, were more serious than that === Dandy Stuff === Beau Brummell: "Men began to aspire to fine cutting, tailoring, and perfect fit in their clothes, flawless grooming and manners in themselves. The Englishman George Bryan (Beau) Brummell deserves much credit for the ideal of meticulous masculine appearance."<ref name=":25">{{Cite book|title=The History of Costume: From Ancient Mesopotamia Through the Twentieth Century|last=Payne|first=Blanche|last2=Winakor|first2=Geitel|last3=Farrell-Beck|first3=Jane|publisher=Addison-Wesley Longman|year=1992|isbn=0-06-047141-7|edition=2nd|location=New York, New York}}</ref> (458) "From 1796 to 1816, the Beau set the pattern for cleanliness and liberal use of starch." (459) === Gender Ideologies === '''Victorian gender ideologies''' relegated men to the "public sphere" and women to the "domestic sphere." (??) More than that “the cult of youth . . . the secularization of sport and the influence of modern warfare in generating men’s fashions, the tendency to prefer modern ideas of comfort and convenience, and the transformation of fashion knowledge and fashion urbanism from print to hyperreality”<ref name=":26">McNeil, Peter. "Men's Fashion: 1800–2022." Chapter 22. ''The Routledge History of Fashion and Dress, 1800 to the Present''. Routledge, 2024. https://opus.lib.uts.edu.au/bitstream/10453/182707/2/Men%27s%20Fashion%20200822_24_12_20_09_29_44.pdf DOI: 10.4324/9780429295607-27.</ref> (McNeil 1 or 411) changed the perceptions of masculinity and femininity. === Class === The growing middle class involved among other things more and more jobs and careers for young men as clerks, office workers, they were junior, subordinate, and they were commuting on the railroads from suburbs. Carolyn Kirby:<blockquote>In western Europe the fashion for plain dark suits coincided with the rise of the affluent middle-classes in a world where the pace of industrialisation and the globalisation of trade was accelerating as never before. The sharp, dark business suit became the last word in male power-dressing. And so it remains to this day.<ref name=":27">{{Cite web|url=https://historiamag.com/invent-masculine-fashion/|title=The invention of masculine fashion|last=Kirby|first=Carolyn|date=3 December 2025|website=Historia: Magazine of the Historical Writers' Association|access-date=25 August 2026}}</ref></blockquote> === Bertie === Albert Edward, Prince of Wales was very concerned with fashion and authoritative about it, with a very specific eye to small details. He was increasingly influential in men's fashion from the last third of the 19th and into the 20th century, and many noticed what he did and copied him.<blockquote>When that great lover of pleasure, Edward VII, visited Marienbad incognito as the Duke of Lancaster, he was followed by tailors from Paris, Budapest, Vienna, and Berlin who photographed him and took notes about his clothes. Edward VII introduced many [423–424] novelties into men’s fashion. ... He is also credited with making fashionable the creased trouser in 1909 (his groom dried them with a board weight after heavy rain, resulting in the line), turned-up cuff trouser (after hitching his trouser bottoms at a dirty racing track) and, as his girth grew, undoing the bottom button of his waistcoat."<ref name=":26" /> (424)</blockquote>Cowles<blockquote>The Prince was not just ‘a swell’. In the jargon of the day he was ‘a heavy swell’, and apparently there was a world of difference between the two terms. A swell was a rich young aristocrat who lived in extreme comfort; but a heavy swell added showmanship to the comfort and lived in a stylish luxury that even the French were obliged to envy. And of course the heavy swell was the acme of sartorial elegance. The Prince did not mind changing his dress half a dozen times a day. He loved clothes, and since whatever he chose to wear became the prevailing fashion overnight, he soon was regarded as an expert on the subject. His tailor-in-chief made a fortune. For many years he patronised a Mr. Poole. He discovered this gentleman by accident. He went to the theatre one night to see a well-known actor by the name of Fecher playing ‘Robert Macaire’. As an impecunious adventurer [129–130] Fechter was obliged to wear a coat that was torn and dirty, but Bertie’s expert eye noticed the elegant cut. At the end of the performance he asked Fechter for the name of his tailor, and Mr. Poole’s future was assured. The Prince had so many clothes he could never travel with less than two valets; and two more valets were left at home cleaning, brushing and pressing his vast wardrobe. There were suits and coats for every variation of every climate the world over. There were over a hundred pieces of headgear; and since Bertie was an honorary admiral and an honorary general of most of the countries of Europe, there was an entire room devoted to uniforms, sashes, epaulettes, belts, buckles, swords, feathers and other regalia. ... Men’s clothes became of such importance that new [130–131] shops sprang up like mushrooms in Savile Row, Clifford Street and Bond Street. Most of the Prince’s innovations were inspired by comfort and convenience. He altered the cut of the evening dress waistcoat, he shortened the tails on the tail coat, he left his frock coat open (due to an increasing girth), he introduced the black homburg, and he attended race meetings, not in the frock coat hitherto ''de regueur'' but in tweeds. He tried having his trousers creased down the sides rather than the front and back, in order to hide his bandy legs, but this idea did not catch on, and he soon discarded it himself. But the prince was not the only arbiter of men’s fashions. The band of  "heavy swells" who followed his lead gave him plenty of competition. Lord Raglan and Lord Petersham invented coats which are still named after them. Lord Dupplin the dinner jacket and Lord Cardigan the button-up sweater. But Lord Hardwicke made the most spectacular contribution. Men’s silk hats were made of beaver which was left in its original rough, shaggy state. Lord Hardwicke polished his hat until he could see his face in it, and consequently was known as "Glossy Top". He is responsible for the top hat as we know it today.<ref name=":28">{{Cite book|title=Gay monarch, the life and pleasures of Edward VII|last=Cowles|first=Virginia|publisher=Harper|year=1956|location=New York, New York|archive-url=https://archive.org/details/gaymonarchlifepl0000cowl/}}</ref> (129–131)</blockquote> === Men’s Clothing 1830–1900 (Intro to the Decades) === * “Tailored wool garments , originally inspired by the clothing of English country gentry, formed the backbone of men’s wardrobes in the early nineteenth century.”<ref name=":25" /> (458) Men’s clothing began to change from styles that were colorful and flamboyant in the eighteenth century to styles that emphasized cut over color. By the end of the century men’s clothing had settled into a kind of uniform with coats and trousers usually made of the same fabric, often with a vest or waistcoat of the same fabric. The uniform included a shirt with a stiff collar, a necktie and very little jewelry. It took the entire nineteenth century to change to a rather static style that is still worn today. (this is middle-class men for day wear?) * The silhouette for men sometimes followed the same line as women’s clothing on the top half of their bodies. For example, as women’s sleeves grew into the huge leg-of-mutton fullness, the shoulders of men’s coats also had added fabric to make a pouf at the shoulder, thus emphasizing a broad shouldered look across the front. Men’s silhouettes changed subtly without the exaggerated fullness of women’s skirts or the raising and lowering of the neckline. Once men began to wear trousers, they did not go back to formal knee breeches (except in European courts or for court wear for Victoria). * The line of men’s garments changed in smaller increments than women’s. Coats and trousers might fit tighter in some decades and looser in others. They were slim, then boxy, flared, then straight and evolved into the contemporary line we see on men today. * Men's fashions in the 19th century have been less researched than women's fashions, in part because of fewer artifacts of what men wore were saved in museums for study. * Photography, the first century that we have actual images (rather than artists' interpretations) of what people wore. ==== Court and Formal Dress ==== * Livery, working class * Formal dress was separate attire, used for specified occasions, varies by decade ==== Shirts ==== * Shirts and collars separated: "by 1827 detachable collars became available."<ref name=":25" /> (477) Shirts were considered more like underwear early in the century but by the end of the 1800s, they were becoming the outside garments popular in the twentieth and twenty-first centuries. * Neck treatments (and shirt cuffs) * Shirts were treated as underwear early in the century but by the end of the 1800s, they were becoming the outside garments popular in the twentieth and twenty-first centuries. ==== Coats and Jackets ==== The 1901 ''Clothes Dictionary'' says coats were "The principal garment of men's attire."<ref name=":30" />{{rp|10}} Several styles of coats were available to fashion-conscious aristocratic and upper-middle-class men. Men’s suits in this period were tailor made, fitted to one specific man. The factories, however, knew that the vast majority of men were working class and began to manufacture and sell ready-made shirts, coats and trousers. More clothes were available for working men and they cost less. The cheaper work clothes enabled upper-working-class men to own more than one set of clothes and to have more formal clothes for church or other events. Ready-made shirts were also available in generic sizes, small, medium and large. Generic sizing speeded up the mass production process and, since little of the shirts could be seen, they did not need custom fitting. Almost all of the coats Victorian men wore could be single or double breasted, with a couple of exceptions like morning coats and Norfolk jackets. On more formal occasions upper-class men wore a '''cutaway morning coat''' with light trousers during the daytime, and a dark '''tail coat''' and trousers in the evening. (define these terms) Morning coats was a general term for the jacket part of "full dress" or formal daytime wear, distinguished from formal evening wear. Most morning coats were cutaways, but the shape or cut is not the distinguishing feature: the bottom was curved or tailed, so the coat was short in front, it did not have a waist seam, single-breasted. Worn over a waistcoat and shirt. * According to Payne, "During the early 1820s tailcoats were the most popular type of coat."<ref name=":25" />{{rp|474}} * "Tailcoats gave way to skirted frock coats by the 1840s."<ref name=":25" /> (491) Tailcoats came down as far as the waist and then had tails (or "claw hammer" tails) attached at the waist in the back. Single-breasted, could be worn buttoned or unbuttoned in the front, over a waistcoat and shirt. * Nowadays, the only tailcoats are tuxedos, a term not in use until the 1880s, and named for Tuxedo Park, a village in New York populated by the elite. Originally only the white coat, ''tuxedo'' gradually came to mean the entire suit. ==== Frock Coats ==== [[File:Frock coat MET CI38.23.39 F.jpeg|alt=Old-fashioned coat hanging on a mannequin|thumb|Frock coat, 1840, American]] * Frock coats were introduced in the 1820s and worn throughout the nineteenth century with only minor changes to the line or fit. Sleeves were full at the top and waists were tight, creating an hourglass form. Initially frock coats were tightly fitted and closed with buttons. The more tailored frock coats had a waist seam and a flared skirt almost to the knees. The rate of fashion change notable in women's fashion was true for men's clothing as well. The frock coat persisted for a century, but it was in a state of constant (but subtle) evolution, with collars and lapels, coat length, the waist seam, the fullness of the coat itself as well as the sleeve, color, patterns like stripes or plaids. * A Prince Albert is a double-breasted, very well-tailored, form-fitting frock coat.<ref name=":30">[Baker, William Henry.] ''Clothes Dictionary''. Chicago, Ill.: Cahn, Wampold, 1901. https://commons.wikimedia.org/wiki/File:Clothes_dictionary;_(IA_clothesdictionar00bake).pdf.</ref>{{rp|33}} * Gentlemen, clergy, military * Men around the world and especially in the British Empire wore frock coats: images and descriptions exist of men wearing frock coats in India<ref>Woodyatt, Nigel Gresley. ''Under Ten Viceroys; The Reminiscences of a Gurkha''. London: H. Jenkins Limited, 1922. Pp. 31, 107, 1774.</ref>, Iran (Persia at that time)<ref>Sparroy, Wilfrid. ''Persian children of the royal family; the narrative of an English tutor at the court of H. I. H. Zillus-Sultán, G.C.S.I''. London and New York, J. Lane, 1902. Pp. 11, 29, 30.[https://upload.wikimedia.org/wikipedia/commons/d/d1/Persian_children_of_the_royal_family%3B_the_narrative_of_an_English_tutor_at_the_court_of_H._I._H._Zillus-Sult%C3%A1n%2C_G._C._S._I_%28IA_persianchildreno00sparrich%29.pdf?utm_source=commons.wikimedia.org&utm_campaign=index&utm_content=original https://upload.wikimedia.org/wikipedia/commons/d/d1/Persian_children_of_the_royal_family%3B_the_narrative_of_an_English_tutor_at_the_court_of_H._I._H._Zillus-Sult%C3%A1n%2C_G._C._S._I_%28IA_persianchildreno00sparrich%29.pdf].</ref>, Japan<ref>Watson, W. Petrie. ''Japan, Aspects and Destinies''. New York, Dutton: 1904. P. 7, 19, 31.</ref>, Egypt and Turkey<ref>Butler, Alfred Joshua. ''Court Life in Egypt''. London: Chapman and Hall, 1887. P. 7.</ref>, ... * Payne: "In the 1720s, Englishmen adopted another coat [than the redingote]: a straighter, functional style that originated among the working classes. Made of plain woollen cloth in practical colors such as brown, it had no pleats, narrow sleeves, a small flat collar, and a minimum of decoration. This became known as the frock coat and was useful for riding, hunting, and overseeing one's lands." (402) ==== Trousers ==== In 1872, in analyzing men's clothing as evolutionary behavior — like coloration for mating among birds in the natural world — George Darwin (Charles's son) says, of trousers:<blockquote>... a form yields to a succeeding form, which is better adapted to the then surrounding conditions; thus, when it ceased to be requisite that men in active life should be ready to ride at any moment, and when riding had for some time ceased to be the ordinary method of travelling, knee breeches and boots yielded to trousers.<ref name=":33" /> (125 of 565)</blockquote> * Working-class men had been wearing trousers (or "trowsers") increasingly since the 16th century in Europe,('''citation''') but after the French revolution the middle and upper classes adopted them because "the safest dress was that of the working man, ... who did not wear knee breeches, but rather trousers."<ref name=":26" /> (416) * Lewandowski: "First worn by the early horse-riding steppe people, introduced to Western world by Italians in 16th century. They were accepted as informal dress in the Directoire (1790–1815 C.E.). They became a part of semi-formal dress around 1815 and became worn commonly after 1830."<ref name=":31">{{Cite book|title=The Complete Costume Dictionary|last=Lewandowski|first=Elizabeth J.|publisher=Scarecrow Press|year=2011}}</ref> (29) * Breeches: Men’s bifurcated garment that ended just below the knee * Trousers (or long pants): men’s bifurcated garment that went down to the ankles or foot * The most sweeping change in men’s fashion before 1820 was in the length of their pants. Trousers that went from waist to foot were so much more comfortable and popular than knee breeches that they replaced what had dominated men’s wear since the sixteenth century. And trousers have been worn for over two centuries now. * They were originally tailored but began to be mass produced. Although the aristocracy were required to wear knee breeches at court, they wore trousers more than they wore breeches, and they could afford to have their suits and shirts hand sewn and individually. Working class men had to contend with small, medium, and large from mass production. * This bifurcated garment was modified in specific places, like the length, the fullness over the hips or the width of the leg. In summer trousers were lighter in color — "grayish blue, aqua and pearl gray." (Payne 2, 505) In winter trousers mixed darker colors like "steel gray and brown, black and brown, and black and green." (Payne 505) More on summer colors, winter colors? * [Find quote about QV's men's court dress and the men being so cold] ==== Waistcoats ==== For most of the century, the waist-length sleeveless coat that men wore over their shirt and under their suit coat was called a waistcoat. By the 1880s, everyday waistcoats were called vests, and what was called a waistcoat was worn with formal, evening attire like tailcoats and tuxedos. Necklines rose and fell and collars and lapels came and went over the century, but by the end of the century, formal waistcoats had low necklines and lapels.<ref name=":25" /> (528) Many more waistcoats exist as artifacts in collections than vests, perhaps because they were special and thus were saved.[[File:Waistcoat MET CI60.22.55 F.jpg|thumb|Men's waistcoat, 1850s]] Over the course of the century the waistcoat evolved from ornate and decorative to something simpler, whose fabric matched the suit coat and trousers — the vest, in fact. Victorian men put their creativity, color and specialty fabrics into their waistcoats. At the beginning of the century men's waistcoats were made from specialty fabrics, like velvet or brocade or a patterned silk. Early in the century, stylish men wore two vivid waistcoats at the same time.<ref name=":25" /> (474) An elaborate waistcoat might have a colorful, patterned base fabric as well as a floral ribbon embellishment (for example, see right). Long after coats and trousers were made from the same dark wool, waistcoats continued to be colorful, making them the focus of the ensemble. By the end of the 19th century, the plain style came to dominate men’s fashion (and continues to dominate it to this day). When waistcoats became vests, they became part of the plain style. [[File:Homosexualitywilde.jpg|thumb|Oscar Wilde and Bosie Douglas, straw boater and derby, 1894]] ==== Overcoats ==== As their name suggests, overcoats or surcoats were coats worn for warmth and for protection from rain and snow on top of the other articles of clothing, including suit coats and frock coats. Overcoats became a significant fashion element, and many styles were seen throughout the century. Victorians used the terms ''overcoat'' and ''great coat''., and sometimes perhaps ''surtout''. A great coat was a "large overcoat, commonly made of wool."<ref name=":31" /> (124) In his 1901 ''Clothes Dictionary'', Baker says a surtout is "An overcoat cut in the style of a frock coat."<ref name=":30" /> (39) Lewandowski says something more generic — "term referring to any man's long cloak or coat."<ref name=":31" /> (282) Victorians also specified which of the many kinds of overcoat: Chesterfield coat, Inverness coat, macintosh (or mackintosh), paletot, redingote, Ulster coat ===== Redingote ===== * Redingotes had a rolled collar, sleeves with fullness at the shoulders and upper arm and narrow at the elbows to the wrist. * It was fitted at the waist and reached mid-calf. existed various forms in the 18th century. * Women's redingote 1865–1890, in France, to cover the bustle. (Lewandowski, 247–248) * Payne: "The greatcoat or redingote evolved into an overcoat ..., but the overcoat lacked the pleats and large cuffs of the redingote." (402) * high, double collar, standing collar, single- or double-breasted. "from riding coat" (Payne 402) * Rebecca Mitchell: "full-length overcoat, usually open at the front and belted, popular in varying designs through the eighteenth and nineteenth centuries."<ref name=":32">Mitchell, Rebecca, ed. ''Fashioning the Victorians: A Critical Sourcebook''. Bloomsbury Visual Arts, 2018.</ref> (527 or 565) [[File:George Stanhope, 7th Earl of Chesterfield.jpg|thumb|George Stanhope, 7th Earl of Chesterfield, wearing a Chesterfield coat, 1860]] ===== Chesterfield Coat ===== * "Chesterfield coats were single-breasted, with fly fronts [regular buttons and button holes, as opposed to fall front, in which a panel of cloth covers the button holes and the placket covers the closure completely] and velvet collars."<ref name=":25" />  Payne2 p. 519 * They were one kind of overcoat, named for George Stanhope, 6th Earl of Chesterfield. 1840s. * Signature velvet collar. * "Man's fitted overcoat with hidden buttons and velvet collar"<ref name=":31" /> (62) * "fitted men’s knee-length overcoat, narrower than the frock coat and without a waist seam, popular from the end of the nineteenth century."<ref name=":32" /> (519 of 565) ===== Ulster Coat ===== In 1872, in analyzing men's clothing as evolutionary behavior — like coloration for mating among birds in the natural world — George Darwin (Charles's son) says, of Ulster coats:<blockquote>... a form yields to a succeeding form, which is better adapted to the then surrounding conditions .... The "Ulster Coat", now so much in vogue, is evidently largely fostered by railway travelling, and could hardly have flourished in the last century, when men either rode or travelled in coaches, where there was no spare room for any very bulky garment.<ref name=":33">Mitchell, Rebecca, ed. "3. 'George H. Darwin, ‘Development in Dress', ''Macmillan’s Magazine'' 26 (September 1872), pp. 410–16." Section 1: Fashion Theory in the Nineteenth Century. ''Fashioning the Victorians: A Critical Sourcebook''. Bloomsbury Visual Arts, 2018.</ref> (125 of 565)</blockquote> * "men’s overcoat with attached waist-length cape, popular in the nineteenth century."<ref name=":32" /> (530 of 565) * ulster overcoat for travel. * "Man's single- or double-breasted overcoat with waist belt. Initially had hood, but by 1870s was detachable. In 1875, the ticket pocket was added on the left sleeve above the cuff."<ref name=":31" /> (304) * "Travel called for an ulster — a long, loose, heavy overcoat — or for an Inverness coat with attached cape."<ref name=":25" /> (527) * "Heavy tweed coats for travel such as the ulster had half-belts and box pleating in back."<ref name=":25" /> (551) * Ulster may have had a shorter cape than the Inverness * Can we find a source linking John McGee (of McGee & Co.) with the invention of the Ulster coat in 1866? (per Wikipedia)? [[File:The man wot drove the sovereign to the last stage trying it on the next. (BM 1868,0808.9200).jpg|thumb|Duke of Wellington, 1830, wearing a coachman's coat, or Inverness coat]] ===== Inverness Coat ===== * "men’s overcoat, with long, full, open sleeves, giving a cape-like appearance from the front; worn from the mid-nineteenth century to the early twentieth century as formal wear."<ref name=":32" /> (523 of 565) * an Inverness coat with up to five cape collars and a high curved collar. * Lewandowski: "(1840–1865 C.E.) to 20th century. Worn after 1858, man's large, loose, knee-length overcoat with fitted collar and long cape."<ref name=":31" /> (143) * An Inverness coat had a high standing collar as well as up to 5 capes over a knee-length, loose overcoat.<ref name=":31" /> (143) * Caricature of the Duke of Wellington in a "coachman's caped great-coat,"<ref>{{Cite web|url=https://www.britishmuseum.org/collection/object/P_1868-0808-9200|title=Duke of Wellington in Coachman's coat. print; satirical print {{!}} British Museum|website=www.britishmuseum.org|language=en|access-date=2026-11-04}}</ref> or Inverness, 1830 * In the 1880s, the Inverness was sometimes sleeveless, so the capes were not merely decorative. * both formal and informal styles exist; also called a coachman's cape. [[File:Carson, Pirie, Scott & Co. Macintosh, 1893.jpg|thumb|360x360px|Gentleman's Macintosh, 1893]] ===== Mackintosh ===== * "Introduced in 1836, short loose overcoat made of Mackintosh's [sic] patent India-rubber cloth. Came in drab or dark green."<ref name=":31" /> (181) * "waterproofed overcoat, named after its inventor [Charles Macintosh]; now used generically to refer to raincoat."<ref name=":32" /> (524 of 565) * [according to Wikipedia, the spelling ''Mackintosh'' is now standard.] ===== Paletot ===== * "short, loose overcoat for men or women."<ref name=":32" /> (526 of 565) <ref name=":24" />Kay, Fiona and Neil R. Storey, ''Victorian Fashions for Women'', Pen & Sword History, Yorkshire - Philadelphia, 2022 ==== Hats ==== The upper-class continued to wear top hats, and [[bowler hats]] (known in America as a derby) were worn by the working class. 1830s This class of men also wore round cloth caps, often seen on male children. The straw boater was the most popular hat by the end of the century and was worn by all of the classes. Men wore [[top hats]], with wide brims in sunny weather. 1830s - 1900 Top hats briefly became the very tall "stovepipe" shape, but a variety of other hat shapes were popular. Homburgs, deerstalkers and “Trilbies” were also popular. Buck, p. 198-199 (George de Maurier novel ''Trilby'') Hats were ''de rigueur'' for all Victorians, all classes, all ages, all genders. ==== Shoes ==== [[File:Men's fashion silhouette of 1837.jpg|thumb|Men's coats, 1837]] [[File:Lev Wittgenstein by F. Kruger.JPG|thumb|Lev Wittgenstein by F. Kruger, 1836, wearing a surtout]] === 1830s and 1840s === For men's fashion, the 1830s and 1840s were dominated by a defined waist, lean limbs and tall hats. Working-class men had been wearing trousers for some time, but by the 1830s and '40s, middle- and upper-class men generally wore frock coats and trousers for casual or less formal attire in the city. In the country, what they wore depended on what they were doing — hunting, riding, shooting and so onen[[File:Man's Frock Coat LACMA M.2007.211.61 (1 of 4).jpg|alt=Male mannequin wearing a light-colored thigh-length coat similar to a suit jacket and a top hat|thumb|Men's frock coat, c. 1845]] [[File:Stockholms mode-journal- Tidskrift för den eleganta werlden 1847, illustration nr 2.jpg|thumb|Stockholm's fashion magazine, 1847]]Some men wore corsets during this time, so the waist was nipped in and the skirt of the frock coat flared out. The sleeves of their coats and jackets were slightly puffed at the shoulders. The trouser legs were straight with a strap under the instep. The figure on the left in the fashion plate from ''Modes de Paris Journal des Tailleurs'' (right) illustrates this silhouette very clearly as well as the frock coat, the strap under the instep, and the puffed shoulders. ==== Coats and Jackets ==== Frock coats had become fashionable for day wear in the 1820s<ref name=":25" />{{rp|474}} and were worn throughout the century. In the 1830s frock coats were tightly fitted and closed with buttons. The more tailored frock coats had a waist seam and a flared skirt almost to the knees. The main changes in frock coats in these two decades were in the colors and the neck treatments (collar and lapels). Initially frock coats were not necessarily black. They could be colorful — like "plum, mulberry, brown, bright blue, and bottle green" — with "contrasting velvet collars."<ref name=":25" />{{rp|474}} The truly bright, saturated colors made possible by synthetic dyes, however, were a couple of decades away still and were never used for frock coats. Middle- and upper-class men limited their use of tailcoats to evening wear and more formal occasions. (In the 1847 Stockholm fashion magazine, below right, the figure in the center is wearing a tailcoat.) A cutaway morning coat was worn with light trousers for formal events during the daytime, and a dark tail coat and trousers were worn in the evening. ==== Shirts ==== Shirts were made of linen or cotton with low collars, occasionally turned down, and were worn with wide [[cravats]] or neck ties. Before they were mass produced, shirts made of finer linen or cotton were tailored for upper and wealthy middle-class men. Depending on occupation, working-class men's shirts were made from a rougher, home-woven linen or cotton, often shaped loosely like a smock and sometimes without a front closure. Shirts were sometimes as long as knee-length. ==== Trousers ==== Men had adopted trousers over knee breeches for semi-formal dress "around 1815"<ref name=":31" /> (298) beginning after the French Revolution.<ref name=":26" /> (416) They were wearing trousers both for formal and semi-formal dress by the 1830s. Initially trousers "fit quite snugly ..., but in the 1840s they became looser."<ref name=":25" /> (477) Upper-class and aristocratic men still wore knee breeches at court and loose-fitting breeches (in the form of knickerbockers) for country wear and some sports, like riding and golf. Generally, however, they wore trousers more than they wore breeches. Men flocked to them once trousers became a stylish option because the looser fit was warmer in cold weather and more comfortable than the tight-fitting knee breeches from earlier in the century. Ready-made men's clothing was available in the US early, but by the 1840s ready-made suits were making an impact in Europe and the UK,<ref name=":26" /> (419–420) especially outside the upper classes. Aristocrats and upper middle class could afford the well-tailored cut and fit of individually hand-sewn trousers and coats. ==== Waistcoats ==== Waistcoats were both single- or double-breasted, with shawl or notched collars, and might be finished in double points at the lowered waist. Collars gradually lowered over the course of the two decades. Often the colorful vests were decorated with fur or velvet collars. “Waistcoats remained colorful, decorative items in men’s wardrobes after the rest of their costumes had become plain.” (Payne 2 460-462) <nowiki>:</nowiki> "During the 1820s and 1830s, men sometimes wore two waistcoats, in combinations like white velvet over rose-and-gold brocade."<ref name=":25" /> (474) ==== "Overcoats ==== Men wore a variety of topcoats or overcoats. Surtouts were overcoats built like long frock coats, like the one worn by Lev Wittgenstein in the portrait (above right). The redingote was most popular in the 1830s and 1840s. ==== Original Text ==== During the [[1840s in fashion|1840s]], men wore tight-fitting, calf length [[frock coat]]s and a [[waistcoat]] or vest. Sleeves were full at the top and waists were tight, creating an hourglass form. For more formal occasions, a cutaway morning coat was worn with light trousers during the daytime, and a dark tail coat and trousers was worn in the evening. Shirts were made of linen or cotton with low collars, occasionally turned down, and were worn with wide [[Cravat (early)|cravat]]s or neck ties. Men wore [[top hat]]s, with wide brims in sunny weather. === 1850s === The 1850s revealed several industrial inventions that significantly impacted clothing. First, the Industrial Revolution began (and still does today) with machines to weave fabric and machines to sew fabrics. Manufacturers quickly learned that the majority of the population was working class. Theirs were the simplest, least trimmed garments made in the nineteenth century. Second, mass production of shirts, trousers and coats changed how most people shopped for clothes. The aristocracy and upper middle class folks could still have their clothes made to specifically fit their bodies. Manufacturers knew that they could not mass produce clothing for each individual’s body. They invented generic sizing in small, medium, and large. They could, therefore, produce many more items of clothing with generic sizing. Third, Isaac Singer patented and began to manufacture home or domestic sewing machines (treadle machines) enabling more form-fitted '''dresses''' to be constructed for an individual’s specific shape and size. The fourth item to impact fashion and clothing was the invention of the paper pattern. These were introduced in fashion magazines targeting women. Almost all women were taught to sew by hand, whether they were aristocrats or working class. Sewing machines enormously speeded up the '''process of constructing from hours to minutes'''. Finally, aniline dyes made colors brighter and longer lasting. Prints and stripes, checks and plaids, fabrics had much more vibrant color with aniline dyes. The Industrial Revolution impacted the working class and changed their lives the most. children's clothes? Women made men's clothing too According to Judith Flanders, working-class men's clothes could be colorful. For example drivers of hansom cabs () dressed more up-scale, and drivers of hackneys () did not.<blockquote>While hackney drivers were also considered to be stereotypically shabby, hansom-cab drivers were generally represented as smartly dressed. A print in 1850 showed a driver in a snappy brown coat instead of the coachman’s heavy multiple-caped outfit, pale green striped trousers, short boots and top hat, the [167–168] reins held daintily in his gloved hands. Both cab and coach drivers wore top hats, but cabbies of a sporting bent later switched to bowlers, and in summer donned bright checked outfits.<ref name=":23">{{Cite book|title=The Victorian City: Everyday Life in Dickens' London|last=Flanders|first=Judith|publisher=Thomas Dunne Books|year=2012|location=New York, New York}}</ref> (167–168 [of 972])</blockquote> A variety of silhouettes was seen in the 1850s. Trousers were long and slim with a strap under the instep of the shoe. ==== Coats and Jackets ==== "Although the frock coat was worn throughout the century, it lost its dominant position in the 1850s, when the morning coat began to replace it."<ref name=":29" /> A jacket shorter than the frock coat or morning coat that had visible pockets began to be worn in about 1850. Cutaway tail coats were still reserved for formal occasions. All of these coats could be single or double breasted. They were called “ditto suits” by those who wore them because the coat, trousers, and waistcoat were all made from the same fabric. There were morning coats and cutaway morning coats; there were Norfolk jackets and ditto suits. The most formal were the cutaway tail coats for evening wear and balls; and the least formal were the boxy shaped sack coats. By the 1860s, it had become the most popular casual coat. ==== Shirts ==== During the [[1850s in fashion|1850s]], men started wearing shirts with high upstanding or turnover [[collar (clothing)|collars]] and [[necktie#Four-in-hand|four-in-hand necktie]]s tied in a bow, or tied in a knot with the pointed ends sticking out like "wings." ==== Trousers ==== Trousers had fly fronts and breeches were used for formal functions and knickers (short pants fuller cut than breeches) for horseback riding. Men's choice to wear trousers is an instance of the importance of comfort and convenience outweighing style and presentation, a value that increased in importance through the 20th century. ==== Waistcoats ==== Waistcoats remained the one garment that could be brightly colored and copiously trimmed on exotic fabrics. Waistcoats show that men were concerned with decorative elements in their ensemble. ==== Notes & Quotes ==== * transition from frock coats to ditto suits, 1850s (Payne, 463) ==== Hats ==== The upper-class continued to wear top hats, and [[bowler hat]]s were worn by the working class. Caps ==== Overcoats ==== Overcoats were cut fuller with wider sleeves and no extra fullness at the shoulder. Unlike the redingote, the overcoats had straight seams to the knee length. Many had wide lapels that reached below the waist. Working class men and their families benefitted more from the industrial revolution in regards to clothing.  Manufacturers marketed to them because because they were the largest class. [[File:Gazette of Fashion, February 1865, RP-P-2009-2675.jpg|thumb|1865, Frock coats, etc. (Paris)]] [[File:Gazette of Fashion, October 1864, RP-P-2009-2673.jpg|thumb|1864, Frock coats, etc., Paris]] [[File:Broadway fashions, New York LCCN2003677917.jpg|thumb|Butterick, Spring-Summer 1866, Broadway Fashions]] [[File:Four young men wearing frock coats - DPLA - 4706a21c9267c45034d51c8fc5309251 (page 1).jpg|thumb|1860s, working class?, ill fitting, frock coats, trousers, shirts, vest, derby hats]] === 1860s === [[File:Corporal frock coat 1862 med.jpg|thumb|Corporal frock coat 1862, 9-botton military]]The 1860s saw fewer inventions that influenced or changed the way people were acquiring clothes. But the changes begun by mass production continued with the working class. Little changed for the higher classes, except maybe paper patterns and sewing machines. The different shape and size in clothing continued to provide slightly different silhouettes for men in the upper and middle classes. As women's clothes grew more complex in cut and style, men's clothes became simpler in cut and style. ==== Coats and Jackets ==== Individualized fittings continued for upper class and wealthy middle class clients. (Working class as well as middle and upper class.) Working class was well over 60% of the population. For the largest class, ready-made garments were ideal for men and women who were working class. For the two upper classes morning coats still led the way but frock coats were still in demand and worn in business and more casual situations. Leisure wear like knickers and ditto suits became more popular in the 1860s. Haute couture wanted every occasion or event to have special clothing. In some ways men dressed more alike (coat, trousers, waistcoat) and in some ways there was more variety in the cut of their coats and in the manufactured clothing they could buy. Many upper class men still had their shirts custom made. Frock coats were shortened to knee-length and were worn for business, while the mid-thigh length [[sack coat]] slowly displaced the frock coat for less-formal occasions, with the overall effect of a looser silhouette. The Prince of Wales: "For the countryside such as at Sandringham, he permitted an informal dress code. The Henry Poole ledger marked as 'HRH 1865' is for an evening coat without tails, the first 'dinner jacket.'"<ref name=":26" /> (424) ==== Shirts ==== Ready-made shirts rarely were seen since shirts were more underwear than outerwear. Factories began to make and sell detachable collars and cuffs. It was more convenient to replace collars and cuffs each day than the time it took a servant (working class) to wash and starch enough to look like the celluloid many of those collars were made of. ==== Trousers ==== Trousers were cut full and boxy during most of the 1850s and 1860s. The strap under the shoe disappeared and trouser legs reached the back of the heel in length. Knickers were still worn for hunting or riding and breeches were still ''de rigueur'' at court. ==== Waistcoats ==== Waistcoats remained decorative and worn with most of the coats that were popular. The young “dandies” wore the most colorful and decorated waistcoats. These “dandies” gave their highest priority to their appearance. They were the avant garde of haute couture for men. ==== Necks Treatments ==== In the [[1860s in fashion|1860s]], men started wearing wider neckties that were tied in a bow or looped into a loose knot and fastened with a stickpin. ==== Hats ==== Top hats briefly became the very tall "stovepipe" shape, but a variety of other hat shapes were popular. ==== Overcoats ==== The single-breasted redingote continued in popularity among the aristocracy and upper middle-class.  These coats were often “in colors such as plum, mulberry, brown, bright blue, and bottle green . . . [with] contrasting velvet collars.” Payne2 p. 474 [[File:Mens Coats 1872 Fashion Plate.jpg|thumb|upright|Drawing of Victorian men 1870s]] [[File:Ashurst frock coat.jpg|thumb|Ashurst Frock coat]] [[File:Grand Duke Alexis Alexandrovich of Russia Wearing a Frock Coat; Standing (fbc266c2feca43f6ad5b44e031635a8d).tif|alt=Old photograph of a man wearing a dark thigh-length coat similar to a suit coat with light-colored trousers|thumb|Grand Duke Alexis Alexandrovich of Russia Wearing a Frock Coat, c. 1872]] [[File:Caillebotte Hugot.jpg|thumb|Portrait of Paul Hugot, 1878]] [[File:Oscar Wilde (1854-1900), by Hills & Saunders, Rugby & Oxford 3 april 1876.jpg|thumb|Oscar Wilde, 1876, ditto suit]] [[File:FrockCoat.jpg|thumb|English frock coat, 1870s]] === 1870s === To correct and prevent errors being made in court dress, in 1875 the Lord Chamberlain published ''Dress Worn by Gentleman at Her Majesty's Court'', "a summary of regulations for court uniform and dress."<ref>{{Cite book|url=https://www.google.com/books/edition/Dress_worn_by_Gentlemen_at_Her_Majesty_s/pvrbQCXq0MEC?hl=en|title=Dress worn by Gentlemen at Her Majesty's Court|last=Britain)|first=Victoria (Queen of Great|date=1875|language=en}}</ref> This is the kind of thing Bertie really cared about. Define court dress. A portly, boxy silhouette was sought for most of the 1870s. Older men whose stomachs had spread outward and thus looked prosperous to the Victorians were the ideal for men’s high fashion. Sack coats and trousers were cut a little wider in the legs. Corduroy for trousers and muslin for shirts was often used in working class clothing. ==== Coats and Jackets ==== There were minor changes to some of the coats Victorian men wore. Frock coats were shortened to the knee. Sack coats (or ditto coats — the forerunner of the suits worn by men in the western cultures of the twentieth and twenty-first centuries) were found to be so comfortable that most men of the upper and middle classes had a ditto suit. Men still wore morning coats and the aristocracy still wore cutaway coats with trousers to parties and balls but cutaway coats with knee breeches at court. ==== Shirts ==== Although ready-made shirts were available, the upper and middle classes still wore bespoke clothing hand-made, with better fabric and an excellent fit but stitched on machines. The suits and shirts made for the working class were of cheaper fabrics. ==== Trousers ==== Most men were wearing trousers  all of the time (except at court) and have worn them ever since. Trousers had button flies by the 1870s which most men found to be much more convenient than laces. Trousers did not “ride up” the leg in the way that knee breeches did when dancing or sitting. If a man did not have fashionable legs, they could be obscured behind trousers. ==== Waistcoats ==== White waistcoats began to be worn by men. Black were also worn when mourning. Changes in the waistcoat could be seen but were very subtle. For example, collars were lowered as the neckline grew deeper and deeper in front, revealing more of the fancy vest. ==== Overcoats ==== A longer version of the standard overcoat, “sometimes with fur collar, cuffs, and lining” was often worn for warmth. Payne2 p. 519 ==== Original Text ==== During the [[1870s in fashion|1870s]], three-piece suits grew in popularity along with patterned fabrics for shirts. Neckties were the four-in-hand and, later, the [[Ascot tie]]s. A narrow ribbon tie was an alternative for tropical climates, especially in the Americas. Both frock coats and sack coats became shorter and more form fitting. Flat straw boaters were worn when boating. [[File:Elliott & Fry10a.jpg|thumb|William Gladstone, 1887, morning coat]] [[File:British politicians gambling at Ascot Wellcome V0050378.jpg|thumb|British Politicians Gambling at Ascot, 1884]] [[File:Scribner's magazine (1887) (14595087238).jpg|thumb|1887, types of American servants]] === 1880s === The silhouette narrowed and straightened. That is, the line went straight down without the flare of skirts on the coats. This smooth line had few decorations. Men wore hardly any jewelry. They might have a boutonniere or a ring.  Suits were not  decorated. All coats were straighter and shorter. Depending on the time of day, the Prince of Wales might change his clothes (thus, his silhouette) up to six times per day,<ref name=":28" /> (129) and some aristocrats and upper-class men would follow his lead. ==== Coats and Jackets ==== Although the frock coat was worn throughout the nineteenth century, the single fabric sack suit was worn for casual wear and was beginning to be seen as a businessman’s suit.  Morning coats were still in use for some events as well as casual occasions. The modern tuxedo or Cowes jacket appeared and was worn for formal occasions. The tuxedo is still worn as formal dress today. ==== Shirts ==== Shirts were still custom made for the rich and the working class bought ready-made shirts. The shirt remained basically the same but it could handle the different neck treatments.  Even mass produced shirts could have buttons or button holes for detachable collars and cuffs. ==== Trousers ==== Trouser legs began to decrease in width as the coat became narrower and emphasized strong vertical lines. ==== Waistcoats ==== During the 1880s waistcoats were called vests [is it regional?] and much of their bold trim and colorful fabric disappeared and vests were constructed from the same fabric as the coat and trousers. The business suit for men developed in the 1880s and was firmly established in the 1890s. ==== Overcoats ==== Versions of topcoats increased, adding an ulster overcoat for travel and an Inverness coat with up to five cape collars and a high curved collar. Double breasted topcoats were popular and some were knee-length and some calf-length. ==== Original Text ==== During the [[1880s in fashion|1880s]], formal evening dress remained a dark tail coat and trousers with a dark waistcoat, a white bow tie, and a shirt with a winged collar. In mid-decade, the dinner jacket or [[tuxedo]], was used in more relaxed formal occasions. The [[Norfolk jacket]] and tweed or woolen breeches were used for rugged outdoor pursuits such as shooting. Knee-length topcoats, often with contrasting velvet or fur collars, and calf-length overcoats were worn in winter. Men's shoes had higher heels and a narrow toe. [[File:American fashions - The Jno. J. Mitchell Co. Litho., New York. LCCN2006676665.jpg|thumb|Fashion plate from the Jno. J. Mitchell Co., New York, 1899]] [[File:Alfred Milner Vanity Fair 1897-04-15.jpg|thumb|Alfred Milner, 1897, Leslie Ward ("Spy"), for Vanity Fair]] [[File:Nobl-08.jpg|thumb|Sherlock Holmes, 1892, Sidney Paget]] [[File:PUK1894page380rector poor.jpg|thumb|Rector and Drinker, 1894]] === 1890s === The dominant silhouette for men became the sack suit in the 1890s and while changes in the sack have been subtle, it remains the dominant silhouette today. ==== Coats and Jackets ==== “Sacks became ''the'' favored style of suit coat,”(Payne2, p. 532) especially among the younger men. Frock coats were associated with older men along with the cutaway coat. There were many more choices for sports wear and leisure wear than ever before. Norfolk jackets were often worn while hunting. Most informal styles could be purchased ready made. The rich still had their clothing custom made by tailors but even working class men could purchase most ready-made versions of the coats, trousers, and vests. ==== Shirts ==== Shirts could be made of patterned fabric as well as solid colors and white.  Shirts were beginning to be acceptable as outer wear especially when worn as sporting clothing. ==== Trousers ==== Early in the 1890s, “slim, straight trousers were news.” (Payne 2, p. 532) Trousers could be plaid or checked as well as the same fabric as the coat and vest. ==== Waistcoats ==== Fancy fabrics and bold trim were not seen much as the more subdued vest dominated the fashion scene. ==== Neck Treatments ==== In the 1860s, men started wearing wider neckties that were tied in a bow or looped into a loose knot and fastened with a stickpin. ==== Overcoats ==== “Topcoats included fingertip-length styles made of twill covert cloth. Single-breasted, fly-front overcoats were fashionable too. Roomy overcoats with fur linings and collars warmed the wealthy. Inverness overcoats remained in vogue.” Payne2, p.532 ==== Original Text ==== Starting from the [[1890s in fashion|1890s]], the [[blazer]] was introduced, and was worn for sports, sailing, and other casual activities.<ref>{{cite web|last=Landow|first=George|url=http://www.victorianweb.org/art/costume/90s/2.html|title=Men's informal sporting dress, late 1880s and '90s}}</ref> Throughout much of the Victorian era most men wore fairly short hair. This was often accompanied by various forms of facial hair including moustaches, side-burns, and full beards. A clean-shaven face did not come back into fashion until the end of the 1880s and early 1890s.<ref>{{cite web|url=http://www.victorianweb.org/art/costume/nunn21.html|title=Victorian Men's Fashions, 1850–1900: Hair}}</ref> Distinguishing what men really wore from what was marketed to them in periodicals and advertisements is difficult, as reliable records do not exist.<ref name="shannon597">{{cite journal|last=Shannon|first=Brent|title=Refashioning Men: Fashion, Masculinity, and the Cultivation of the Male Consumer in Britain, 1860–1914|journal=Victorian Studies|year=2004|volume=46|issue=4|pages=597–630|doi=10.1353/vic.2005.0022}}</ref> === Notes === *Men's suits buttoned higher up than today (Payne, 467) * Norfolk jackets and sack suits (Payne, 471) * formal attire, tuxedos with tails, cutaways (Payne, 469) * Keith Middlemas (https://archive.org/details/storyoffiesta00huxf/page/200/mode/2up?q=fashion) *Brent Shannon. "Refashioning Men: Fashion, Masculinity, and the Cultivation of the Male Consumer in Britain, 1860–1914." Victorian Studies 46, no. 4 (Summer 2004): 597–630. == Hats and headwear == [[File:Ford.madox.brown.last.emma.study.jpg|thumb|''Emma Hill'' by [[Ford Madox Brown]] (1853), a woman wearing a later version of the [[poke bonnet]]]] [[File:Hoed,_objectnr_KA_1237.tif|left|thumb|Perched bonnet style of the early 1870s.]] Hats were crucial to a respectable appearance for both men and women. === Men's Hats === The top hat, for example, was standard formal wear for upper- and middle-class men.[Payne] According to Blanche Payne, "The high top hat, usually black or dark gray, had reached its characteristic shape by 1798 and dominated the entire nineteenth century." (457–58) Although top hats were the dominant hat in the 19th century, other hats became popular for working classes and lower income middle class. “The style of an individual’s hat varied, depending on fashion and their social position, as well as their profession or chosen activity..” ''Goodman 53 of 460'' Other hats that became necessary and popular include the Derby, the straw Boater, and a flat cap with a short brim. In many cases the class, work activity and income could be determined by what kind of hat was on the head of the wearer. For some men, a hat supporting a particular sport or team was important. The Derby or Bowler hat was designed by William and Thomas Bowler, brother shopkeepers in 1849. ''(Goodman 55 of 460)'' It cost less than a top hat but  lasted longer and was soon worn by middle class bankers and clerks. Straw boaters were worn by the aristocracy for casual events and working class factory workers and agricultural laborers. By 1901 working class men had changed their preference to the flat caps which became the most popular hat for the workers. Headdress for men was an essential part of dress for the entire period of the Victorian age, from the 1830s through the end of the century. Judith Flanders describes the hats worn by men in London, <blockquote>It is difficult to bear in mind the importance of hats as not only markers of class and income, but also as indicators of respectability. [509–519] [George Augustus] Sala commented that "every" man throughout the history of the world "must, necessarily and habitually, wear some kind of covering to his head". Postmen wore hats, small children wore hats, field labourers and market gardeners wore hats, cricketers, skaters — all sportsmen — wore hats. It was, self-evidently, impossible to go outdoors without one. ... Those in professional occupations wore pot hats, as did clerks and all those with pretensions to middle-class status. Even doctors' delivery boys wore battered hand-me-down pot hats: "the nap rusty, the band a mournful strip of tarnished lace; but still a Hat", which "stamps him as being associated, in however slender a manner, with a learned profession". Cloth caps were for labourers, for costers and for boys. ... Artisans wore caps made out of paper, which they folded [510–511] themselves and so could easily replace as they became dirty.<ref name=":23" /> (509–511 [of 972]) </blockquote>A pot-hat is a kind of derby or bowler in men's hats. (Lewandowski, 237) ==== Original Text ==== Hats were crucial to a respectable appearance for both men and women. The top hat, for example, was standard formal wear for upper- and middle-class men.<ref name=":4">{{Cite book |last=Steele |first=Valerie |url=https://archive.org/details/fashioneroticism0000stee |title=Victorian Fashion. Fashion and Eroticism: Ideals of Feminine Beauty from the Victorian Era to the Jazz Age |publisher=Oxford University Press |year=1985 |isbn=978-0-19-503530-8 |pages=[https://archive.org/details/fashioneroticism0000stee/page/51 51]–84 |url-access=registration}}</ref> For women, the styles of hats changed over time and were designed to match their outfits. === Women's Hats === For a discussion of the history of plumes and feathers, see [[Social Victorians/Victorian Things#Ostrich Plumes and Prince of Wales's Feathers|Ostrich Plumes and Prince of Wales's Feathers in ''Victorian Things'']]. ==== Original Wikipedia Text ==== During the early Victorian decades, hats were modest in size and design, straw and fabric bonnets being the popular choice. [[Poke bonnet]]s, which had been worn during the late [[Regency period]], had high, small crowns and brims that grew larger until the 1830s, when the face of a woman wearing a poke bonnet could only be seen directly from the front. They had rounded brims, echoing the rounded form of the bell-shaped hoop skirts. Bonnets shrunk at the end of the 1860s and moved to a perched position in the early 1870s as hairstyles grew in scale and intricacy. This led to the popularization of hats, which became the headwear of choice for the remainder of the Victorian era.<ref name="g4223">{{cite book |last=Cunnington |first=Cecil Willett |title=English Women's Clothing in the Nineteenth Century |date=1990-05-01 |publisher=Courier Corporation |isbn=0-486-26323-1 |publication-place=New York |page=}}</ref> [[File:The_London_and_Paris_ladies'_magazine_(Apr_1885)_03.png|thumb|Flower pot style hat of 1885.]] The 1880s saw a hat inspired by the top hat for women known as the flowerpot hat, and the 1890s saw the popularity of the boater. The hats of the late Victorian era were covered with elaborate creations of silk flowers, ribbons, and above all, exotic plumes; hats sometimes included entire exotic birds that had been stuffed. Many of these plumes came from birds in the Florida everglades, which were nearly made entirely extinct by overhunting. By 1899, early environmentalists like [[Adeline Knapp]] were engaged in efforts to curtail the hunting for plumes. By 1900, more than five million birds a year were being slaughtered, and nearly 95 per cent of Florida's shore birds had been killed by [[Plume hunting|plume hunter]]s.<ref>{{cite web|title=Everglades National Park|url=https://www.pbs.org/nationalparks/parks/everglades/|archive-url=https://web.archive.org/web/20090927085907/http://www.pbs.org/nationalparks/parks/everglades/|url-status=dead|archive-date=27 September 2009|publisher=PBS|access-date=7 November 2011}}</ref> == Shoes == The women's shoes of the early Victorian period were narrow and heelless, in black or white satin. By 1850s and 1860s, they were slightly broader with a low heel and made of leather or cloth. Ankle-length laced or buttoned boots were also popular. From the 1870s to the twentieth century, heels grew higher and toes more pointed. Low-cut pumps were worn for the evening.<ref name=":4" /> == Cosmetics == [[Victorian-era cosmetics]] were typically less obvious than ours. However, small amounts of pale face powder or powdered blush were widely used.<ref>{{Cite book |last=Goodman |first=Ruth |title=How to be a Victorian |date=2014 |publisher=Penguin Books |isbn=978-0-670-92136-2 |location=London}}</ref> Tints were sometimes added to face creams. Some cosmetics contained toxic or caustic ingredients like lead, mercury, ammonia, and arsenic {{Citation needed|date=October 2025}}. Hair color ==Mourning black== {{See also |Mourning stationery}} [[File:The royal children in mourning Mar 1862.jpg|thumb|Victoria's five daughters (Alice, Helena, Beatrice, Victoria and Louise), photographed wearing mourning black beneath a bust of their late father, Prince Albert (1862)]] [[File:Mourning dress MET 50.40.3a-b front CP4.jpg|alt=Black Victorian mourning dress|thumb|Mourning Dress, 1894–95]] In Britain, black is the colour traditionally associated with mourning for the dead. The customs and etiquette expected of men, and especially women, were rigid but evolving during much of the Victorian era. The expectations depended on a complex hierarchy of close or distant relationship with the deceased. (Davidoff) The closer the relationship, the longer the mourning period and the wearing of black. The wearing of full black was known as First Mourning, which had its own expected attire, including fabrics, and an expected duration of 4 to 18 months. Following the initial period of First Mourning, the mourner would progress to Second Mourning, a transition period of wearing less black, which was followed by Ordinary Mourning, and then Half-mourning. Some of these stages of mourning were shortened or skipped completely if the mourner's relationship to the deceased was more distant. Half-mourning was a transition period when black was replaced by acceptable colours such as lavender and mauve, possibly considered acceptable transition colours because of the tradition of [[Church of England]] (and [[Catholic Church|Catholic]]) clergy wearing lavender or mauve [[Stole (vestment)|stoles]] for funeral services, to represent the [[Passion (Christianity)|Passion of Christ]].<ref>{{cite web|title=The Colors of the Church Year|url=http://fullhomelydivinity.org/articles/colors.htm|publisher=Consortium of Country Churches|access-date=6 November 2011|archive-date=13 November 2011|archive-url=https://web.archive.org/web/20111113075214/http://fullhomelydivinity.org/articles/colors.htm|url-status=dead}}</ref> The mourning dress worn by Queen Victoria (below, right) "shows the traditional touches of mourning attire, which she wore from the death of her husband, Prince Albert (1819–1861), until her own death."<ref>{{Cite web|url=https://www.metmuseum.org/art/collection/search/155839?&searchField=All&sortBy=Relevance&deptids=8&ft=queen+victoria&offset=0&rpp=20&amp;pos=2|title=Mourning Dress, 1894–95|last=The Metropolitan Museum of Art|date=7 September 2019|website=The Metropolitan Museum of Art|access-date=7 September 2019}}</ref> Dating from 1894–95, Queen Victoria wore this dress as a result of the death of the eldest son of the Prince and Princess of Wales, Eddy, in line to the throne. === Norms for mourning=== ''Manners and Rules of Good Society, or, Solecisms to be Avoided'' (London, Frederick Warne & Co., 1887) gives clear instructions, such as the following:<ref>{{cite book|last=Flanders|first=Judith|title=The Victorian House|year=2003|publisher=Harper Perennial|location=London|isbn=0-00-713189-5|pages=378–83}}</ref> {| class="wikitable" |- ! Relationship to deceased !! First mourning !! Second mourning !! Ordinary mourning !! Half-mourning |- | Wife for husband || 1-year, 1-month; [[bombazine]] fabric covered with [[Crape|crepe]]; [[widow's cap]], [[lawn cuff]]s, collars || 6 months: less crepe || 6 months: no crepe, silk or wool replaces bombazine; in last 3 months jet jewellery and ribbons can be added || 6 months: colours permitted are grey, lavender, mauve, and black-and-grey |- | Daughter for parent || 6 months: black with black or white crepe (for young girls); no linen cuffs and collars; no jewellery for first 2 months || 4 months: less crepe || – || 2 months as above |- | Wife for husband's parents || 18 months in black bombazine with crepe || – || 3 months in black || 3 months as above |- | Parent for son- or daughter-in-law's parent || – Black armband in representation of someone lost || – || 1-month black || – |- | Second wife for parent of a first wife || – || – || 3 months black || – |} The complexity of these etiquette rules extends to specific mourning periods and attire for siblings, step-parents, aunts and uncles distinguished by blood and by marriage, nieces, nephews, first and second cousins, children, infants, and "connections" (who were entitled to ordinary mourning for a period of "1–3 weeks, depending on level of intimacy"). Men were expected to wear mourning black to a lesser extent than women, and for a shorter mourning period. After the mid-19th century, men would wear a black hatband and black suit, but for only half the prescribed period of mourning expected of women. Widowers were expected to mourn for a mere three months, whereas the proper mourning period expected for widows was up to four years.<ref>{{cite book|last=Flanders|first=Judith|title=The Victorian House|year=2003|publisher=Harper Perennial|location=London|isbn=0-00-713189-5|pages=378–9}}</ref> Women who mourned in black for longer periods were accorded great respect in public for their devotion to the departed, the most prominent example being Queen Victoria herself. It was not uncommon for a widow who did not remarry to wear half-mourning for the rest of her life, except when another death necessitated full mourning. For example, Alexandra, Princess of Wales wore half-mourning for the rest of her life after her eldest son Eddy died in 1894. Empress Elisabeth of Austria did the same, as did Empress Eugénie of France. They reverted to full mourning when appropriate, but they never wore less than half-mourning after their sons' deaths. Women with lesser financial means tried to keep up with the example being set by the middle and upper classes by dyeing their daily dress. Dyers made most of their income during the Victorian period by dyeing clothes black for mourning.<ref>{{cite book|last=Flanders|first=Judith|title=The Victorian House|year=2003|publisher=Harper Perennial|location=London|isbn=0-00-713189-5|page=341}}</ref> == Technological advancement == The technological changes that affected the manufacture and consumption of clothing in the Victorian age included the following: * the mass production of fabrics — for example, "by the early 1850s there were thousands of steam-powered looms churning out millions of miles of fabric every year"  [62] * the invention of aniline dyes, which were much more vibrantly colored and resistant to fading than the natural dyes that had been used. — . Invented by chemist [[William Henry Perkin]] in 1856, the first aniline dye mauveine (or mauve) "wash[ed] the fashionable landscape in a haze of purple."<ref name=":22">{{Cite book|title=The Dress Diary: Secrets from a Victorian Woman's Wardrobe|last=Strasdin|first=Kate|publisher=Pegasus Books|year=2023|location=New York, New York}}</ref> (247) Other intense and, to the Victorians, intensely exciting colors followed, but the new synthetic additions to fabric sometimes included chemicals harmful to their wearers. For example, a "bright-magenta hue was achieved by adding arsenical-based chemicals to existing aniline dyes, brightening the already luminous shades – but these left residues themselves, along with a toxic labour trail in their wake."<ref name=":22" /> (255) Perhaps the most famous of these is arsenic green, used on fabrics, wallpapers, and trim: "The craze for artificial foliage to adorn the heads and dresses of women of fashion in the mid-nineteenth century had seen the proliferation of flower workshops, where young women in their hundreds laboured to produce the lifelike green leaves and blooms that would make a fetching headdress or would trail becomingly across the bodice of a gown. The lushness of the green was achieved by the application of a powder, a pigment that was created by mixing copper and the highly toxic chemical, arsenic trioxide. The physical effects of working with this poisonous compound were horrific. Contemporary medical drawings depict the green hue of the skin and dreadful open lesions on the hands of the maker, whilst the daily gradual ingestion of the powder by the flower girls was eventually fatal."<ref name=":22" /> (255) * the invention of a sewing machine that could be used in the home. Although sewing machines were already in use in the clothing industry, in 1858 Isaac Merritt Singer began to sell "lightweight domestic machines" for home sewing, radically increasing women's control over their own dress.<ref name=":24">{{Cite book|title=Victorian Fashions for Women|last=Kay|first=Fiona|last2=Storey|first2=Neil R.|publisher=Pen & Sword History|year=2022|isbn=978 1 39900 416 9|location=Yorkshire and Philadelphia|pages=}}</ref> (91 [of 298]) * the spread of journalism for women and fashion journalism * the Jacquard loom: "French inventor Joseph-Marie Jacquard’s loom attachment mechanized the weaving of figured fabrics by 1804."<ref name=":25" /> (454) Perhaps not at the same scale as these but as important in 1850s designs was a technology that turned iron into steel, which could then be drawn into fine wires.<ref name=":3">{{Cite book|title=The Culture of Fashion|last=Breward|first=Christopher|publisher=Manchester University Press|year=1995|pages=145–180}}</ref> Steel was refined to a malleable state so that thin blades could be curved into concentric circles (called hoops) and connected with wires to form the cage. Technological advancements not only influenced the economy but brought a major change in the fashion styles worn by men and women. As the Victorian era was based on the principles of gender, race and class.<ref>{{cite journal|last1=Graham|first1=P|title=The Victorian Era|url=https://archive.org/details/in.ernet.dli.2015.261548|journal=Digital Library of India}}</ref> Much advancement was in favor of the upper class as they were the ones who could afford the latest technology and change their fashion styles accordingly. In 1830s there was introduction of horse hair crinoline that became a symbol of status and wealth as only the upper-class women could wear it. In 1850s there were more fashion technological advancements hence 1850s could rightly be called a revolution in the Victorian fashion industry such as the innovation of artificial cage crinoline that gave women an artificial hourglass silhouette without layers of petticoats, which was lighter and more hygienic.<ref>{{cite book|last1=Shrimpton|first1=J|title=Victorian Fashion|publisher=Bloomsbury Shire Publications}}</ref> Synthetic dyes, such as [[mauveine]] (aniline purple), were introduced in 1856, adding bright colours to garments. In 1855's ''[[Haute couture]]'' was introduced as tailoring became more mainstream in years to follow.<ref>{{cite book|last1=Aspelund|first1=Karl|title=Fashioning Society|publisher=Fairchild Books}}</ref> Charles Frederick Worth, a prominent English designer, became popular amongst the upper class though its city of destiny always is Paris. Haute couture became popular at the same time that sewing machines were invented.<ref name="Haute Couture">{{cite book|last1=Martin|first1=Richard|last2=Koda|first2=Harold|title=Haute Couture|publisher=The Metropolitan Museum of Art}}</ref> Princess [[Eugénie de Montijo|Eugenie]] of France wore the Englishman dressmaker, Charles Frederick Worth's couture and he instantly became famous in France though he had just arrived in Paris a few years ago. In 1855, Queen Victoria and Prince Albert of Britain welcomed [[Napoleon III]] and Eugenie of France to a full state visit to England. Eugenie was considered a fashion icon in France. Queen Victoria, who had been the fashion icon for European high fashion, was inspired by Eugenie's style and the fashions she wore.{{Citation needed|date=October 2025}} Later, Queen Victoria also appointed Charles Frederick Worth as her dress maker and he became a prominent designer amongst the European upper class. Charles Frederick Worth is known as the father of the haute couture as later the concept of labels were also invented in the late 19th century as custom, made to fit tailoring became mainstream.<ref>{{cite book|last1=Saillard|first1=Olivier|last2=Zazzo|first2=Anne|title=Paris Haute Couture|publisher=Skira Flammarion}}</ref> By the 1860s, when made-to-fit tailoring was popular in Europe, crinolines were considered impractical. In the 1870s, women preferred more slimmer silhouettes, hence bodices grew longer and the polonaise, a skirt and bodice made together, was introduced. In 1870s the Cuirass Bodice, a piece of armour that covers the torso and functions like a corset, was invented. Towards the end of Victoria's reign, dresses were flared naturally as crinolines were rejected by middle-class women. Designers such as Charles Frederick Worth were also against them. All these inventions and changes in fashion led to women's liberation as tailored looks improved posture and were more practical.<ref name="Haute Couture"/> dressmakers, couturiers, modistes == Home decor == {{main|Victorian decorative arts}} Home decor started spare, veered into the elaborately draped and decorated style we today regard as Victorian, then embraced the retro-chic of [[William Morris]] as well as pseudo-[[Japonaiserie]]. == Myths and Oversimplifications == === Modesty === {{main|Victorian morality}} {{Original research|section|date=May 2008}} [[File:1868-skirt-lengths-girl-ages-Harpers-Bazar.gif|thumb|upright|"The proper length for little girls' skirts at various ages", from ''[[Harper's Bazaar]]'', showing a 1900 idea of how the hemline should descend towards the ankle as a girl got older]]Many myths and exaggerations about the period persist to the modern day. Examples include the idea of men's clothing is seen as formal and stiff, women's as elaborate and over-done; clothing covered the entire body, and even the glimpse of an ankle was scandalous. Critics contend that [[corset]]s constricted women's bodies and lives. Homes are described as gloomy, dark, cluttered with massive and over-ornate furniture and proliferating [[bric-a-brac]]. Myth has it that even piano legs were scandalous, and covered with tiny [[pantalette]]s. === Tight Lacing === Tight-lacing, which was not possible until the development of the grommet in 1828, was famously controversial in the Victorian age, generating many column inches of profitable newspaper copy, in part because it was (and still is) fetishistic and subversive in that adolescent girls used it as a means of rebellion and upper-working- or lower-middle-class shop girls saw it as a means of upward mobility.<ref name=":21">{{Cite book|title=Fashion and Fetishism: Corsets, Tight-Lacing and Other Forms of Body-sculpture|last=Kunzle|first=David|publisher=History Press|year=2013|isbn=978 0 7524 9545 3|location=Stroud, Gloucestershire|pages=}}</ref> (71 [of 1182]) No evidence exists that tight lacing was widespread or particularly dangerous.<ref name=":21" /> () In truth, men's formal clothing may have been less colourful than it was in the previous century, but brilliant [[waistcoat]]s and [[cummerbund]]s provided a touch of colour, and [[smoking jacket]]s and [[robe|dressing gown]]s were often of rich Oriental [[brocade]]s. This phenomenon was the result of the growing textile manufacturing sector, developing mass production processes, and increasing attempts to market fashion to men.<ref name="shannon597"/> Corsets stressed a woman's sexuality, exaggerating hips and bust by contrast with a tiny waist. Women's [[evening gown]]s bared the shoulders and the tops of the breasts. The [[jersey dress]]es of the 1880s may have covered the body, but the stretchy novel fabric fit the body like a glove.<ref>{{cite book |last=Gernsheim |first=Alison |title=Victorian & Edwardian Fashion: A Photographic Survey |year=1981 |publisher=Dover Publications |location=New York |page=65|edition=New |isbn=0-486-24205-6}}</ref> Home furnishing was not necessarily ornate or overstuffed. However, those who could afford lavish draperies and expensive ornaments, and wanted to display their wealth, would often do so. Since the Victorian era was one of increased social mobility, there were ever more ''[[nouveaux riches]]'' making a rich show. The items used in decoration may also have been darker and heavier than those used today, simply as a matter of practicality. London was noisy and its air was full of [[soot]] from countless coal fires. Hence those who could afford it draped their windows in heavy, sound-muffling curtains, and chose colours that didn't show soot quickly. When all washing was done by hand, curtains were not washed as frequently as they might be today. There is no actual evidence that piano legs were considered scandalous. Pianos and tables were often draped with [[shawl]]s or cloths—but if the shawls hid anything, it was the cheapness of the furniture. There are references to lower-middle-class families covering up their [[pine]] tables rather than show that they couldn't afford [[mahogany]]. The piano leg story seems to have originated in the 1839 book, ''A Diary in America'' written by Captain [[Frederick Marryat]], as a satirical comment on American prissiness.<ref>{{cite book |last1=Marryat |first1=C.B. |title=A Diary in America: With Remarks on Its Institutions |date=1839 |publisher=Longman, Orme, Brown, Green, and Longmans |location=London, England |volume=2 |pages=246–247 |url=https://books.google.com/books?id=2-VEAAAAIAAJ&pg=PA246}} From pp. 246-247: "I was requested by a lady to escort her to a seminary for young ladies, and on being ushered into the reception-room, conceive my astonishment at beholding a square piano-forte with four ''limbs''. However, that the ladies who visited their daughters, might feel in its full force the extreme delicacy of the mistress of the establishment, and her care to preserve in their utmost purity the ideas of the young ladies under her charge, she had dressed all these four limbs in modest little trousers, with frills at the bottom of them!"</ref> Victorian manners may have been as strict as imagined—on the surface. One simply did not speak publicly about sex, childbirth, and such matters, at least in the respectable middle and upper classes. However, as is well known, discretion covered a multitude of sins. Prostitution flourished. Upper-class men and women indulged in [[adultery|adulterous]] liaisons. == Gallery == {{gallery |2=A mid-Victorian interior: ''Hide and Seek'' by [[James Tissot]], c. 1877 Image:Winterhalter Elisabeth.jpg|3=Dress designed by [[Charles Frederick Worth]] for [[Elisabeth of Bavaria|Elisabeth of Austria]] painted by [[Franz Xaver Winterhalter]].|4=File:Frith A Private View detail.jpg|5=[[William Powell Frith]]'s painting of 1883 contrasts women's [[Aesthetic dress]] (left and right) with fashionable attire (center).|6=File:Tissot lilacs 1875.jpg|7=Day dress, c. 1875 [[James Tissot]] painting.|8=File:James Abbot McNeill Whistler 011.jpg|9=[[James McNeill Whistler|Whistler]]'s [[Portrait of Lady Meux]], 1882 Image:Jeanna_Samary-Renoir.png|10=[[Pierre-Auguste Renoir|Renoir]]'s portrait of [[Jeanne Samary]] in an [[evening gown]], 1878|11=File:Melville_-_Queen_Victoria.jpg|12=Portrait by [[Alexander Melville (artist)|Alexander Melville]] of [[Victoria of the United Kingdom|Queen Victoria]], 1845|13=File:Henry Treffry Dunn Rossetti and Dunton at 16 Cheyne Walk.jpg|14=An artistic interior: [[Dante Gabriel Rossetti]] reading to [[Theodore Watts-Dunton]] in the drawing room at No. 16 [[Cheyne Walk]], 1882|15=File:Punch - Masculine beauty retouched1.png|16=Men's swimwear: Cartoon from ''[[Punch (magazine)|Punch]]'' by [[George du Maurier]]}} == See also == * [[Emily Clapham]] * [[Victorian decorative arts]] * [[Victorian dress reform]] * [[Victorian morality]] * [[Victoriana]] * [[Women in the Victorian Era]] * [[Charles Frederick Worth]] === Time periods === * [[1830s in fashion]] * [[1840s in fashion]] * [[1850s in fashion]] * [[1860s in fashion]] * [[1870s in fashion]] * [[1880s in fashion]] * [[1890s in fashion]] === Women's clothing === * [[Corset]] * [[Corset controversy]] * [[Tightlacing]] * [[Bloomers (clothing)|Bloomers]] * [[Bodice]] === Contemporary interpretations === * [[Steampunk]] * [[Neo-Victorian]] * [[Lolita Fashion|Lolita]] == References == {{Reflist}} == Further reading == *{{cite book |author=Phipps, Elena| title= ''From Queen to Empress: Victorian dress 1837-1877'' | location=New York | publisher=The Metropolitan Museum of Art | year=1988 | isbn=0870995340| url= http://libmma.contentdm.oclc.org/cdm/compoundobject/collection/p15324coll10/id/69547/rec/235 | display-authors=etal}} * Sweet, Matthew – ''Inventing the Victorians'', St. Martin's Press, 2001 {{ISBN|0-312-28326-1}} == External links == * [http://www.victorians.co.uk/victorian-fashion Victorian Fashion] {{Webarchive|url=https://web.archive.org/web/20180407223711/http://www.victorians.co.uk/victorian-fashion |date=7 April 2018 }} * [https://www.victorianvoices.net/topics/fashion/index.shtml VictorianVoices.net] – Fashion articles and illustrations from Victorian periodicals; extensive fashion image gallery * [http://www.cracked.com/article_19575_5-ridiculous-sex-myths-from-history-you-probably-believe.html Victorian myths] * [http://www.victorianstation.com/lifestylemenu.htm Victorian fashion, etiquette, and sports] {{Webarchive|url=https://web.archive.org/web/20180103162620/http://www.victorianstation.com/lifestylemenu.htm |date=3 January 2018 }} * [http://www.thesmartset.com/article/article12180701.aspx Background on "A Diary in America"] * [http://www.mccord-museum.qc.ca/en/keys/webtours/VQ_P2_17_EN.html Form and Fashion] — the evolution of women's dress during the 19th century (many photographs) * [http://www.mccord-museum.qc.ca/en/keys/games/jeu2/ Educational Game: Mix and Match] — build a 19th-century dress using a virtual mannequin * {{cite web |publisher= [[Victoria and Albert Museum]] |url= http://www.vam.ac.uk/content/articles/v/victorian-dress-at-v-and-a/ |title= Victorian Dress |work= Fashion, Jewellery & Accessories |date= 14 January 2011 |access-date= 2011-04-03}} *[http://cv.vic.gov.au/stories/creative-life/fashion-detective-fashion-fiction-and-forensics/ Fashion detective: Fashion, Fiction and Forensics in nineteenth century Australian fashion] on Culture Victoria {{Timeline of clothing and fashion|state=collapsed}}{{Victorian era|state=collapsed}} [[Category:Victorian fashion| ]] [[Category:19th-century fashion|*]] [[Category:1900s fashion]] [[Category:History of Western fashion]] [[Category:19th century in the arts]] =From ''Women in the Victorian era''= ===Victorian women's fashion=== {{Multiple issues|{{tone|date=March 2023}} {{more footnotes needed|date=March 2023}}|section=y}}{{Further|Victorian fashion}} The ideal Victorian woman was pure, chaste, refined, and modest. This ideal was supported by etiquette and manners. The etiquette extended to the pretension of never acknowledging the use of undergarments (sometimes generically referred to as "unmentionables"). The discussion of such a topic, it was feared, would gravitate towards unhealthy attention on anatomical details. As one Victorian lady expressed it: "[those] are not things, my dear, that we speak of; indeed, we try not even to think of them", in contrast to current norms.<ref>{{cite book |last=Cunnington |first=C. Willett |title=English Women's Clothing in the Nineteenth Century: A Comprehensive Guide with 1,117 Illustrations |publisher=Dover Publications |year=1990 |isbn=978-0-486-26323-6 |pages=20}}</ref> The pretence of avoiding acknowledgement of anatomical realities met with embarrassing failure on occasion. In 1859, the Hon. Eleanor Stanley wrote about an incident where the [[Louisa Cavendish, Duchess of Devonshire|Duchess of Manchester]] moved too quickly while manoeuvring over a [[stile]], tripping over her large [[hoop skirt]]: {{blockquote|[the Duchess] caught a hoop of her cage in it and went regularly head over heels lighting on her feet with her cage and whole petticoats above, above her head. They say there was never such a thing seen – and the other ladies hardly knew whether to be thankful or not that a part of her undergarments consisted in a pair of scarlet tartan [[knickerbockers (clothing)|knickerbockers]] (the things Charlie shoots in) which were revealed to the view of all the world in general and the [[Aimable Pélissier|Duc de Malakoff]] in particular".<ref>{{cite book|last=Cunnington|first=C. Willett|title=English Women's Clothing in the Nineteenth Century: A Comprehensive Guide with 1,117 Illustrations|year=1990|publisher=Dover Publications|isbn=978-0-486-26323-6|pages=20–1}}</ref>}} However, despite the fact that Victorians considered the mention of women's undergarments in mixed company unacceptable, men's entertainment made great comedic material out of the topic of ladies' [[bloomers (clothing)|bloomers]], including men's magazines and music hall skits.<ref>{{cite book |last=Cunnington |first=C. Willett |title=English Women's Clothing in the Nineteenth Century: A Comprehensive Guide with 1,117 Illustrations |publisher=Dover Publications |year=1990 |isbn=978-0-486-26323-6 |pages=22}}</ref> Victorian women's clothing followed trends that emphasised elaborate dresses, skirts with wide volume created by the use of layered material such as [[crinoline]]s, hoop skirt frames, and heavy fabrics. Because of the impracticality and health impact of the era's fashions, a [[Victorian dress reform|dress reform movement]] began among women. The ideal silhouette of the time demanded a narrow waist, which was accomplished by constricting the abdomen with a laced [[corset]]. While the silhouette was striking, and the dresses themselves were often exquisitely detailed creations, the fashions were cumbersome. At best, they restricted women's movements and at worst, they had a harmful effect on women's health. Physicians turned their attention to the use of corsets and determined that they caused several medical problems: compression of the thorax, restricted breathing, organ displacement, poor circulation, and prolapsed uterus.<ref name="O'Connor"/> Articles advocating the reform of women's clothing by the British National Health Society, the Ladies' Dress Association, and the [[Rational Dress Society]] were reprinted in ''The Canada Lancet'', Canada's medical journal. In 1884, Dr J. Algernon Temple of Toronto even voiced concern that the fashions were having a negative impact on the health of young women from the working classes. He pointed out that a young working-class woman was likely to spend a large part of her earnings on fine hats and shawls, while "her feet are improperly protected, and she wears no flannel petticoat or woollen stockings".<ref name="O'Connor"/> [[File:Bloomers.jpg|thumb|1850s illustration of a woman wearing [[bloomers]]]] [[Florence Pomeroy]], Lady Haberton, was president of the Rational Dress movement in Britain. At a National Health Society exhibition held in 1882, Viscountess Haliburton presented her invention of a "[[divided skirt]]", which was a long skirt that cleared the ground, with separate halves at the bottom made with material attached to the bottom of the skirt. She hoped that her invention would become popular by supporting women's freedom of physical movement, but the British public was not impressed by the invention, perhaps because of the negative "unwomanly" association of the style with the American [[Bloomers]] movement.<ref>{{cite book|last=Murray|first=Janet Horowitz|title=Strong-Minded Women and Other Lost Voices from 19th Century England|year=1982|publisher=Pantheon Books|location=New York|isbn=0-394-71044-4|pages=[https://archive.org/details/strongmindedwome00jane/page/68 68–70]|url=https://archive.org/details/strongmindedwome00jane/page/68}}</ref> [[Amelia Jenks Bloomer]] had encouraged the wearing of visible bloomers by feminists to assert their right to wear comfortable and practical clothing, but it was no more than a passing fashion itself among radical feminists. The movement to reform women's dress would persist and have long-term success, however; by the 1920s, [[Coco Chanel]] was successful at selling a progressive, far less restrictive silhouette that abandoned the corset and raised hemlines. The new silhouette symbolised modernism for trendy young women and became the 20th century standard. Other Paris designers continued reintroducing pants for women and the trend was gradually adopted over the next century. Fashion trends, in one sense, travelled "full circle" over the course of the Victorian era. The popular women's styles during the [[Georgian era]], and at the very beginning of Victoria's reign, emphasized a simple style influenced by flowing gowns worn by women in [[Ancient Greek clothing|Ancient Greece]] and [[Clothing in ancient Rome|Rome]]. The [[Empire waist]] silhouette was replaced by a trend towards ornate styles and an artificial silhouette, with the restrictiveness of women's clothing reaching its low point during the mid-century passion for narrow corseted waists and hoop skirts. The iconic wide-brimmed women's hats of the later Victorian era also followed the trend towards ostentatious display. Hats began the Victorian era as simple [[Bonnet (headgear)|bonnets]]. By the 1880s, milliners were tested by the competition among women to top their outfits with the most creative (and extravagant) hats, designed with expensive materials such as silk flowers and exotic plumes such as ostrich and peacock. As the Victorian era drew to a close, however, fashions were showing indications of a popular backlash against excessive styles. Model, actress and socialite [[Lillie Langtry]] took London by storm in the 1870s, attracting notice for wearing simple black dresses to social events. Combined with her natural beauty, the style appeared dramatic. Fashions followed her example (as well as Queen Victoria's wearing of mourning black later in her reign). According to [[Harold Koda]], the former Curator-in-chief of the [[Costume Institute at The Met|Metropolitan Museum of Art's Costume Institute]],<ref>{{cite web|url=http://www.metmuseum.org/about-the-museum/press-room/exhibitions/2014/death-becomes-her|title=Death Becomes Her: A Century of Mourning Attire : October 21, 2014-February 1, 2015|website=Metmuseuim.org|access-date=7 November 2021}}</ref> "The predominantly black palette of [[mourning]] dramatizes the evolution of period silhouettes and the increasing absorption of fashion ideals into this most codified of etiquettes," said Koda, "The veiled widow could elicit sympathy as well as predatory male advances. As a woman of sexual experience without marital constraints, she was often imagined as a potential threat to the social order." ====Evolution of Victorian women's fashion==== <gallery> File:Fashion plate December 1844.jpg|Ladies' December Fashions (1844). Hand-coloured steel engraving from a women's magazine. File:Thegalleryofhmscalcutta james tissot 1876.jpg|''[[The Gallery of HMS Calcutta]]'' by [[James Tissot]] (1876). [[Bustle]]s were fashionable in the 1870s and 1880s. File:Mrs lillie langtry george frederic watts 1880.jpg|''Mrs. Lillie Langtry'' by [[George Frederic Watts]] (1880). File:Five-women-on-queenslander-steps-r.jpg|Fashionable women in [[Queensland]], Australia around 1900. </gallery> {{Short description|Irish writer (born 1963)}} {{Use Irish English|date=August 2025}} {{Use dmy dates|date=August 2025}} {{Infobox writer | name = Darach Ó Scolaí | image = Darach Ó Scolaí.JPG | alt = Man holding prize-winning book | caption = Ó Scolaí in 2019 | birth_name = Darach Ó Scolaí | birth_date = {{Birth date and age|1963|df=y}} | birth_place = [[County Galway]], The Republic of Ireland | death_date = | death_place = | occupation = Writer, artist, publisher | alma_mater = [[University of Galway]] | years_active = 1998–present | genre = Novel, retelling, translation, play, screenplay, illustrated book for children and adults | other_names = | spouse = | children = 3 | awards = [[Awards and Honors received by Darach Ó Scolaí|Awards and Honors]] | signature = | website = }}[[File:Darach Ó Scolaí.JPG|thumb|Darach Ó Scolaí, holding ''Oileán an Órchiste'' (his translation of Robert Louis Stevenson's ''Treasure Island'')]] == Darach Ó Scolaí == Darach Ó Scolaí (<small>Irish:</small> [/ˈda.rax/ /oː/ /sˠkˠoː/l̪ˠəi/]; born 1963<ref>{{Cite web|url=https://portraidi.ie/en/darach-o-scolai/|title=Darach Ó Scolaí|date=20 October 2017|website=Portráidí (Portraits of Irish-Language Writers)|access-date=1 August 2025}}</ref>) is an Irish author who works in a number of genres, from novels, plays and screenplays to illustrated books for children and adults. He began his literary career in 1998 writing screenplays, stage plays, retellings and translations; he began to publish novels in 2008. Ó Scolaí is widely recognized as a leading figure in contemporary Irish literature, known as “one of the most important Irish language writers of his generation”<ref>{{Cite journal|last=Poirtéir|first=Cathal|date=2022|title=? Suil an Daill: Constant Tensions and Shifting Allegiances|url=https://booksirelandmagazine.com/suil-an-daill-constant-tensions-and-shifting-allegiances/|journal=Books Ireland}}</ref> and "one of the great Irish language novelists [duine d’úrscéalaithe móra na Gaeilge]."<ref name=":17" /> His writing has been called “the high literature of the Irish language.”<ref>Ó Coimín, Maitiú. ''Nós'' 2 February 2018). Qtd. in "Táin Bó Cuailnge." ''Leabhar Breac''. Retrieved 25 August 2025.</ref> Much of his fiction is based on a knowledge of traditional Irish tales and narrative practices as well as Irish history. He specializes in literary and [[wikipedia:Historical_fiction|historical fiction]], or as novelist Alan Titley says, Ó Scolaí’s “peak (for now), or at least his greatest imaginative interest, is the historical novel [tá an chuma air gurb é a bhuaic (go fóill), nó ar a laghad, a mhórspéis samhlaíochta, an t-úrscéal staire].”<ref name=":11">{{Cite journal|last=Titley|first=Alan|date=Fall 2020|title=An Stíl Go Deo!: Soather Dharach Uí Scolaí (The style would be forever!: Worker Darach Ó Scolaí)|url=https://www.jstor.org/stable/27046090|journal=Comhar|volume=80, No. 10|pages=27|via=JSTOR}}</ref> His retellings of old stories and tales from their original Middle and Early-Modern Irish into Modern Irish ([[wikipedia:Irish_language|Gaeilge]]) are respected for their accessibility to students and language learners as well as for their artistry. Ó Scolaí also regularly reviews books and lectures and writes on literature and culture. Beyond his writing, Ó Scolaí is a publisher and has co-produced a number of film, television shows and stage plays. == Life == Ó Scolaí was born in Dublin and raised in the Galway [[wikipedia:Gaeltacht#Galway Gaeltacht|Gaeltacht]] (Irish-speaking) regions of Cois Fharraige on the north shore of Galway Bay, in the Republic of Ireland, where he lives now with his wife and children in Lochán Beag (Indreabhán).<ref name=":7">{{Cite journal|date=30 October 2024|title=Duais don úrscéal liteartha is fearr buaite ag Darach Ó Scolaí ag Oireachtas na Samhna|url=https://tuairisc.ie/duais-don-ursceal-liteartha-is-fearr-buaite-ag-darach-o-scolai-ag-oireachtas-na-samhna/|journal=Tuairisc}}</ref><ref>{{Cite journal|last=Ní Scolaí|first=Aifric|date=2024|title=Darach Ó Scolaí|url=https://www.taiscecf.ie/ealaiontoiri?category=Scr%C3%ADbhneoir|journal=Taisce Chois Fharraige}}</ref> He graduated the [[wikipedia:University_of_Galway|University of Galway]] (then University College Galway) with a B.A. in 1983.<ref>{{Cite web|url=https://www.linkedin.com/in/darach-ó-scolaí-20026920/|title=Darach Ó Scolaí|last=Ó Scolaí|first=Darach|date=August 2025|website=LinkedIn}}</ref> === Writing and Publishing === Ó Scolaí writes in Irish ([[wikipedia:Irish_language|Gaeilge]]), his native language, and lives in an area defined for the predominant presence of Irish as the vernacular language, the language spoken at home. Irish was the language of his parents' home and is the language of children as well. He is fluent in Irish and English and conversant in French. None of his works has been translated into English. ==== Leabhar Breac ==== In 1995 Darach Ó Scolaí and his brother Caomhán Ó Scolaí — a [[wikipedia:Typography|typographer]] and designer — founded the publishing house Leabhar Breac at Indreabhán (Inverin), County Galway. Their father “Séamas Ó Scolaí was an editor at An Gúm and worked on the Irish-English dictionary team [bhí a n-athair Séamas Ó Scolaí ina eagarthóir sa Ghúm agus d’oibrigh sé ar fhoireann an fhoclóra Gaeilge-Béarla].”<ref name=":0">{{Cite web|url=https://leabharbreac.com/en/about-us/|title=About Us|date=2024|website=Leabhar Breac|access-date=1 July 2025}}</ref> Darach Ó Scolaí has been publisher and literary editor at Leabhar Breac since its founding. Named for [[wikipedia:An_Leabhar_Breac|An Leabhar Breac (The Speckled Book)]], Leabhar Breac publishing house has more than 140 books in print.<ref name=":0" /> Leabhar Breac aims to publish Irish-language books that meet “a high literary and artistic standard.”<ref name=":0" /> Besides the content, Leabhar Breac is known for the typically "superb [thar cionn]" quality of the design and production of the "physical book [leabhar fisiciúil]."<ref name=":8">{{Cite journal|last=Ní Mhuilneoir|first=Gráinne|date=30 July 2024|title=‘Bláthnaid’ – leabhar álainn i sraithín álainn faoi mhná|url=https://tuairisc.ie/blathnaid-leabhar-alainn-i-sraithin-alainn-faoi-mhna/|journal=Tuairisc}}</ref> Its books regularly win awards for literary and artistic quality. Leabhar Breac also publishes translations for children and adults from various early versions of Irish as well as from French and English (and has published translations of books for young readers from Spanish, Catalan, and Italian as well). Leabhar Breac prints its books in Ireland. === Stage and Screen === ==== Rosg ==== In 1998 along with Ciarán Ó Cofaigh,<ref name=":1">{{Cite web|url=http://www.rosg.ie/en/about/History_6/|title=About Us: History|date=July 2025|website=Rosg|access-date=1 August 2025}}</ref> Ó Scolaí co-founded the film and television production company [http://www.rosg.ie/en/ Rosg] and was co-director until 2006. Rosg produced Ó ScolaÍ’s films ''Cosa Nite'' (1999), ''An Leabhar'' (2001) and ''Na Cloigne'' (2010). He left Rosg in 2006 to devote his time to other artistic activities. ==== Ealaín ar Oileán ==== In 2004, along with Val Balance, Ó Scolaí co-founded the annual artists' symposium Ealaín ar Oileán (trans., Art on an Island). The Irish-language symposium was held annually in the Áras Éanna arts and cultural center on Inis Oírr ([[wikipedia:Inisheer|Inisheer]], the smallest of the [[wikipedia:Aran_Islands|Aran Islands]]) from 2004 to 2013. Ó Scolaí was its co-director from its founding<ref>{{Cite web|url=https://ga.wikipedia.org/wiki/Darach_Ó_Scolaí.|title=Darach Ó Scolaí|date=3 February 2024|website=Vicipéid|access-date=1 July 2025}}</ref> until 2013. Besides being its co-director, Ó Scolaí has taken part in this conference as an artist<ref>{{Cite journal|date=16 January 2005|title=Darach Ó Scolaí|url=https://web.archive.org/web/20050116163252/http://bliainiris.com/authors/darach_oscolai.html|journal=Bliainiris}}</ref> and writer<ref name=":2">{{Cite web|url=http://ealainaroilean.ie/ealainaroilean.html|title=The Conference|date=7 September 2013|website=Ealaín ar Oileán|archive-url=https://web.archive.org/web/20130907083744/http://ealainaroilean.ie/ealainaroilean.html|archive-date=7 September 2013|access-date=1 August 2025}}</ref>. ==== Salamandar ==== In 2006 Ó Scolaí founded the stage production company Salamandar and directed his own play ''An Braon Aníos''. His plays ''An tSeanbhróg'' (2009) and ''Craos'' (2008) were also produced by Salamandar.<ref name=":19">{{Cite web|url=https://leabharbreac.com/en/product-category/darach-o-scolai/|title=Darach Ó Scolaí|date=2024|website=Leabhar Breac|access-date=1 July 2025}}</ref> == Works == === Novels === * [[wikipedia:An_Cléireach|''An Cléireach'' (trans., ''The Clerk'')]], Leabhar Breac, 2007. The Oireachtas Prize for Literary Fiction, 2007; The Ó Súilleabháin Award (Book of the Year) in 2008, and "named as ‘the best novel since the turn of the Century’ by Comhar."<ref>{{Cite web|url=https://leabharbreac.com/en/shop/fiction/an-cleireach/|title=An Cléireach - Leabhar Breac - Irish language novel|website=Leabhar Breac|language=en-US|access-date=2025-10-24}}</ref> * ''Na Comharthaí'' (trans., ''The Signs''), Leabhar Breac, 2014. * ''Súil an Daill'' (trans., ''The Eye of the Blind''), Leabhar Breac, 2021. The Oireachtas Prize for Literary Fiction, 2019.<ref name=":4">{{Cite web|url=https://leabharbreac.com/en/shop/fiction/suil-an-daill/|title=Súil an Daill|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> * ''Bódléar'', Leabhar Breac, 2024. The Oireachtas Prize for Literary Fiction, 2024<ref name=":7" />; The Ó Súilleabháin Award (Book of the Year) in 2025; featured in the 2025 Listen-Up Irish Summer Challenge for students of the Irish language.<ref>{{Cite news|url=https://connachttribune.ie/novel-approach-helps-people-learn-irish-in-a-creative-way/|title=Novel approach helps people learn Irish in a creative way|last=Murphy|first=Judy|date=3 October 2025|work=Connaught Tribune|access-date=24 October 2025}}</ref> === Retellings, Translations and Editions === The retellings and translations are into modern Irish. * ''Feis Tigh Chonáin'' (trans., ''The Feast of Conán's House''), Leabhar Breac, 2000; a retelling of a 15^th-century tale from the [[wikipedia:Fenian_Cycle|Fenian Cycle]].<ref>{{Cite web|url=https://leabharbreac.com/en/shop/fiction/feis-tigh-chonain/|title=Feis Tigh Chonáin|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> * ''An Ceithearnach Caolriabhach'' (trans., ''The Narrow-Striped Kern''), Leabhar Breac, 2002; a retelling from c. 1500, also illustrated by Darach Ó ScolaÍ.<ref>{{Cite web|url=https://leabharbreac.com/en/shop/fiction/an-ceithearnach-caolriabhach/|title=An Ceithearnach Caolriabhach|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> * ''Táin Bó Cuailnge'' (trans., ''The Cattle Raid of Cooley''), Leabhar Breac, 2017, both a modern edition of an 11th-century epic and an annotated edition.<ref name=":3">{{Cite web|url=https://leabharbreac.com/en/shop/darach-o-scolai/tain-bo-cuailnge-2-2/|title=Táin Bó Cuailnge|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> "''Táin Bó Cuailnge'' won the Aodán Mac Poilín Memorial Prize 2017."<ref name=":19" /> * ''Deirdre'', Leabhar Breac, 2023, a “picture book for adults” with artist Anastasia Melnykova.<ref>{{Cite web|url=https://leabharbreac.com/en/shop/darach-o-scolai/deirdre/|title=Deirdre|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> Part of the [[wikipedia:Ulster_Cycle|Ulster Cycle]], ''Deirdre'' is a retelling of the story of possibly the most widely known Irish figure from the early tales and sagas.<ref>{{Cite book|title=A Dictionary of Celtic Mythology|last=MacKillop|first=James|publisher=Oxford University Press|year=2004|isbn=9780198609674|pages=181}}</ref> * ''Bláthnaid'', Leabhar Breac, 2024, a “picture book for adults” with artist Anastasia Melnykova; “one of the great stories of the [[wikipedia:Ulster_Cycle|Ulster Cycle]].”<ref name=":4" /> * ''Sadhbh,'' Leabhar Breac, 2025, a picture book for adult readers, illustrated by Alé Mercado; a retelling of the medieval tale ''Ceasacht Inghine Ghuile (''trans., ''The Complaint of Guile's Daughter'').<ref name=":5">{{Cite web|url=https://leabharbreac.com/en/tales-of-wonder/|title=Tales of Wonder|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> * ''Eoghan Béal'', Leabhar Breac, 2025, a picture book for adult readers illustrated by Alé Mercado<ref name=":5" />; a retelling of the medieval tale ''[https://ga.wikipedia.org/wiki/Caithr%C3%A9im_Cellaig Cathréim Ceallaigh]'' from ''The Yellow Book of Leacan.''<ref name=":5" /> === For Young Readers === Ó Scolaí has written illustrated books for young readers (8–10 years old) in two series, the Fionn Series and the Scéalta Staire series, and translated a large number of classics and popular books for children of all ages. The number of these written and translated works suggests a commitment to children and their literacy in Irish. The Fionn Series “is a retelling ... of the great legends of the Fianna for the young Irish readers of today.”<ref name=":6">{{Cite web|url=https://leabharbreac.com/en/shop/oige-en/8-9/doiteoir-na-samhna/|title=Dóiteoir na Samhna|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> [[wikipedia:The_Boyhood_Deeds_of_Fionn|Macgnímartha Finn (The Boyhood Deeds of Fionn)]] is a medieval story in the [[wikipedia:Fenian_Cycle|Fenian Cycle]]. * ''An Bradán Feasa'' (trans., ''The Salmon of Knowledge''), Leabhar Breac, 2010, “shortlisted for the Réics Carlo award 2010.”<ref name=":9">{{Cite web|url=https://leabharbreac.com/en/shop/oige-en/8-9/an-bradan-feasa/|title=An Bradán Feasa|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> * ''Dóiteoir na Samhna'' (trans., ''The Halloween Burner''), 2010.<ref name=":6" /> * ''Bodach an Chóta Lachna'' (trans., ''The Churl in the Dun Coat''), 2011.<ref>{{Cite web|url=https://leabharbreac.com/en/shop/oige-en/7-8/bodach-an-chota-lachna/|title=Bodach an Chóta Lachna|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> The Scéalta Staire (Historical Stories) series<ref name=":9" /> * ''Mánas Ó Dónaill'', 2000. * ''Seán Ó Néill'', Leabhar Breac, 2000. * ''Gráinne Mhaol Ní Mháille'', Leabhar Breac, 2003. * ''Tadhg Dall Ó hUiginn'', Leabhar Breac, 2003. ==== Translations ==== * Robert Louis Stevenson, ''Oileán an Órchiste'' (trans. of ''Treasure Island''), Leabhar Breac, 2014.<ref>{{Cite journal|date=2025-06-19|title=Oireachtas na Gaeilge|url=https://en.wikipedia.org/w/index.php?title=Oireachtas_na_Gaeilge&oldid=1296394643|journal=Wikipedia|language=en}}</ref> * Robert Louis Stevenson, ''An Fuadach'' (trans. of ''Kidnapped''), Leabhar Breac, 2016. * Clement Clarke Moore, ''Cuairt San Nioclás'' (trans. of ''A Visit from St. Nicholas'', or "'Twas the Night Before Christmas"), Leabhar Breac, 2022. '''''The Corto Maltese Graphic Novels''''' Written in Italian by Hugo Pratt and translated by Ó Scolaí, both adults and teenagers read this series of Italian adventure graphic novels.<ref>{{Cite journal|date=2025-07-01|title=Corto Maltese|url=https://en.wikipedia.org/w/index.php?title=Corto_Maltese&oldid=1298285365|journal=Wikipedia|language=en}}</ref> Ó Scolaí's '''translation of ''Corto Maltese''''' was listed in 2017 among "The 30 Irish books that Irish people love."<ref>{{Cite journal|last=Ó Murchú|first=Eoin P.|date=09/06/2017|title=Na 30 leabhar Gaeilge is fearr leis na Gaeil [The 30 Irish books that Irish people love]|url=https://nos.ie/cultur/leabhair/an-30-leabhar-gaeilge-is-fearr-leis-na-gaeil/|journal=Nós}}</ref> * Hugo Pratt, ''Corto: Port na Farraige Goirt'', Leabhar Breac, 2013. * Hugo Pratt, ''Corto: The Golden House in Samarkand'', 2014. * Hugo Pratt, ''Corto: Na Liopard-Fhir ó Rufiji'' (trans. of ''Corto: The Leopard Men of Rufiji''), Leabhar Breac, 2014. * Hugo Pratt, ''Corto: In Ainm Dé Uilthrócairigh'' (trans. of ''Corto: In the Name of God All-Merciful''), Leabhar Breac, 2014. * Hugo Pratt, ''Corto: Tóraíocht Eile'' (trans. of ''Corto: Another Quest''), Leabhar Breac, 2014. * Hugo Pratt, ''Corto: Sa tSibéir'' (trans. of ''Corto: In Siberia''), Leabhar Breac, 2016. '''''Other Translations for Children''''' Ó Scolaí has translated into Irish six books from the ''Le Pavillon Noir'' (trans., ''Jolly Roger'') series by Alain Surget; four books from the ''Catalan First Steps'' series by Enric Lluch Girbés and the ''Caitlín & Cormac'' series by Joan Carles; three books from the ''Louisette le Taupe'' series by Bruno Heitz, and three books from the ''Loup'' series by Orianne Lallemand. === Plays and Screenplays === ==== Stage Plays ==== Ó Scolaí was writer and director of the original productions of two plays in the ''Trí Bhraon'' (trans., ''Three Drops'') trilogy; ''Coinneáil Orainn'' was directed by Darach Mac Con Iomaire and staged by An Taibhdhearc. All three plays have been published in book form by Leabhar Breac. * ''Coinneáil Orainn'' (trans., ''We'll Keep Going''), 2005.<ref>{{Cite web|url=https://irishplayography.com/play?playid=32553|title=Coinneáil Orainn|date=2025|website=PlayographyIreland: A Comprehensive Database of New Irish Plays Produced Professionally Since 1904.|access-date=25 August 2025}}</ref> The first play in the ''Trí Bhraon'' (''Three Drops'') trilogy. [[wikipedia:Taibhdhearc_na_Gaillimhe|An Taibhdhearc]], the national Irish-language theatre of Ireland, toured the country in 2005 with ''Coinneáil Orainn''.<ref>{{Cite web|url=https://leabharbreac.com/en/shop/darach-o-scolai/coinneail-orainn/|title=Coinneáil Orainn|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> Walter Macken Prize, 2005; BBC Stewart Parker Award, 2006.<ref>{{Cite web|url=https://irishplayography.com/person/darach-scola|title=Darach Ó Scolaí|date=2025|website=PlayographyIreland: A Comprehensive Database of New Irish Plays Produced Professionally Since 1904.|access-date=25 August 2025}}</ref> * ''Branwen'', 2006, by Darach Ó Scolaí and Ifor ap Glyn, in Irish, Welsh and English, co-produced by Project Arts Centre and Llwyfan Gogledd Cymru, toured the Republic of Ireland and Wales.<ref>{{Cite web|url=https://irishplayography.com/play?playid=32418|title=Branwen|date=2025|website=PlayographyIreland: A Comprehensive Database of New Irish Plays Produced Professionally Since 1904|access-date=25 August 2025}}</ref> * ''An Braon'' Aníos (trans., ''Rising Damp''), 2006, directed by Ó Scolaí.<ref>{{Cite web|url=https://irishplayography.com/play?playid=32461|title=An Braon Aníos|date=2025|website=PlayographyIreland: A Comprehensive Database of New Irish Plays Produced Professionally Since 1904|access-date=25 August 2025}}</ref> The second play in the ''Trí Bhraon'' (''Three Drops'') trilogy. “The Salamandar company toured the country in 2006-07 with this play, and Salamandar also produced a radio version of the play for RTÉ Raidió na Gaeltachta in 2009.”<ref>{{Cite web|url=https://leabharbreac.com/en/shop/darach-o-scolai/an-braon-anios/|title=An Braon Aníos|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> * ''Craos'' (trans., ''Gluttony''), 2008, directed by Ó Scolaí.<ref name=":13">{{Cite web|url=https://irishplayography.com/play?playid=32867|title=Craos|date=2025|website=PlayographyIreland: A Comprehensive Database of New Irish Plays Produced Professionally Since 1904|access-date=25 August 2025}}</ref> The third play in the ''Trí Bhraon'' (''Three Drops'') trilogy, it toured to Cork and Belfast.<ref name=":13" /> A review of the 2008 Salamander performance in the ''Irish Times'' says, “a humorous play which offers plenty to think about, fine acting, and sparklingly witty dialogue.”<ref>{{Cite web|url=https://leabharbreac.com/en/shop/darach-o-scolai/craos-2/|title=Craos|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> * ''A+E'', 2008, by Ríonach Ní Néill and Darach Ó Scolaí, "dance and music drama," co-produced by Ciotóg and Salamandar.<ref>{{Cite web|url=https://irishplayography.com/play?playid=32962|title=A+E|date=2025|website=PlayographyIreland: A Comprehensive Database of New Irish Plays Produced Professionally Since 1904|access-date=25 August 2025}}</ref> * ''An tSeanbhróg'' (trans., ''The Old Shoe''), 2009, produced by Salamander<ref>{{Cite web|url=https://irishplayography.com/play?playid=33042|title=An tSeanbhróg|date=2025|website=PlayographyIreland: A Comprehensive Database of New Irish Plays Produced Professionally Since 1904|access-date=25 August 2025}}</ref> and staged in the Axis Arts Centre, Dublin, and the Letterkenny Arts Centre. * '''In ''Mhuir Fhíondorcha/The Wine-Dark Sea: The Homer Project'', Ó Scolaí's translation of Homer's Cyclops story, performed at the 2019 IMRAM festival'''.<ref>{{Cite news|url=https://www.irishtimes.com/culture/books/imram-a-festival-celebrating-the-irish-language-1.4047610|title=Imram: a festival celebrating the Irish language. Liam Carson reveals the myths and legends appearing in this year’s programme|last=Carson|first=Liam|date=11 October 2019|work=The Irish Times|access-date=15 October 2025}}</ref> ==== Screenplays ==== * ''Cosa Nite'' (trans., ''Washed Feet''), short film, 1998 (dir. Dearbhla Walsh, prod. Ciarán Ó Cofaigh, Rosg); "a prose version of ''Cosa Nite'' was published (Rosg 2000)."<ref name=":9" /> Nominated for an Irish Film and Television Award.<ref>{{Citation|title=Cosa Nite (Short 1998) - Awards - IMDb|url=https://www.imdb.com/title/tt0191917/awards/|accessdate=2025-08-25|language=en-US}}</ref> * ''Na Glúnta'' (trans., ''The Generations''), 2001<ref>{{Cite web|url=https://www.iftn.ie/production/production_companies/production_sub/feature/?act1=record&aid=70&rid=3917&tpl=filmography_dets&only=1&force=1|title=Na Glúnta {{!}} The Irish Film & Television Network|website=www.iftn.ie|access-date=2025-08-25}}</ref>, co-directors Ciarán Ó Cofaigh & Darach Ó Scolaí, prod. Ciarán Ó Cofaigh, Rosg. * ''An Leabhar'' (trans., ''The Book''), short film, 2000, (dir. Robert Quinn, prod. Ciarán Ó Cofaigh, Rosg) Rosg, 2000.<ref>{{Citation|title=An Leabhar|url=https://www.imdb.com/title/tt0963767/|publisher=Bord Scannán na hÉireann / The Irish Film Board, ROSG|accessdate=2025-08-25|first=Robert|last=Quinn|others=Colm O&apos;Maonlai, Peadar O&apos;Treasaigh, Diarmuid Mac an Adhastair}}</ref> * ''Na Cloigne'' [trans., The Heads], 3-episide series, 2010 (dir. Robert Quinn, prod. Ciarán Ó Cofaigh, Rosg), TG4.<ref>{{Cite web|url=https://www.imdb.com/title/tt1607924/|title=Na cloigne|date=2010|website=IMDb|access-date=25 August 2025}}</ref> === Nonfiction === Ó Scolaí's essays and lectures are published and his interviews are broadcast regularly, making for a large body of nonfiction critical and analytical work. Here are a few, almost all published in [https://comhar.ie/iris/scribhneoiri/darach-o-scolai/ Comhar]: * “Ceol Ciúin na nÉagmaise” (trans., “The Silent Music of Absence ['''the Fall?''']”), an essay on the 2014 Nobel Prize winner for literature, [[wikipedia:Patrick_Modiano|Patrick Modiano]], ''Comhar'', December 2014. * The Ó Cadhain Lecture: [https://leachtaiuichadhain.clo.ie/leachtai/2014 “Cuimhne agus Díchuimhne (trans., “Memory & Forgetfulness"]), 2014. * “Rithim agus Réim” ("Rhythm and Register"), a public lecture in the University College Dublin lecture series “Ó Thrácht go Twitter” (trans., "From Talk to Twitter"), 2014. * Review of Pádraig Ó Cíobháin’s ''Dréachta Chrích Fodla'', '''Comhar?, ??'''. * “Na Geilt i mBun an Tí” (trans., "The Madmen in Charge"), a talk at the Merriman Winter School, Comhar April 2012.<ref name=":18">{{Cite web|url=http://darachoscolai.ie/beathaisneis.html|title=Darach Ó Scolaí: Beathaisnéis|website=darachoscolai.ie|access-date=2025-09-26}}</ref> * The EFACIS podcast: Síle Ní Choincheannain talks to Darach Ó Scolaí about the historical novel. == Critical Reception == Ó Scolaí’s style has been called “crisp and elegant, and rich in language while being highly readable,”<ref>{{Cite journal|last=Heussaf|first=Anna|date=Summer 2025|title=Bláthnaid—A tale of love, violence and sorcery retold for readers today|url=https://booksirelandmagazine.com/blathnaid-a-tale-of-love-violence-and-sorcery/|journal=Books Ireland}}</ref> with “an unsurpassed richness and precision of language.”<ref name=":12">{{Cite journal|last=Ó Cróinín|first=Breandán|date=Summer 2025|title=unknown|journal=The Limerick Leader}}</ref> “Whimsical, hilarious, and subtly learned” is how Éilis Ní Dhuibhne described his writing.<ref name=":20" /> === Original Works === Ó Scolaí’s first novel, the 2007 ''An Cléireach'' (''The Clerk'') won two prizes and was described as “one of the great historical novels in the Irish language and among the best books written in the language since the beginning of this century.”<ref name=":12" /> Novelist Alan Titley says, “In ''An Cléireach'' Ó Scolaí creates the Ireland of war in the 17th century more fully than any other Irish writer on the subject of war since ''L’Attaque'' Eoghain Ó Thuairisc around 1798 [In ''An Cléireach'' cruthaíonn Ó Scolaí Éire an chogaidh san 17ú haois níos iomláine ná mar a dhein aon scríbhneoir Gaeilge eile ar ábhar cogaidh ó ''L’Attaque'' Eoghain Uí Thuairisc timpeall ar 1798].”<ref name=":11" />{{rp|25, Col. 1a}} Not all the reviews of this first novel were so positive, however; Proinsias O' Drisceoil says for the Irish Times says,<blockquote>This then is a novel in search of a plot, a story that attempts to attain a significance that eludes it.<ref>{{Cite news|url=https://www.irishtimes.com/news/a-disaffected-clerk-in-the-confederates-1.943070|title=A disaffected clerk in the confederates|last=O' Drisceoil|first=Proinsias|date=5 July 2008|work=The Irish Times|access-date=16 October 2025}}</ref></blockquote> In the ''Oxford Handbook of Modern Irish Fiction'' Pádraig Ó Siadhail analyzes rather than reviews ''An Cléireach'': <blockquote>In ''An Cléireach'', Ó Scolaí revisits the trauma of Cromwellian Ireland. The primary narrative device is once again the first-hand account, in this case by Tadhg Ó Dúbháin, a clerk and quartermaster in the Confederate Army in 1650. We sample the hardships, the friendships, the tensions, the rivalries, and the petty jealousies amongst comrades in arms, including remnants of the Gaelic literary class, as the Confederate soldiers, increasingly a rabble more than a cohesive unit, retreat in advance of Cromwell’s forces. ''An Cléireach'' concludes with the narrator and his family in exile in continental Europe. But along the retreat route, and central to the novel, members of the Confederate army camp, rest up, and tell versions of a story about the keeper of the treasured manuscript "Saltair an Easpaig" (The Bishop’s Psalter). Their versions raise issues about memory construction, the limitations of individual perspectives, personal agendas, and how minor changes in the telling of a story can alter our understanding of history, Thus, ''An Cléireach'' complements ''Fontenoy'' in moving beyond more realistic recreation of a historical event or period to interrogate the notion of history as construct.<ref>{{Cite book|title=The Oxford Handbook of Modern Irish Fiction|last=Ó Siadhail|first=Pádraig|publisher=Oxford University Press|year=2020|isbn=9780198754893|editor-last=Harte|editor-first=Liam|pages=598–99|chapter=Contemporary Irish Fiction}}</ref> </blockquote> Of ''Súil an Daill,'' in ''Nós'', Cathal Seoighe says, "The book deserves a significant place among the collection of high-quality books published in recent years that would make you feel sorry for someone who does not speak Irish [Tá áit shuntasach ag dul don leabhar i measc an chnuasaigh leabhair ar ardchaighdeán a foilsíodh le roinnt blianta anuas a d’fhágfadh trua agat don té atá gan Ghaeilge]."<ref>{{Cite journal|last=Seoighe|first=Cathal|date=09/26/2022|title=‘Dar leathmhagairle an diabhail, is leabhar den scoth é seo!’ ['According to the devil’s half-wit, this is a great book!’]|url=https://nos.ie/cultur/leabhair/dar-leathmhagairle-an-diabhail-is-leabhar-den-scoth-e-seo/|journal=Nós}}</ref> ''Bódléar'', Ó Scolaí's most recent book, is a “beautiful novel. There is magic and craftsmanship in it. A small miracle of a book and it is highly recommended.”<ref>{{Cite web|url=https://leabharbreac.com/bodlear-mioruilt-bheag-de-leabhar/|title=Bódléar: Míorúilt bheag de leabhar (Bódléar: A Small Miracle of a Book)|last=Ní Ghairbhí|first=Róisín|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> Éilis Ní Dhuibhne in the ''Irish Times'' says,<blockquote>what a gem! An affectionately gentle satire of the Irish poetic scene during one creatively fluid 19th-century year, the story focuses on a Maigue poet and schoolteacher who goes on a trip to France and returns with camembert, a cafetiere, ‘Fleurs du Mal’, and a mission to convert the local traditionalists to la modernité. Whimsical, hilarious, and subtly learned, it’s absolutely delightful!<ref name=":20">{{Cite journal|last=Ní Dhuibhne|first=Éilis|date=30 June 2025|title=Éilís Ní Dhuibhne on the best Irish language books of 2025 so far: Including a history of the Gaeltacht Civil Rights Movements, a gem of a novel by Darach Ó Scolaí and Joe McHugh’s entertaining account of learning Irish|url=https://www.irishtimes.com/culture/books/review/2025/06/30/eilis-ni-dhuibhne-on-the-best-irish-language-books-of-2025-so-far/|journal=The Irish Times|pages=22}}</ref></blockquote> === Retellings and Translations === ==== ''Táin Bó Cuailnge'' ==== ''Táin Bó Cuailnge'' [''The Cattle Raid of Cooley''] is a modern edition of an 11th-century epic into modern Irish.<ref name=":3" /> Gearóid Denvir reviewed ''Táin Bó Cuailnge'' for ''Comhar'':<blockquote>Darach Ó Scolaí has ​​achieved a feat in this challenging reworking. He has found a high level of the Irish language to tell his story – as he has done before in his groundbreaking novel An Cléireach (2007, Leabhar Breac) and in his other prose works. This book is a decoration of the language, literature and culture of the Irish language, following the path of the old storytellers and writers and presenting material from the tradition to his own generation according to the understandings of his own time. The book will be a classic that will be of great interest to all readers of the Irish language, both ordinary readers, students, scholars and writers, and there should be a copy in every home in the country. [Tá éacht déanta ag Darach Ó Scolaí san athleagan dúshlánach seo. Tá réim ard den teanga Ghaeilge aimsithe aige lena scéal a inseacht – mar a rinne sé cheana ina úrscéal ceannródaíoch An Cléireach (2007, Leabhar Breac) agus i saothair eile phróis dá chuid. Is maisiú ar an teanga agus ar litríocht agus cultúr na Gaeilge an leabhar seo a leanas conair na seanscéalaithe agus na seanscríobhaithe agus ábhar de chuid an traidisiúin á chur i láthair a ghlúine féin aige de réir thuiscintí a linne féin. Clasaic a bheas sa leabhar a gcuirfidh léitheoirí uilig na Gaeilge, idir ghnáthléitheoirí, mhic léinn, scoláirí agus scríbhneoirí spéis thar na bearta ann, agus ba cheart cóip a bheith i chuile theach sa tír.]<ref name=":15">{{Cite journal|last=Denvir|first=Gearóid|date=April 2018|title=Táin Bó Cuailgne|url=https://comhar.ie/iris/78/4/leirmheas/|journal=Comhar|via=JSTOR}}</ref> </blockquote>Cathal Poirtéir says, "The freshness and richness of Ó Scolaí’s version are a joy …. The author delights us with the linguistic and stylistic richness of the ancient epic in a modern-Irish version that reflects the original’s spirit and language."<ref>{{Cite journal|last=Poirtéir|first=Cathal|date=May/June 2018|title=Leabhair Idir Lámha|url=https://www.jstor.org/stable/26564180|journal=Books Ireland|pages=46–47|via=JSTOR}}</ref>{{rp|47}} Novelist and academic Alan Titley calls Ó Scolaí's "a wonderful gutsy telling" of ''Táin Bó Cuailnge''.<ref>{{Cite news|url=https://www.irishtimes.com/culture/2023/03/11/the-tain-retold-maeve-and-ailills-spat-could-be-out-of-a-soap-opera/|title=The Táin retold: ‘Maeve and Ailill’s spat could be out of a soap opera’|last=Titley|first=Alan|date=11 March 2023|work=The Irish Times|access-date=16 October 2025}}</ref> ==== ''Deirdre'' ==== Marie Whelton, in "Léann Teanga" ("Language Studies"), in the 2024 ''An Reiviú'' says,<blockquote>this version [of ''Deirdre''] by Darach Ó Scolaí succeeds in skillfully capturing and portraying the complexity of gender and power issues in the ‘Deirdre’ tradition [éiríonn leis an leagan seo le Darach Ó Scolaí castacht cheisteanna na hinscne agus na cumhachta i dtraidisiún scéal Dheirdre a ghabháil agus a léiriú go sciliúil]. … There is no doubt that this new version greatly contributes to the legacy of the story and that it revives that legacy thoughtfully and artistically [Níl amhras faoi ach go gcuireann an leagan úr seo go mór le hoidhreacht an scéil agus go ndéanann sé an oidhreacht sin a athbheochan go tuisceanach agus go healaíonta.].<ref name=":16">{{Cite web|url=https://www.tara.tcd.ie/tara8/server/api/core/bitstreams/3c20175a-7631-44b2-8b0f-f454edd712b4/content|title=An Artistic Retelling of Deirdre's Tale and the Defeat of Conor Review of Deirdre or the Ship of Mac Uisnigh by Darach Ó Scolaí [Athinsint Ealaíonta ar Oidhe Dheirdre agus ar Ansmacht Chonchúir Léirmheas ar Deirdre nó Loingeas Mhac Uisnigh le Darach Ó Scolaí]|last=Whelton|first=Marie|date=2024|website=The Review [An Reiviú], Language Studies [Léann Teanga]|access-date=25 September 2025}}</ref></blockquote> === Works for Young Readers === Meadhbh Ní Eadhra said of ''Bodach an Chóta Lachna'' that it was "Beautiful Irish, but easy to understand for young readers."<ref>Ní Eadhra, Meadhbh. In ''Gaelscéal'', qtd. in "Bodach an Chóta Lachna" https://leabharbreac.com/en/shop/oige-en/7-8/bodach-an-chota-lachna/.</ref> == Awards and Honors == Ó Scolaí's works are regularly nominated and make the short list for prizes, an honor in itself, but they are generally not listed here unless they are named as the first-place winner in their category. === Oireachtas Prize === The Oireachtas Prize is the literary prize awarded by [[wikipedia:Oireachtas_na_Gaeilge|Oireachtas na Gaeilge]], the annual arts festival dedicated to Irish language, arts and culture. Darach Ó Scolaí has won the Oireachtas Prize for Literary Fiction three times, once for ''An Cléireach'' (''The Clerk'') in 2007, for ''Súil an Daill'' (''The Eye of the Blind'') in 2021 and for ''Bódléar'' in 2024. * 2007, for ''An Cléireach'' (trans., ''The Clerk'') — “(a special prize commemorating the 400th anniversary of the foundation of Coláiste na nGael in Louvain, awarded under the auspices of the Franciscan Province of Ireland). The prize of €10,000 was the largest prize ever awarded to an Irish language novel [(duais speisialta chomórtha 400 bliain bhunú Choláiste na nGael i Lobháin a bronnadh faoi urraíocht Phroibhinse Phroinsiasach na hÉireann). Ba é an duais €10,000 sin an duais ba mhó a bronnadh riamh ar úrscéal Gaeilge].”<ref name=":18" /> * 2021, for ''Súil an Daill'' (''The Eye of the Blind'') * 2024, for ''Bódléar'' === Ó Shúilleabháin Award, Irish language “Book of the Year” === The first prize of this award includes €5,000 to the publisher and €2,500 to the author of the winning work.<ref name=":10">{{Cite journal|date=15 August 2023|title=20 saothar san iomaíocht do ‘Leabhair Ghaeilge na Bliana 2023’|url=https://tuairisc.ie/20-saothar-san-iomaiocht-do-leabhair-ghaeilge-na-bliana-2023/|journal=Tuairisc}}</ref> * ''An Cléireach'' (''The Clerk'').<ref>{{Cite web|url=http:/www.gaelport.com/uploads/documents/edition19.html|title=Eagrán / Edition 19 - 04 11 2008|date=4/11/2008|website=Internet Archive|archive-url=https://web.archive.org/web/20130525011340/http:/www.gaelport.com/uploads/documents/edition19.html|archive-date=25 May 2013|access-date=25 August 2025}}</ref> * ''Táin Bó Cuailnge'', 2018. * ''Bódléar'', 2025. ==== De Bhaldraithe Award ==== The Gradam de Bhaldraithe is awarded to the best work in translation.<ref name=":10" /> * ''Cuairt San Nioclás,'' a translation of Clement Clarke Moore's ''A Visit from St. Nicholas'', or "'Twas the Night Before Christmas."<ref name=":10" /> ==== Other ==== * Walter Macken Prize, for ''Coinneáil Orainn'' (trans., ''We'll Keep Going''), 2005 * Bháiteir Uí Mhaicín Memorial Award, for ''Coinneáil Orainn'' (trans., ''We'll Keep Going''), 2005<ref>{{Cite news|url=https://www.irishtimes.com/gaeilge/tuarascail/duais-oireachtais-1.501571|title=Oireachtas Prize: Over €50,000 was awarded to writers in the Oireachtas Literary Competitions at an event in Dublin last night. Winners… [Duais Oireachtais: Bronnadh breis agus €50,000 ar scríbhneoirí i gComórtais Liteartha an Oireachtais ar ócáid i mBaile Átha Cliath aréir. Bhuaigh…]|work=5 October 2005|access-date=15 October 2025}}</ref> * BBC Stewart Parker Award, for ''Coinneáil Orainn'' (trans., ''We'll Keep Going''), 2006 * The Aodán Mac Póilín Commemorative Prize, for ''Táin Bó Cuailnge'' (trans., ''The Cattle Raid of Cooley''), 2017 == External Links == * Leabhar Breac website: https://leabharbreac.com/en/ * Leabhar Breac Facebook pages: * Rosg website: [http://www.rosg.ie/en/ <nowiki>http://ww</nowiki>w.rosg.ie/en/] * Art on the Island (Ealaín ar Oileán) website, archived at the Wayback Machine: https://web.archive.org/web/20130601000520/http://ealainaroilean.ie/ 31 March 2012, 1 June 2013 and 8 January 2014 * Darach Ó Scolaí's website Archived 25 September 2015 at the Wayback Machine: https://web.archive.org/web/20150925103456/http://darachoscolai.ie/ * Youtube video of [https://www.youtube.com/watch?v=OlP2AmSBzXc Breandán Ó Cróinin introducing Deirdre at the book launch] in the pub Tigh Mholly (Molly’s House). == Primordial Ooze == * Known for his sensitivity to language and voices. * Finish scanning through JSTOR * Scan through Irish Times, 56 hits * Check Goodreads * Check YouTube (In the spring of 2013, the arts programme Imeall interviewed the author on TG4.) * Check both Wikipedias for pages on the origins of the retold tales (like Deirdre) and link to this article * Propose link from University of Galway page once Darach’s is up * Write Irish National Biography (<nowiki>https://www.dib.ie</nowiki>) to propose an article about Darach once the Wikip article is done? See what they say. * Link to Ó Scolaí from the Wikipedia * Make sure links '''to''' Wikipedia in the actual encyclopedia work right === Not Placed Yet === * "So here are the books that Irish people love the most! [Mar sin seo iad na leabhair is gile leis na Gaeil!]" — "32. An Cléireach – Darach Ó Scolaí (2)" [18 books got 2 votes, and then they're alphabetized by author's last name, so the 32 of 34 doesn't signify the specificity it seems to]<ref name=":14">{{Cite journal|last=Ó Murchú|first=Eoin P.|date=9 June 2017|title=Na 30 leabhar Gaeilge is fearr leis na Gaeil. [The 30 best Irish books for Irish people]|url=https://nos.ie/cultur/leabhair/an-30-leabhar-gaeilge-is-fearr-leis-na-gaeil/|journal=Nós}}</ref> * "Below is a list of those 111 works – a list that shows a great deal of diversity in the reading habits of Irish speakers.Here is a list of those 111 works – a list that shows a great deal of diversity in the reading habits of Irish speakers [Anseo thíos tá liosta den 111 saothar sin – liosta a léiríonn éagsúlacht an-mhór i nósanna léitheoireachta Gaeilgeoirí.Anseo thíos tá liosta den 111 saothar sin – liosta a léiríonn éagsúlacht an-mhór i nósanna léitheoireachta Gaeilgeoirí]." "Corto Maltese – Hugo Pratt (aistrithe ag Darach Ó Scolaí)"<ref name=":14" /> * "Ceann eile de bhuaicphointí na hÉigse a bheidh sa seisiún le Darach Ó Scolaí, duine d’úrscéalaithe móra na Gaeilge, agus duine de chomhbhunaitheoirí teach foilsitheoireachta Leabhar Breac. [Another highlight of the Éigse will be the session with Darach Ó Scolaí, one of the great Irish language novelists, and one of the co-founders of the publishing house Leabhar Breac.]"<ref name=":17">{{Cite journal|last=Nós|date=4 May 2023|title=Éigse na Bruiséile le filleadh i mí na Bealtaine. [Éigse na Bruséile to return in May]|url=https://nos.ie/cultur/eigse-na-bruiseile-le-filleadh-i-mi-na-bealtaine/|journal=Nós}}</ref> === Things Taken Out for Now === “’The play is a comedy about language, lies, bureaucracy and Gaeltacht grants, in the tradition of Myles na Gcopaleen,’ according to Norma-Jean Kenny in the ''Galway Advertizer'', ‘in which the author comments and criticizes the institutions of the Irish language in Ireland without ceasing.’" Supposedly a quotation by Gearóid Denvir reviewing ''Táin Bó Cuailnge'' for ''Comhar'' (but I don't find it in the article): This book has long been needed by Irish language readers and there is no doubt that it will become a classic in time and surpass Thomas Kinsella’s English version. This version remains faithful to the language of the original while at the same time finding an appropriate language in today’s Irish. Ó Scolaí masterfully overcomes the difficulties of the original’s rhetorical difficulties and the versions of the original poetic texts are extremely effective.[supposedly <ref name=":15" />] “The biggest prize ever awarded for a novel in Irish was presented at a special ceremony in the National Concert Hall in Dublin, today (Thursday, 4 October 2007). Darach Ó Scolaí, writer, artist & playwright from Casla, Co. Galway, was awarded €10,000 for his literary novel, ‘An Ardscoil’. This work, under the new title ‘An Cléireach’, will be launched at Oireachtas na Samhna in Westport in November. This is the first novel from his pen, a story set in the late seventeenth century. This competition was sponsored by the Franciscan Province of Ireland.” (archive, Oireachtas na Gaeilge site, 04 October, 2007) ''Súil an Daill'' (trans., ''The Eye of the Blind''), Leabhar Breac, 2021. number 2 in ''Comhar'' literary magazine’s list of best books of 2021. '''{6}.''' *William Shakespeare, ''Romeo agus Juliet'' (trans. of ''Romeo and Juliet''), Leabhar Breac, 2016. *Jonathan Swift, ''Camchuairt Ghuilivéir'' (trans. of ''Gulliver's Travels''), Leabhar Breac, 2016. *Hugo Pratt, ''Corto Maltese'' '''''Flag of Bones (Bratach na gCnámh) Series''''' Leabhar Breac published the Bratach na gCnámh series of books for young readers. Written in French by Alain Surget, illustrated by Annette Marnat and translated by Darach Ó Scolaí, this series uses the history of Caribbean Sea pirates<ref>{{Cite web|url=https://leabharbreac.com/en/product-category/alain-surget/|title=Alain Surget Archives|website=Leabhar Breac|language=en-US|access-date=2025-09-30}}</ref>: *Alain Surget, ''Éalú as Páras'' (''Escape from Paris''), Annette Marnat (Illustr.), Leabhar Breac, 2011. * Alain Surget, ''Oilean na Siorcanna'' (''Shark Island''), Annette Marnat (Illustr.), Leabhar Breac, 2011. * Alain Surget, ''Long na dTaibhsi'' (''Ship of the Ghosts''), Annette Marnat (Illustr.), Leabhar Breac, 2011. * Alain Surget, ''San Ochtapas Dubh'' (''In the Black Octopus''), Annette Marnat (Illustr.), Leabhar Breac, 2013. * Alain Surget, ''San Ionsai ar Veracruz'' (''The Attack on Veracruz''), Annette Marnat (Illustr.), Leabhar Breac, 2013. '''''For "First Readers" (children to 6 years old or so)''''' These books were written originally in Catalan by Spanish author Enric Lluch Girbés and translated into Irish by Ó ScolaÍ: *Enric Lluch, ''Ag Péinteáil an Tí'' (''Painting the House''), Anna Clariana (Illustr.), Leabhar Breac, 2017. *Enric Lluch, ''An Colúr Bacach'' (''The Lazy Dove''), Anna Clariana (Illustr.), Leabhar Breac, 2017. *Enric Lluch, ''An Phluais'' (''The Cave''), Anna Clariana (Illustr.), Leabhar Breac, 2017. *Enric Lluch, ''Madra Dhaideo'' (''Grandpa's Dog''), Anna Clariana (Illustr.), Leabhar Breac, 2017. *Enric Lluch, ''Fiacail Mháire'' (''Mary's Tooth''), Anna Clariana (Illustr.), Leabhar Breac, 2017. '''''Bruno Heitz''''' Leabhar Breac published a series of 3 Heitz books for small children. Published originally in French, this series of three comic books is about a blind mole named Cáitín Chaoch in Irish (and ''Louisette la taupe'' in French).<ref>{{Cite web|url=https://leabharbreac.com/en/product-category/bruno-heitz-en/|title=Bruno Heitz Archives|website=Leabhar Breac|language=en-US|access-date=2025-10-02}}</ref> Ó Scolaí translated these: *Bruno Heitz (author and illustr.), ''Práinneach'' (''Urgent''), Leabhar Breac, 2020 *Bruno Heitz (author and illustr.), ''Preab san Aer'' (''Bounce in the Air''), Leabhar Breac, 2020. '''''Books for Toddlers''''' Leabhar Breac has published 14 books written by French author Orianne Lallemand's and illustrated by Eleonore Thuillier, about Lallemmand's popular character Loup, Wolf. These are translated by Ó Scolaí: *Orianne Lallemand, ''An Mac Tire a Raibh Faitios an Domhain Air'' (trans. of ''The Son Who Saw the World in His Eyes''), Eleonore Thuillier  Illustr.), Leabhar Breac, 2018. *Orianne Lallemand, ''Macan agus an Goban'' (trans. of ''Macan and the Goblin''), Eleonore Thuillier (Illustr.), Leabhar Breac, 2018. * Orianne Lallemand, ''A Mac Tíre a Chuaigh go Tóin na Farraige'' (trans. of ''The Wolf Who Went to the Bottom of the Sea''), Éléanore Thuillier  (Illustr.), Leabhar Breac, 2019. '''''Board Books (for babies)''''' J. C. (Joan Carles) Girbés Aparisi is a Catalan author and editor. These books were written in Catalan and translated by Ó Scolai. *J. C. Girbés, ''An Phicnic'' (''The Picnic''), Silvia Ortega (Illustr.), Leabhar Breac, 2013. * J. C. Girbés, ''An Chóisir'' (''The Party''), Silvia Ortega (Illustr.), Leabhar Breac, 2013. *J. C. Girbés, ''Lá Mór Fada'' (''A Long Day''), Silvia Ortega (Illustr.), Leabhar Breac, 2014. *J. C. Girbés, ''Tabhair Leat do Leabhar'' (''Bring Your Book''), Silvia Ortega (Illustr.), Leabhar Breac, 2014. ==== Gradam Réics Carló ==== The Réics Carló prize is awarded for the best book in the Irish language for young readers. It is named for one of the characters of 20th-century writer [[wikipedia:Cathal_Ó_Sándair|Cathal Ó Sándair (Charles Saunders)]]. * ''An Bradán Feasa'' was “shortlisted for the Réics Carlo award 2010.”<ref name=":9" /> == References == {{reflist}} iclhznw0n1mv5ujwn74rqufqrhtb3ek 0 285380 2844476 2843969 2026-11-04T13:53:58Z Young1lim 21186 /* Applications */ 2844476 wikitext text/x-wiki === Introduction === * Overview ([[Media:C01.Intro1.Overview.1.A.20170925.pdf |A.pdf]], [[Media:C01.Intro1.Overview.1.B.20170901.pdf |B.pdf]], [[Media:C01.Intro1.Overview.1.C.20170904.pdf |C.pdf]]) * Number System ([[Media:C01.Intro2.Number.1.A.20171023.pdf |A.pdf]], [[Media:C01.Intro2.Number.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro2.Number.1.C.20170914.pdf |C.pdf]]) * Memory System ([[Media:C01.Intro2.Memory.1.A.20170907.pdf |A.pdf]], [[Media:C01.Intro3.Memory.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro3.Memory.1.C.20170914.pdf |C.pdf]]) === Handling Repetition === * Control ([[Media:C02.Repeat1.Control.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat1.Control.1.B.20170918.pdf |B.pdf]], [[Media:C02.Repeat1.Control.1.C.20170926.pdf |C.pdf]]) * Loop ([[Media:C02.Repeat2.Loop.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat2.Loop.1.B.20170918.pdf |B.pdf]]) === Handling a Big Work === * Function Overview ([[Media:C03.Func1.Overview.1.A.20171030.pdf |A.pdf]], [[Media:C03.Func1.Oerview.1.B.20161022.pdf |B.pdf]]) * Functions & Variables ([[Media:C03.Func2.Variable.1.A.20161222.pdf |A.pdf]], [[Media:C03.Func2.Variable.1.B.20161222.pdf |B.pdf]]) * Functions & Pointers ([[Media:C03.Func3.Pointer.1.A.20161122.pdf |A.pdf]], [[Media:C03.Func3.Pointer.1.B.20161122.pdf |B.pdf]]) * Functions & Recursions ([[Media:C03.Func4.Recursion.1.A.20161214.pdf |A.pdf]], [[Media:C03.Func4.Recursion.1.B.20161214.pdf |B.pdf]]) === Handling Series of Data === ==== Background ==== * Background ([[Media:C04.Series0.Background.1.A.20180727.pdf |A.pdf]]) ==== Basics ==== * Pointers ([[Media:C04.S1.Pointer.1A.20240524.pdf |A.pdf]], [[Media:C04.Series2.Pointer.1.B.20161115.pdf |B.pdf]]) * Arrays ([[Media:C04.S2.Array.1A.20240514.pdf |A.pdf]], [[Media:C04.Series1.Array.1.B.20161115.pdf |B.pdf]]) * Array Pointers ([[Media:C04.S3.ArrayPointer.1A.20240208.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]]) * Multi-dimensional Arrays ([[Media:C04.Series4.MultiDim.1.A.20221130.pdf |A.pdf]], [[Media:C04.Series4.MultiDim.1.B.1111.pdf |B.pdf]]) * Array Access Methods ([[Media:C04.Series4.ArrayAccess.1.A.20190511.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]]) * Structures ([[Media:C04.Series3.Structure.1.A.20171204.pdf |A.pdf]], [[Media:C04.Series2.Structure.1.B.20161130.pdf |B.pdf]]) ==== Examples ==== * Spreadsheet Example Programs :: Example 1 ([[Media:C04.Series7.Example.1.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.1.C.20171213.pdf |C.pdf]]) :: Example 2 ([[Media:C04.Series7.Example.2.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.2.C.20171213.pdf |C.pdf]]) :: Example 3 ([[Media:C04.Series7.Example.3.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.3.C.20171213.pdf |C.pdf]]) :: Bubble Sort ([[Media:C04.Series7.BubbleSort.1.A.20171211.pdf |A.pdf]]) ==== Applications ==== * Address-of and de-reference operators ([[Media:C04.SA0.PtrOperator.1A.20261031.pdf |A.pdf]]) * Applications of Pointers ([[Media:C04.SA1.AppPointer.1A.20241121.pdf |A.pdf]]) * Applications of Arrays ([[Media:C04.SA2.AppArray.1A.20240715.pdf |A.pdf]]) * Applications of Array Pointers ([[Media:C04.SA3.AppArrayPointer.1A.20240210.pdf |A.pdf]]) * Applications of Multi-dimensional Arrays ([[Media:C04.Series4App.MultiDim.1.A.20210719.pdf |A.pdf]]) * Applications of Array Access Methods ([[Media:C04.Series9.AppArrAcess.1.A.20190511.pdf |A.pdf]]) * Applications of Structures ([[Media:C04.Series6.AppStruct.1.A.20190423.pdf |A.pdf]]) === Handling Various Kinds of Data === * Types ([[Media:C05.Data1.Type.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data1.Type.1.B.20161212.pdf |B.pdf]]) * Typecasts ([[Media:C05.Data2.TypeCast.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data2.TypeCast.1.B.20161216.pdf |A.pdf]]) * Operators ([[Media:C05.Data3.Operators.1.A.20161219.pdf |A.pdf]], [[Media:C05.Data3.Operators.1.B.20161216.pdf |B.pdf]]) * Files ([[Media:C05.Data4.File.1.A.20161124.pdf |A.pdf]], [[Media:C05.Data4.File.1.B.20161212.pdf |B.pdf]]) === Handling Low Level Operations === * Bitwise Operations ([[Media:BitOp.1.B.20161214.pdf |A.pdf]], [[Media:BitOp.1.B.20161203.pdf |B.pdf]]) * Bit Field ([[Media:BitField.1.A.20161214.pdf |A.pdf]], [[Media:BitField.1.B.20161202.pdf |B.pdf]]) * Union ([[Media:Union.1.A.20161221.pdf |A.pdf]], [[Media:Union.1.B.20161111.pdf |B.pdf]]) * Accessing IO Registers ([[Media:IO.1.A.20141215.pdf |A.pdf]], [[Media:IO.1.B.20161217.pdf |B.pdf]]) === Declarations === * Type Specifiers and Qualifiers ([[Media:C07.Spec1.Type.1.A.20171004.pdf |pdf]]) * Storage Class Specifiers ([[Media:C07.Spec2.Storage.1.A.20171009.pdf |pdf]]) * Scope === Class Notes === * TOC ([[Media:TOC.20171007.pdf |TOC.pdf]]) * Day01 ([[Media:Day01.A.20171007.pdf |A.pdf]], [[Media:Day01.B.20171209.pdf |B.pdf]], [[Media:Day01.C.20171211.pdf |C.pdf]]) ...... Introduction (1) Standard Library * Day02 ([[Media:Day02.A.20171007.pdf |A.pdf]], [[Media:Day02.B.20171209.pdf |B.pdf]], [[Media:Day02.C.20171209.pdf |C.pdf]]) ...... Introduction (2) Basic Elements * Day03 ([[Media:Day03.A.20171007.pdf |A.pdf]], [[Media:Day03.B.20170908.pdf |B.pdf]], [[Media:Day03.C.20171209.pdf |C.pdf]]) ...... Introduction (3) Numbers * Day04 ([[Media:Day04.A.20171007.pdf |A.pdf]], [[Media:Day04.B.20170915.pdf |B.pdf]], [[Media:Day04.C.20171209.pdf |C.pdf]]) ...... Structured Programming (1) Flowcharts * Day05 ([[Media:Day05.A.20171007.pdf |A.pdf]], [[Media:Day05.B.20170915.pdf |B.pdf]], [[Media:Day05.C.20171209.pdf |C.pdf]]) ...... Structured Programming (2) Conditions and Loops * Day06 ([[Media:Day06.A.20171007.pdf |A.pdf]], [[Media:Day06.B.20170923.pdf |B.pdf]], [[Media:Day06.C.20171209.pdf |C.pdf]]) ...... Program Control * Day07 ([[Media:Day07.A.20171007.pdf |A.pdf]], [[Media:Day07.B.20170926.pdf |B.pdf]], [[Media:Day07.C.20171209.pdf |C.pdf]]) ...... Function (1) Definitions * Day08 ([[Media:Day08.A.20171028.pdf |A.pdf]], [[Media:Day08.B.20171016.pdf |B.pdf]], [[Media:Day08.C.20171209.pdf |C.pdf]]) ...... Function (2) Storage Class and Scope * Day09 ([[Media:Day09.A.20171007.pdf |A.pdf]], [[Media:Day09.B.20171017.pdf |B.pdf]], [[Media:Day09.C.20171209.pdf |C.pdf]]) ...... Function (3) Recursion * Day10 ([[Media:Day10.A.20171209.pdf |A.pdf]], [[Media:Day10.B.20171017.pdf |B.pdf]], [[Media:Day10.C.20171209.pdf |C.pdf]]) ...... Arrays (1) Definitions * Day11 ([[Media:Day11.A.20171024.pdf |A.pdf]], [[Media:Day11.B.20171017.pdf |B.pdf]], [[Media:Day11.C.20171212.pdf |C.pdf]]) ...... Arrays (2) Applications * Day12 ([[Media:Day12.A.20171024.pdf |A.pdf]], [[Media:Day12.B.20171020.pdf |B.pdf]], [[Media:Day12.C.20171209.pdf |C.pdf]]) ...... Pointers (1) Definitions * Day13 ([[Media:Day13.A.20171025.pdf |A.pdf]], [[Media:Day13.B.20171024.pdf |B.pdf]], [[Media:Day13.C.20171209.pdf |C.pdf]]) ...... Pointers (2) Applications * Day14 ([[Media:Day14.A.20171226.pdf |A.pdf]], [[Media:Day14.B.20171101.pdf |B.pdf]], [[Media:Day14.C.20171209.pdf |C.pdf]]) ...... C String (1) * Day15 ([[Media:Day15.A.20171209.pdf |A.pdf]], [[Media:Day15.B.20171124.pdf |B.pdf]], [[Media:Day15.C.20171209.pdf |C.pdf]]) ...... C String (2) * Day16 ([[Media:Day16.A.20171208.pdf |A.pdf]], [[Media:Day16.B.20171114.pdf |B.pdf]], [[Media:Day16.C.20171209.pdf |C.pdf]]) ...... C Formatted IO * Day17 ([[Media:Day17.A.20171031.pdf |A.pdf]], [[Media:Day17.B.20171111.pdf |B.pdf]], [[Media:Day17.C.20171209.pdf |C.pdf]]) ...... Structure (1) Definitions * Day18 ([[Media:Day18.A.20171206.pdf |A.pdf]], [[Media:Day18.B.20171128.pdf |B.pdf]], [[Media:Day18.C.20171212.pdf |C.pdf]]) ...... Structure (2) Applications * Day19 ([[Media:Day19.A.20171205.pdf |A.pdf]], [[Media:Day19.B.20171121.pdf |B.pdf]], [[Media:Day19.C.20171209.pdf |C.pdf]]) ...... Union, Bitwise Operators, Enum * Day20 ([[Media:Day20.A.20171205.pdf |A.pdf]], [[Media:Day20.B.20171201.pdf |B.pdf]], [[Media:Day20.C.20171212.pdf |C.pdf]]) ...... Linked List * Day21 ([[Media:Day21.A.20171206.pdf |A.pdf]], [[Media:Day21.B.20171208.pdf |B.pdf]], [[Media:Day21.C.20171212.pdf |C.pdf]]) ...... File Processing * Day22 ([[Media:Day22.A.20171212.pdf |A.pdf]], [[Media:Day22.B.20171213.pdf |B.pdf]], [[Media:Day22.C.20171212.pdf |C.pdf]]) ...... Preprocessing <!----------------------------------------------------------------------> </br> See also https://cprogramex.wordpress.com/ == '''Old Materials '''== until 201201 * Intro.Overview.1.A ([[Media:C.Intro.Overview.1.A.20120107.pdf |pdf]]) * Intro.Memory.1.A ([[Media:C.Intro.Memory.1.A.20120107.pdf |pdf]]) * Intro.Number.1.A ([[Media:C.Intro.Number.1.A.20120107.pdf |pdf]]) * Repeat.Control.1.A ([[Media:C.Repeat.Control.1.A.20120109.pdf |pdf]]) * Repeat.Loop.1.A ([[Media:C.Repeat.Loop.1.A.20120113.pdf |pdf]]) * Work.Function.1.A ([[Media:C.Work.Function.1.A.20120117.pdf |pdf]]) * Work.Scope.1.A ([[Media:C.Work.Scope.1.A.20120117.pdf |pdf]]) * Series.Array.1.A ([[Media:Series.Array.1.A.20110718.pdf |pdf]]) * Series.Pointer.1.A ([[Media:Series.Pointer.1.A.20110719.pdf |pdf]]) * Series.Structure.1.A ([[Media:Series.Structure.1.A.20110805.pdf |pdf]]) * Data.Type.1.A ([[Media:C05.Data2.TypeCast.1.A.20130813.pdf |pdf]]) * Data.TypeCast.1.A ([[Media:Data.TypeCast.1.A.pdf |pdf]]) * Data.Operators.1.A ([[Media:Data.Operators.1.A.20110712.pdf |pdf]]) <br> until 201107 * Intro.1.A ([[Media:Intro.1.A.pdf |pdf]]) * Control.1.A ([[Media:Control.1.A.20110706.pdf |pdf]]) * Iteration.1.A ([[Media:Iteration.1.A.pdf |pdf]]) * Function.1.A ([[Media:Function.1.A.20110705.pdf |pdf]]) * Variable.1.A ([[Media:Variable.1.A.20110708.pdf |pdf]]) * Operators.1.A ([[Media:Operators.1.A.20110712.pdf |pdf]]) * Pointer.1.A ([[Media:Pointer.1.A.pdf |pdf]]) * Pointer.2.A ([[Media:Pointer.2.A.pdf |pdf]]) * Array.1.A ([[Media:Array.1.A.pdf |pdf]]) * Type.1.A ([[Media:Type.1.A.pdf |pdf]]) * Structure.1.A ([[Media:Structure.1.A.pdf |pdf]]) go to [ [[C programming in plain view]] ] [[Category:C programming language]] </br> 3tbigba6s34ronxp472m6opwl3peyg1 2844478 2844476 2026-11-04T13:54:59Z Young1lim 21186 /* Applications */ 2844478 wikitext text/x-wiki === Introduction === * Overview ([[Media:C01.Intro1.Overview.1.A.20170925.pdf |A.pdf]], [[Media:C01.Intro1.Overview.1.B.20170901.pdf |B.pdf]], [[Media:C01.Intro1.Overview.1.C.20170904.pdf |C.pdf]]) * Number System ([[Media:C01.Intro2.Number.1.A.20171023.pdf |A.pdf]], [[Media:C01.Intro2.Number.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro2.Number.1.C.20170914.pdf |C.pdf]]) * Memory System ([[Media:C01.Intro2.Memory.1.A.20170907.pdf |A.pdf]], [[Media:C01.Intro3.Memory.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro3.Memory.1.C.20170914.pdf |C.pdf]]) === Handling Repetition === * Control ([[Media:C02.Repeat1.Control.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat1.Control.1.B.20170918.pdf |B.pdf]], [[Media:C02.Repeat1.Control.1.C.20170926.pdf |C.pdf]]) * Loop ([[Media:C02.Repeat2.Loop.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat2.Loop.1.B.20170918.pdf |B.pdf]]) === Handling a Big Work === * Function Overview ([[Media:C03.Func1.Overview.1.A.20171030.pdf |A.pdf]], [[Media:C03.Func1.Oerview.1.B.20161022.pdf |B.pdf]]) * Functions & Variables ([[Media:C03.Func2.Variable.1.A.20161222.pdf |A.pdf]], [[Media:C03.Func2.Variable.1.B.20161222.pdf |B.pdf]]) * Functions & Pointers ([[Media:C03.Func3.Pointer.1.A.20161122.pdf |A.pdf]], [[Media:C03.Func3.Pointer.1.B.20161122.pdf |B.pdf]]) * Functions & Recursions ([[Media:C03.Func4.Recursion.1.A.20161214.pdf |A.pdf]], [[Media:C03.Func4.Recursion.1.B.20161214.pdf |B.pdf]]) === Handling Series of Data === ==== Background ==== * Background ([[Media:C04.Series0.Background.1.A.20180727.pdf |A.pdf]]) ==== Basics ==== * Pointers ([[Media:C04.S1.Pointer.1A.20240524.pdf |A.pdf]], [[Media:C04.Series2.Pointer.1.B.20161115.pdf |B.pdf]]) * Arrays ([[Media:C04.S2.Array.1A.20240514.pdf |A.pdf]], [[Media:C04.Series1.Array.1.B.20161115.pdf |B.pdf]]) * Array Pointers ([[Media:C04.S3.ArrayPointer.1A.20240208.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]]) * Multi-dimensional Arrays ([[Media:C04.Series4.MultiDim.1.A.20221130.pdf |A.pdf]], [[Media:C04.Series4.MultiDim.1.B.1111.pdf |B.pdf]]) * Array Access Methods ([[Media:C04.Series4.ArrayAccess.1.A.20190511.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]]) * Structures ([[Media:C04.Series3.Structure.1.A.20171204.pdf |A.pdf]], [[Media:C04.Series2.Structure.1.B.20161130.pdf |B.pdf]]) ==== Examples ==== * Spreadsheet Example Programs :: Example 1 ([[Media:C04.Series7.Example.1.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.1.C.20171213.pdf |C.pdf]]) :: Example 2 ([[Media:C04.Series7.Example.2.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.2.C.20171213.pdf |C.pdf]]) :: Example 3 ([[Media:C04.Series7.Example.3.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.3.C.20171213.pdf |C.pdf]]) :: Bubble Sort ([[Media:C04.Series7.BubbleSort.1.A.20171211.pdf |A.pdf]]) ==== Applications ==== * Address-of and de-reference operators ([[Media:C04.SA0.PtrOperator.1A.20261102.pdf |A.pdf]]) * Applications of Pointers ([[Media:C04.SA1.AppPointer.1A.20241121.pdf |A.pdf]]) * Applications of Arrays ([[Media:C04.SA2.AppArray.1A.20240715.pdf |A.pdf]]) * Applications of Array Pointers ([[Media:C04.SA3.AppArrayPointer.1A.20240210.pdf |A.pdf]]) * Applications of Multi-dimensional Arrays ([[Media:C04.Series4App.MultiDim.1.A.20210719.pdf |A.pdf]]) * Applications of Array Access Methods ([[Media:C04.Series9.AppArrAcess.1.A.20190511.pdf |A.pdf]]) * Applications of Structures ([[Media:C04.Series6.AppStruct.1.A.20190423.pdf |A.pdf]]) === Handling Various Kinds of Data === * Types ([[Media:C05.Data1.Type.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data1.Type.1.B.20161212.pdf |B.pdf]]) * Typecasts ([[Media:C05.Data2.TypeCast.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data2.TypeCast.1.B.20161216.pdf |A.pdf]]) * Operators ([[Media:C05.Data3.Operators.1.A.20161219.pdf |A.pdf]], [[Media:C05.Data3.Operators.1.B.20161216.pdf |B.pdf]]) * Files ([[Media:C05.Data4.File.1.A.20161124.pdf |A.pdf]], [[Media:C05.Data4.File.1.B.20161212.pdf |B.pdf]]) === Handling Low Level Operations === * Bitwise Operations ([[Media:BitOp.1.B.20161214.pdf |A.pdf]], [[Media:BitOp.1.B.20161203.pdf |B.pdf]]) * Bit Field ([[Media:BitField.1.A.20161214.pdf |A.pdf]], [[Media:BitField.1.B.20161202.pdf |B.pdf]]) * Union ([[Media:Union.1.A.20161221.pdf |A.pdf]], [[Media:Union.1.B.20161111.pdf |B.pdf]]) * Accessing IO Registers ([[Media:IO.1.A.20141215.pdf |A.pdf]], [[Media:IO.1.B.20161217.pdf |B.pdf]]) === Declarations === * Type Specifiers and Qualifiers ([[Media:C07.Spec1.Type.1.A.20171004.pdf |pdf]]) * Storage Class Specifiers ([[Media:C07.Spec2.Storage.1.A.20171009.pdf |pdf]]) * Scope === Class Notes === * TOC ([[Media:TOC.20171007.pdf |TOC.pdf]]) * Day01 ([[Media:Day01.A.20171007.pdf |A.pdf]], [[Media:Day01.B.20171209.pdf |B.pdf]], [[Media:Day01.C.20171211.pdf |C.pdf]]) ...... Introduction (1) Standard Library * Day02 ([[Media:Day02.A.20171007.pdf |A.pdf]], [[Media:Day02.B.20171209.pdf |B.pdf]], [[Media:Day02.C.20171209.pdf |C.pdf]]) ...... Introduction (2) Basic Elements * Day03 ([[Media:Day03.A.20171007.pdf |A.pdf]], [[Media:Day03.B.20170908.pdf |B.pdf]], [[Media:Day03.C.20171209.pdf |C.pdf]]) ...... Introduction (3) Numbers * Day04 ([[Media:Day04.A.20171007.pdf |A.pdf]], [[Media:Day04.B.20170915.pdf |B.pdf]], [[Media:Day04.C.20171209.pdf |C.pdf]]) ...... Structured Programming (1) Flowcharts * Day05 ([[Media:Day05.A.20171007.pdf |A.pdf]], [[Media:Day05.B.20170915.pdf |B.pdf]], [[Media:Day05.C.20171209.pdf |C.pdf]]) ...... Structured Programming (2) Conditions and Loops * Day06 ([[Media:Day06.A.20171007.pdf |A.pdf]], [[Media:Day06.B.20170923.pdf |B.pdf]], [[Media:Day06.C.20171209.pdf |C.pdf]]) ...... Program Control * Day07 ([[Media:Day07.A.20171007.pdf |A.pdf]], [[Media:Day07.B.20170926.pdf |B.pdf]], [[Media:Day07.C.20171209.pdf |C.pdf]]) ...... Function (1) Definitions * Day08 ([[Media:Day08.A.20171028.pdf |A.pdf]], [[Media:Day08.B.20171016.pdf |B.pdf]], [[Media:Day08.C.20171209.pdf |C.pdf]]) ...... Function (2) Storage Class and Scope * Day09 ([[Media:Day09.A.20171007.pdf |A.pdf]], [[Media:Day09.B.20171017.pdf |B.pdf]], [[Media:Day09.C.20171209.pdf |C.pdf]]) ...... Function (3) Recursion * Day10 ([[Media:Day10.A.20171209.pdf |A.pdf]], [[Media:Day10.B.20171017.pdf |B.pdf]], [[Media:Day10.C.20171209.pdf |C.pdf]]) ...... Arrays (1) Definitions * Day11 ([[Media:Day11.A.20171024.pdf |A.pdf]], [[Media:Day11.B.20171017.pdf |B.pdf]], [[Media:Day11.C.20171212.pdf |C.pdf]]) ...... Arrays (2) Applications * Day12 ([[Media:Day12.A.20171024.pdf |A.pdf]], [[Media:Day12.B.20171020.pdf |B.pdf]], [[Media:Day12.C.20171209.pdf |C.pdf]]) ...... Pointers (1) Definitions * Day13 ([[Media:Day13.A.20171025.pdf |A.pdf]], [[Media:Day13.B.20171024.pdf |B.pdf]], [[Media:Day13.C.20171209.pdf |C.pdf]]) ...... Pointers (2) Applications * Day14 ([[Media:Day14.A.20171226.pdf |A.pdf]], [[Media:Day14.B.20171101.pdf |B.pdf]], [[Media:Day14.C.20171209.pdf |C.pdf]]) ...... C String (1) * Day15 ([[Media:Day15.A.20171209.pdf |A.pdf]], [[Media:Day15.B.20171124.pdf |B.pdf]], [[Media:Day15.C.20171209.pdf |C.pdf]]) ...... C String (2) * Day16 ([[Media:Day16.A.20171208.pdf |A.pdf]], [[Media:Day16.B.20171114.pdf |B.pdf]], [[Media:Day16.C.20171209.pdf |C.pdf]]) ...... C Formatted IO * Day17 ([[Media:Day17.A.20171031.pdf |A.pdf]], [[Media:Day17.B.20171111.pdf |B.pdf]], [[Media:Day17.C.20171209.pdf |C.pdf]]) ...... Structure (1) Definitions * Day18 ([[Media:Day18.A.20171206.pdf |A.pdf]], [[Media:Day18.B.20171128.pdf |B.pdf]], [[Media:Day18.C.20171212.pdf |C.pdf]]) ...... Structure (2) Applications * Day19 ([[Media:Day19.A.20171205.pdf |A.pdf]], [[Media:Day19.B.20171121.pdf |B.pdf]], [[Media:Day19.C.20171209.pdf |C.pdf]]) ...... Union, Bitwise Operators, Enum * Day20 ([[Media:Day20.A.20171205.pdf |A.pdf]], [[Media:Day20.B.20171201.pdf |B.pdf]], [[Media:Day20.C.20171212.pdf |C.pdf]]) ...... Linked List * Day21 ([[Media:Day21.A.20171206.pdf |A.pdf]], [[Media:Day21.B.20171208.pdf |B.pdf]], [[Media:Day21.C.20171212.pdf |C.pdf]]) ...... File Processing * Day22 ([[Media:Day22.A.20171212.pdf |A.pdf]], [[Media:Day22.B.20171213.pdf |B.pdf]], [[Media:Day22.C.20171212.pdf |C.pdf]]) ...... Preprocessing <!----------------------------------------------------------------------> </br> See also https://cprogramex.wordpress.com/ == '''Old Materials '''== until 201201 * Intro.Overview.1.A ([[Media:C.Intro.Overview.1.A.20120107.pdf |pdf]]) * Intro.Memory.1.A ([[Media:C.Intro.Memory.1.A.20120107.pdf |pdf]]) * Intro.Number.1.A ([[Media:C.Intro.Number.1.A.20120107.pdf |pdf]]) * Repeat.Control.1.A ([[Media:C.Repeat.Control.1.A.20120109.pdf |pdf]]) * Repeat.Loop.1.A ([[Media:C.Repeat.Loop.1.A.20120113.pdf |pdf]]) * Work.Function.1.A ([[Media:C.Work.Function.1.A.20120117.pdf |pdf]]) * Work.Scope.1.A ([[Media:C.Work.Scope.1.A.20120117.pdf |pdf]]) * Series.Array.1.A ([[Media:Series.Array.1.A.20110718.pdf |pdf]]) * Series.Pointer.1.A ([[Media:Series.Pointer.1.A.20110719.pdf |pdf]]) * Series.Structure.1.A ([[Media:Series.Structure.1.A.20110805.pdf |pdf]]) * Data.Type.1.A ([[Media:C05.Data2.TypeCast.1.A.20130813.pdf |pdf]]) * Data.TypeCast.1.A ([[Media:Data.TypeCast.1.A.pdf |pdf]]) * Data.Operators.1.A ([[Media:Data.Operators.1.A.20110712.pdf |pdf]]) <br> until 201107 * Intro.1.A ([[Media:Intro.1.A.pdf |pdf]]) * Control.1.A ([[Media:Control.1.A.20110706.pdf |pdf]]) * Iteration.1.A ([[Media:Iteration.1.A.pdf |pdf]]) * Function.1.A ([[Media:Function.1.A.20110705.pdf |pdf]]) * Variable.1.A ([[Media:Variable.1.A.20110708.pdf |pdf]]) * Operators.1.A ([[Media:Operators.1.A.20110712.pdf |pdf]]) * Pointer.1.A ([[Media:Pointer.1.A.pdf |pdf]]) * Pointer.2.A ([[Media:Pointer.2.A.pdf |pdf]]) * Array.1.A ([[Media:Array.1.A.pdf |pdf]]) * Type.1.A ([[Media:Type.1.A.pdf |pdf]]) * Structure.1.A ([[Media:Structure.1.A.pdf |pdf]]) go to [ [[C programming in plain view]] ] [[Category:C programming language]] </br> 3ucm2l67p9bmtbda28747jya4rthp9x 2 289273 2844540 2844438 2026-11-05T00:07:22Z Dc.samizdat 2856930 /* Summary */ 2844540 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]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space. * Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space. * Our entire 3-space manifold is moving at the speed of light <math>c</math> in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is <math>c\prime = 2c</math> measured on a straight line in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space. * The universe may properly be viewed as a <math>4</math>-sphere whose center is wherever there is intelligence. == A theory of the Euclidean cosmos == <blockquote>If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.<ref>{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}</ref></blockquote> 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 detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <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 bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <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 bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <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 bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <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 center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <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 free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity <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 local product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<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 direct dimensional 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. 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 <math>\mathrm{QT}</math> or a simple rotation <math>\mathrm{Q}</math> lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. 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), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not <math>2\pi r</math>. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In 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, but it is a 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 enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be seen as 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>. Objects with mass of any size (including atoms and their subatomic particles) 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>, such as an atom in its own proper reference frame. 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>. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel hyperplanes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting hyperplanes, 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 two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation <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 == Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity <math>c</math>, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.) The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply: <blockquote>''Time ticks at the same rate everywhere in space.''</blockquote> This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than <math>c</math> in any stationary place in 4-space. None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <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 distinct screw translations <math>QT</math>. 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 <math>T</math> through 4-space. The velocity of that vector 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 bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we refer to as isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference, or 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 mirror 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 mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <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 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, since they depend on the shape of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with its 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 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant <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 == Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the six regular convex 4-polytopes. The geometry of the [[Polyscheme#Regular convex 4-polytopes|regular convex 4-polytopes]] is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are called the 5-point 4-simplex, the 8-point 4-orthoplex, the 16-point 4-hypercube, the 24-point 24-cell, the 120-point 600-cell, and the 600-point 120-cell. The smallest regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. The rotational geometry of the proton is that of the 4-simplex. The largest regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is geometrically comprehensive, containing instances of the relationships among all these polytopes. The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each element includes that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, there are only 120 possible atomic elements. The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding, although the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. .... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <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 Euclidean and spherical <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 reference frame, not only from their own proper inertial 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 discovery of higher dimensional space: <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 immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <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:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine "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."<ref name=Einstein1923/> 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}} p0tko283d53o0zby3k7t3yya4nqcjro 2844541 2844540 2026-11-05T00:26:51Z Dc.samizdat 2856930 /* Revolutions */ 2844541 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]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space. * Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space. * Our entire 3-space manifold is moving at the speed of light <math>c</math> in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is <math>c\prime = 2c</math> measured on a straight line in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space. * The universe may properly be viewed as a <math>4</math>-sphere whose center is wherever there is intelligence. == A theory of the Euclidean cosmos == <blockquote>If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.<ref>{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}</ref></blockquote> 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 detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <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 bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <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 bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <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 bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <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 center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <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 free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity <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 local product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<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 direct dimensional 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. 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 <math>\mathrm{QT}</math> or a simple rotation <math>\mathrm{Q}</math> lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. 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), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not <math>2\pi r</math>. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In 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, but it is a 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 enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be seen as 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>. Objects with mass of any size (including atoms and their subatomic particles) 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>, such as an atom in its own proper reference frame. 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>. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel hyperplanes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting hyperplanes, 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 two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation <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 == Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity <math>c</math>, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.) The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply: <blockquote>''Time ticks at the same rate everywhere in space.''</blockquote> This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than <math>c</math> in any stationary place in 4-space. None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <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 distinct screw translations <math>QT</math>. 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 <math>T</math> through 4-space. The velocity of that vector 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 bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we refer to as isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference, or 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 mirror 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 mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <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 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, since they depend on the shape of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with its 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 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant <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 == Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the six regular convex 4-polytopes. The geometry of the [[Polyscheme#Regular convex 4-polytopes|regular convex 4-polytopes]] is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are called the 5-point 4-simplex, the 8-point 4-orthoplex, the 16-point 4-hypercube, the 24-point 24-cell, the 120-point 600-cell, and the 600-point 120-cell. The smallest regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. The rotational geometry of the proton is that of the 4-simplex. The largest regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is geometrically comprehensive, containing instances of the relationships among all these polytopes. The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each element includes that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, there are only 120 possible atomic elements. The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding, although the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. .... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <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 Euclidean and spherical <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 reference frame, not only from their own proper inertial 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 discovery of higher dimensional space: <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, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <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:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine "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."<ref name=Einstein1923/> 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}} edtziap4hinx1e7dkhqtgcecnsiutx0 2844542 2844541 2026-11-05T00:50:18Z Dc.samizdat 2856930 /* Revolutions */ 2844542 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]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space. * Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space. * Our entire 3-space manifold is moving at the speed of light <math>c</math> in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is <math>c\prime = 2c</math> measured on a straight line in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space. * The universe may properly be viewed as a <math>4</math>-sphere whose center is wherever there is intelligence. == A theory of the Euclidean cosmos == <blockquote>If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.<ref>{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}</ref></blockquote> 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 detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <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 bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <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 bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <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 bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <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 center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <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 free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity <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 local product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<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 direct dimensional 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. 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 <math>\mathrm{QT}</math> or a simple rotation <math>\mathrm{Q}</math> lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. 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), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not <math>2\pi r</math>. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In 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, but it is a 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 enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be seen as 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>. Objects with mass of any size (including atoms and their subatomic particles) 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>, such as an atom in its own proper reference frame. 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>. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel hyperplanes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting hyperplanes, 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 two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation <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 == Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity <math>c</math>, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.) The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply: <blockquote>''Time ticks at the same rate everywhere in space.''</blockquote> This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than <math>c</math> in any stationary place in 4-space. None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <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 distinct screw translations <math>QT</math>. 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 <math>T</math> through 4-space. The velocity of that vector 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 bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we refer to as isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference, or 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 mirror 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 mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <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 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, since they depend on the shape of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with its 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 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant <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 == Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the six regular convex 4-polytopes. The geometry of the [[Polyscheme#Regular convex 4-polytopes|regular convex 4-polytopes]] is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are called the 5-point 4-simplex, the 8-point 4-orthoplex, the 16-point 4-hypercube, the 24-point 24-cell, the 120-point 600-cell, and the 600-point 120-cell. The smallest regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. The rotational geometry of the proton is that of the 4-simplex. The largest regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is geometrically comprehensive, containing instances of the relationships among all these polytopes. The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each element includes that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, there are only 120 possible atomic elements. The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding, although the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. .... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <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 Euclidean and spherical <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 reference frame, not only from their own proper inertial 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 discovery of higher dimensional space: <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, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <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 the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine "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."<ref name=Einstein1923/> 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}} gyajr1d2c2yudfo4a4rycl9vafqcsch 2844550 2844542 2026-11-05T01:52:47Z Dc.samizdat 2856930 /* Polycentric spherical relativity */ 2844550 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]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space. * Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space. * Our entire 3-space manifold is moving at the speed of light <math>c</math> in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is <math>c\prime = 2c</math> measured on a straight line in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space. * The universe may properly be viewed as a <math>4</math>-sphere whose center is wherever there is intelligence. == A theory of the Euclidean cosmos == <blockquote>If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.<ref>{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}</ref></blockquote> 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 detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <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 bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <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 bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <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 bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <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 center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <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 free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity <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 local product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<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 direct dimensional 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. 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 <math>\mathrm{QT}</math> or a simple rotation <math>\mathrm{Q}</math> lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. 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), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not <math>2\pi r</math>. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In 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, but it is a 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 enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be seen as 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>. Objects with mass of any size (including atoms and their subatomic particles) 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>, such as an atom in its own proper reference frame. 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>. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel hyperplanes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting hyperplanes, 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 two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation <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 == Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity <math>c</math>, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.) The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply: <blockquote>''Time ticks at the same rate everywhere in space.''</blockquote> This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than <math>c</math> in any stationary place in 4-space. None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <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 distinct screw translations <math>QT</math>. 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 <math>T</math> through 4-space. The velocity of that vector 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 bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we refer to as isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference, or 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 mirror 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 mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <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 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, since they depend on the shape of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with its 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 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant <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 == Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the six regular convex 4-polytopes. The geometry of the [[Polyscheme#Regular convex 4-polytopes|regular convex 4-polytopes]] is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are called the 5-point 4-simplex, the 8-point 4-orthoplex, the 16-point 4-hypercube, the 24-point 24-cell, the 120-point 600-cell, and the 600-point 120-cell. The smallest regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. The rotational geometry of the proton is that of the 4-simplex. The largest regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is geometrically comprehensive, containing instances of the relationships among all these polytopes. The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each element includes that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, there are only 120 possible atomic elements. The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding, although the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. .... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <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 Euclidean and spherical <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 reference frame, not only from their own proper inertial 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 spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space: <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, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <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 the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine "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."<ref name=Einstein1923/> 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}} c8mkgve77kbhmt3nux2ijbcocc9ob4v 2844551 2844550 2026-11-05T01:57:25Z Dc.samizdat 2856930 /* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */ 2844551 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]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space. * Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space. * Our entire 3-space manifold is moving at the speed of light <math>c</math> in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is <math>c\prime = 2c</math> measured on a straight line in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space. * The universe may properly be viewed as a <math>4</math>-sphere whose center is wherever there is intelligence. == A theory of the Euclidean cosmos == <blockquote>If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.<ref>{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}</ref></blockquote> 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 detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <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 bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <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 bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <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 bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <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 center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <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 free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity <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 local product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<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 direct dimensional 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. 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 <math>\mathrm{QT}</math> or a simple rotation <math>\mathrm{Q}</math> lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. 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), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not <math>2\pi r</math>. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In 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, but it is a 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 enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be seen as 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>. Objects with mass of any size (including atoms and their subatomic particles) 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>, such as an atom in its own proper reference frame. 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>. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel hyperplanes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting hyperplanes, 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 two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation <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 == Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity <math>c</math>, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.) The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply: <blockquote>''Time ticks at the same rate everywhere in space.''</blockquote> This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than <math>c</math> in any stationary place in 4-space. None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <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 distinct screw translations <math>QT</math>. 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 <math>T</math> through 4-space. The velocity of that vector 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 bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we refer to as isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference, or 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 mirror 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 mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <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 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, since they depend on the shape of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with its 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 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant <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 == Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are called the 5-point 4-simplex, the 8-point 4-orthoplex, the 16-point 4-hypercube, the 24-point 24-cell, the 120-point 600-cell, and the 600-point 120-cell. The smallest regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. The rotational geometry of the proton is that of the 4-simplex. The largest regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is geometrically comprehensive, containing instances of the relationships among all these polytopes. The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each element includes that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, there are only 120 possible atomic elements. The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding, although the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. .... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <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 Euclidean and spherical <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 reference frame, not only from their own proper inertial 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 spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space: <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, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <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 the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine "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."<ref name=Einstein1923/> 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}} j7vqqayj64tfysf0es504y51co282cp 2844556 2844551 2026-11-05T02:03:57Z Dc.samizdat 2856930 /* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */ 2844556 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]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space. * Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space. * Our entire 3-space manifold is moving at the speed of light <math>c</math> in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is <math>c\prime = 2c</math> measured on a straight line in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space. * The universe may properly be viewed as a <math>4</math>-sphere whose center is wherever there is intelligence. == A theory of the Euclidean cosmos == <blockquote>If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.<ref>{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}</ref></blockquote> 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 detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <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 bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <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 bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <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 bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <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 center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <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 free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity <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 local product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<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 direct dimensional 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. 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 <math>\mathrm{QT}</math> or a simple rotation <math>\mathrm{Q}</math> lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. 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), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not <math>2\pi r</math>. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In 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, but it is a 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 enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be seen as 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>. Objects with mass of any size (including atoms and their subatomic particles) 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>, such as an atom in its own proper reference frame. 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>. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel hyperplanes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting hyperplanes, 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 two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation <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 == Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity <math>c</math>, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.) The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply: <blockquote>''Time ticks at the same rate everywhere in space.''</blockquote> This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than <math>c</math> in any stationary place in 4-space. None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <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 distinct screw translations <math>QT</math>. 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 <math>T</math> through 4-space. The velocity of that vector 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 bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we refer to as isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference, or 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 mirror 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 mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <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 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, since they depend on the shape of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with its 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 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant <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 == Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell). The smallest regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. The rotational geometry of the proton is that of the 4-simplex. The largest regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is geometrically comprehensive, containing instances of the relationships among all these polytopes. The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each element includes that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, there are only 120 possible atomic elements. The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding, although the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. .... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <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 Euclidean and spherical <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 reference frame, not only from their own proper inertial 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 spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space: <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, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <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 the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine "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."<ref name=Einstein1923/> 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}} 4pl10cz83ui07pmuzshh2mqtr6e9fi3 2844560 2844556 2026-11-05T02:12:05Z Dc.samizdat 2856930 /* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */ 2844560 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]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space. * Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space. * Our entire 3-space manifold is moving at the speed of light <math>c</math> in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is <math>c\prime = 2c</math> measured on a straight line in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space. * The universe may properly be viewed as a <math>4</math>-sphere whose center is wherever there is intelligence. == A theory of the Euclidean cosmos == <blockquote>If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.<ref>{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}</ref></blockquote> 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 detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <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 bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <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 bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <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 bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <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 center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <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 free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity <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 local product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<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 direct dimensional 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. 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 <math>\mathrm{QT}</math> or a simple rotation <math>\mathrm{Q}</math> lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. 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), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not <math>2\pi r</math>. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In 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, but it is a 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 enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be seen as 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>. Objects with mass of any size (including atoms and their subatomic particles) 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>, such as an atom in its own proper reference frame. 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>. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel hyperplanes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting hyperplanes, 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 two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation <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 == Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity <math>c</math>, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.) The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply: <blockquote>''Time ticks at the same rate everywhere in space.''</blockquote> This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than <math>c</math> in any stationary place in 4-space. None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <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 distinct screw translations <math>QT</math>. 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 <math>T</math> through 4-space. The velocity of that vector 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 bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we refer to as isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference, or 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 mirror 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 mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <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 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, since they depend on the shape of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with its 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 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant <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 == Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell). The smallest regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. The rotational geometry of the proton is that of the 4-simplex. The largest regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes. The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each element includes that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, there are only 120 possible atomic elements. The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding, although the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. .... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <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 Euclidean and spherical <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 reference frame, not only from their own proper inertial 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 spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space: <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, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <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 the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine "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."<ref name=Einstein1923/> 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}} 6hwpcnq7wyisnwhotqem2izh39phti2 2844565 2844560 2026-11-05T02:16:56Z Dc.samizdat 2856930 /* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */ 2844565 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]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space. * Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space. * Our entire 3-space manifold is moving at the speed of light <math>c</math> in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is <math>c\prime = 2c</math> measured on a straight line in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space. * The universe may properly be viewed as a <math>4</math>-sphere whose center is wherever there is intelligence. == A theory of the Euclidean cosmos == <blockquote>If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.<ref>{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}</ref></blockquote> 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 detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <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 bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <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 bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <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 bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <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 center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <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 free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity <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 local product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<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 direct dimensional 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. 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 <math>\mathrm{QT}</math> or a simple rotation <math>\mathrm{Q}</math> lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. 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), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not <math>2\pi r</math>. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In 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, but it is a 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 enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be seen as 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>. Objects with mass of any size (including atoms and their subatomic particles) 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>, such as an atom in its own proper reference frame. 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>. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel hyperplanes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting hyperplanes, 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 two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation <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 == Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity <math>c</math>, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.) The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply: <blockquote>''Time ticks at the same rate everywhere in space.''</blockquote> This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than <math>c</math> in any stationary place in 4-space. None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <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 distinct screw translations <math>QT</math>. 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 <math>T</math> through 4-space. The velocity of that vector 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 bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we refer to as isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference, or 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 mirror 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 mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <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 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, since they depend on the shape of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with its 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 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant <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 == Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell). The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. The rotational geometry of the proton is that of the 4-simplex. The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes. The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each element includes that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, there are only 120 distinct atomic elements. The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding, although the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. .... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <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 Euclidean and spherical <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 reference frame, not only from their own proper inertial 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 spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space: <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, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <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 the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine "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."<ref name=Einstein1923/> 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}} cr80cwhqck7zlln8f960f59ycvnxqep 2844569 2844565 2026-11-05T02:25:15Z Dc.samizdat 2856930 /* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */ 2844569 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]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space. * Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space. * Our entire 3-space manifold is moving at the speed of light <math>c</math> in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is <math>c\prime = 2c</math> measured on a straight line in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space. * The universe may properly be viewed as a <math>4</math>-sphere whose center is wherever there is intelligence. == A theory of the Euclidean cosmos == <blockquote>If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.<ref>{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}</ref></blockquote> 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 detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <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 bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <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 bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <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 bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <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 center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <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 free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity <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 local product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<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 direct dimensional 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. 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 <math>\mathrm{QT}</math> or a simple rotation <math>\mathrm{Q}</math> lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. 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), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not <math>2\pi r</math>. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In 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, but it is a 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 enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be seen as 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>. Objects with mass of any size (including atoms and their subatomic particles) 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>, such as an atom in its own proper reference frame. 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>. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel hyperplanes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting hyperplanes, 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 two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation <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 == Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity <math>c</math>, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.) The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply: <blockquote>''Time ticks at the same rate everywhere in space.''</blockquote> This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than <math>c</math> in any stationary place in 4-space. None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <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 distinct screw translations <math>QT</math>. 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 <math>T</math> through 4-space. The velocity of that vector 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 bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we refer to as isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference, or 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 mirror 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 mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <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 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, since they depend on the shape of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with its 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 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant <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 == Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell). The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. The rotational geometry of the proton is that of the 4-simplex. The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes. The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each element includes that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, there are only 120 distinct atomic elements. The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series, just not one as straightforward as simple compounding. We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. .... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <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 Euclidean and spherical <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 reference frame, not only from their own proper inertial 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 spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space: <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, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <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 the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine "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."<ref name=Einstein1923/> 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}} avhzwioqppfc94d55f1ez36q5rgajh5 2844570 2844569 2026-11-05T02:26:02Z Dc.samizdat 2856930 /* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */ 2844570 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]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space. * Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space. * Our entire 3-space manifold is moving at the speed of light <math>c</math> in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is <math>c\prime = 2c</math> measured on a straight line in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space. * The universe may properly be viewed as a <math>4</math>-sphere whose center is wherever there is intelligence. == A theory of the Euclidean cosmos == <blockquote>If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.<ref>{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}</ref></blockquote> 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 detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <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 bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <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 bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <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 bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <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 center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <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 free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity <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 local product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<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 direct dimensional 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. 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 <math>\mathrm{QT}</math> or a simple rotation <math>\mathrm{Q}</math> lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. 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), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not <math>2\pi r</math>. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In 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, but it is a 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 enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be seen as 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>. Objects with mass of any size (including atoms and their subatomic particles) 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>, such as an atom in its own proper reference frame. 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>. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel hyperplanes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting hyperplanes, 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 two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation <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 == Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity <math>c</math>, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.) The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply: <blockquote>''Time ticks at the same rate everywhere in space.''</blockquote> This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than <math>c</math> in any stationary place in 4-space. None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <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 distinct screw translations <math>QT</math>. 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 <math>T</math> through 4-space. The velocity of that vector 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 bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we refer to as isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference, or 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 mirror 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 mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <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 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, since they depend on the shape of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with its 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 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant <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 == Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell). The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. The rotational geometry of the proton is that of the 4-simplex. The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes. The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each element includes that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, there are only 120 distinct atomic elements. The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series, just not one as direct as simple compounding. We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. .... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <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 Euclidean and spherical <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 reference frame, not only from their own proper inertial 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 spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space: <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, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <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 the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine "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."<ref name=Einstein1923/> 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}} 0srl6t67ny999dajhejkp9r0w9j4pjn 2844572 2844570 2026-11-05T02:36:17Z Dc.samizdat 2856930 /* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */ 2844572 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]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space. * Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space. * Our entire 3-space manifold is moving at the speed of light <math>c</math> in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is <math>c\prime = 2c</math> measured on a straight line in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space. * The universe may properly be viewed as a <math>4</math>-sphere whose center is wherever there is intelligence. == A theory of the Euclidean cosmos == <blockquote>If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.<ref>{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}</ref></blockquote> 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 detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <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 bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <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 bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <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 bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <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 center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <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 free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity <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 local product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<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 direct dimensional 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. 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 <math>\mathrm{QT}</math> or a simple rotation <math>\mathrm{Q}</math> lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. 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), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not <math>2\pi r</math>. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In 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, but it is a 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 enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be seen as 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>. Objects with mass of any size (including atoms and their subatomic particles) 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>, such as an atom in its own proper reference frame. 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>. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel hyperplanes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting hyperplanes, 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 two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation <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 == Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity <math>c</math>, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.) The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply: <blockquote>''Time ticks at the same rate everywhere in space.''</blockquote> This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than <math>c</math> in any stationary place in 4-space. None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <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 distinct screw translations <math>QT</math>. 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 <math>T</math> through 4-space. The velocity of that vector 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 bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we refer to as isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference, or 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 mirror 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 mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <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 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, since they depend on the shape of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with its 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 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant <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 == Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell). The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. The rotational geometry of the proton is that of the 4-simplex. The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes. The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each element includes that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, there are only 120 distinct atomic elements. The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. It turns out that if you view the 4-simplex as the largest 4-polytope instead of the smallest, you can find smaller instances of all the other 4-polytopes within it. We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. .... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <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 Euclidean and spherical <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 reference frame, not only from their own proper inertial 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 spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space: <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, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <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 the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine "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."<ref name=Einstein1923/> 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}} dpa6646rz6v8hzozkleaq318ij9jq8c 2844574 2844572 2026-11-05T02:44:01Z Dc.samizdat 2856930 /* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */ 2844574 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]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space. * Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space. * Our entire 3-space manifold is moving at the speed of light <math>c</math> in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is <math>c\prime = 2c</math> measured on a straight line in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space. * The universe may properly be viewed as a <math>4</math>-sphere whose center is wherever there is intelligence. == A theory of the Euclidean cosmos == <blockquote>If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.<ref>{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}</ref></blockquote> 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 detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <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 bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <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 bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <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 bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <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 center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <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 free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity <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 local product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<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 direct dimensional 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. 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 <math>\mathrm{QT}</math> or a simple rotation <math>\mathrm{Q}</math> lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. 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), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not <math>2\pi r</math>. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In 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, but it is a 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 enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be seen as 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>. Objects with mass of any size (including atoms and their subatomic particles) 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>, such as an atom in its own proper reference frame. 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>. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel hyperplanes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting hyperplanes, 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 two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation <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 == Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity <math>c</math>, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.) The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply: <blockquote>''Time ticks at the same rate everywhere in space.''</blockquote> This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than <math>c</math> in any stationary place in 4-space. None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <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 distinct screw translations <math>QT</math>. 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 <math>T</math> through 4-space. The velocity of that vector 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 bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we refer to as isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference, or 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 mirror 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 mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <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 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, since they depend on the shape of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with its 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 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant <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 == Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell). The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We observe that the rotational geometry of the proton is that of the 4-simplex. The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes. The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each element includes that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, there are only 120 distinct atomic elements. The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. It turns out that if you view the 4-simplex as the largest 4-polytope instead of the smallest, you can find smaller instances of all the other 4-polytopes within it. We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. .... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <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 Euclidean and spherical <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 reference frame, not only from their own proper inertial 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 spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space: <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, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <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 the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine "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."<ref name=Einstein1923/> 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}} shcf3jbjn46fo1kua0uiqjgorty8bn9 2844575 2844574 2026-11-05T02:46:23Z Dc.samizdat 2856930 /* Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space */ 2844575 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]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space. * Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space. * Our entire 3-space manifold is moving at the speed of light <math>c</math> in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is <math>c\prime = 2c</math> measured on a straight line in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space. * The universe may properly be viewed as a <math>4</math>-sphere whose center is wherever there is intelligence. == A theory of the Euclidean cosmos == <blockquote>If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.<ref>{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}</ref></blockquote> 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 detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <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 bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <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 bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <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 bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <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 center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <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 free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity <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 local product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<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 direct dimensional 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. 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 <math>\mathrm{QT}</math> or a simple rotation <math>\mathrm{Q}</math> lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. 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), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not <math>2\pi r</math>. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In 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, but it is a 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 enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be seen as 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>. Objects with mass of any size (including atoms and their subatomic particles) 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>, such as an atom in its own proper reference frame. 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>. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel hyperplanes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting hyperplanes, 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 two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation <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 == Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity <math>c</math>, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.) The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply: <blockquote>''Time ticks at the same rate everywhere in space.''</blockquote> This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than <math>c</math> in any stationary place in 4-space. None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <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 distinct screw translations <math>QT</math>. 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 <math>T</math> through 4-space. The velocity of that vector 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 bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we refer to as isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference, or 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 mirror 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 mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <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 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, since they depend on the shape of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with its 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 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant <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 == Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell). The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We conclude that the rotational geometry of the proton is that of the 4-simplex. The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes. The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each element includes that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, there are only 120 distinct atomic elements. The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. It turns out that if you view the 4-simplex as the largest 4-polytope instead of the smallest, you can find smaller instances of all the other 4-polytopes within it. We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. .... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <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 Euclidean and spherical <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 reference frame, not only from their own proper inertial 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 spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space: <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, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <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 the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine "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."<ref name=Einstein1923/> 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}} 5w0p9nss2iyvoi2wrsuas1w7efmd7mh 2844593 2844575 2026-11-05T09:33:45Z Dc.samizdat 2856930 /* A theory of the Euclidean cosmos */ 2844593 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]]. An atom's motion is the local product of its discrete self-reflections in Euclidean 4-space. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. It occupies a hyperspherical region in Euclidean 4-space. * Everything of intermediate size between an atom and a star, including ourselves and our planet, lies in a 3-dimensional manifold of ordinary space, embedded in Euclidean 4-space. * Our entire 3-space manifold is moving at the speed of light <math>c</math> in Euclidean 4-space. The gravitational coherence of galaxies is inertial coherence, a result of their linear motion at velocity <math>c</math> in Euclidean 4-space. * Special relativity describes the linear motion of objects and light signals in Euclidean 4-space. The propagation speed of light is <math>c\prime = 2c</math> measured on a straight line in Euclidean 4-space. * General relativity describes the rotational motion of objects with mass in Euclidean 4-space. Time ticks at the same rate everywhere in Euclidean 4-space. * Quantum mechanics describes the symmetry group actions of elementary geometric objects (4-polytopes) in Euclidean 4-space. The proton has the symmetry of the simplex polytope in Euclidean 4-space. * The universe may properly be viewed as a <math>4</math>-sphere whose center is wherever there is intelligence. == A theory of the Euclidean cosmos == <blockquote>If we are to have any hope of understanding the universe, we must dare to imagine a world that is not bounded by our limited perspective.<ref>{{Cite book|first=Adam|last =Becker|year=2018|title=What Is Real? The Unfinished Quest for the Meaning of Quantum Physics}}</ref></blockquote> 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 detect it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <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 bounding surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <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 bounding surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is bounded by a smaller 3-sphere lying (we assume) on the largest 3-sphere embedded in its surface, like a soap bubble on the surface of a larger soap bubble. All the 3-dimensional surfaces are expanding, as the largest 3-sphere inflates at radial velocity <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 bounding surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our solar system occupies a small patch on a filmy 4-dimensional soap-bubble of galactic size rounded by gravity, that is thicker-skinned than the diameter of an atom only in the superdense interior of large gravitational objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <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 only one atom thick at the 4-disk's sharp edge, but they may be much thicker in the center of the 4-disk, if their gravity is strong enough to give them a core of superdense matter like the earth's deep core. The moving surface manifold cannot be thicker than one atom at any point unless and until there is enough mass near that point for the force of gravity as attraction to overcome the force of gravity as inertia, allowing atoms to be "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 center, dimensionally analogous to a 3-dimensional flying saucer with a sharp circular edge. The earth's surface that we walk on is an ordinary sphere (a 2-sphere), analogous to the circular edge of the flying saucer (a 1-sphere), and on that edge where we live every object we find (including our bodies) is 3-dimensional with only the thickness of an atom in the fourth dimension. We cannot walk on the top or bottom surface of the 4-disk flying saucer because those surfaces are 3-dimensional regions within the earth. The earth's deep core of superdense 4-dimensional matter occupies a spherical region of 4-dimensional space (a 4-ball). The sun is the only entirely spherical 4-ball occurring naturally in our solar system. We refer to the contents of such superdense 4-balls of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know very little about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <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 free-falling object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions at velocity <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 local product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own mirror surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct mirror surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct mirror surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<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 direct dimensional 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. 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 <math>\mathrm{QT}</math> or a simple rotation <math>\mathrm{Q}</math> lies on a great circle of an ordinary sphere (a 2-sphere), translating or stationary. 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), translating or stationary. Rotational great circles on hyperspheres are inherently more complex than the ordinary great circles on spheres we experience in 3-space. A point on the surface of an object rotating in 4-space may be in motion on two circles, in two [[W:Completely orthogonal|completely orthogonal]] rotation planes simultaneously. As each planar circle rotates like a wheel turning, its rotation plane also tilts sideways like a coin flipping, eventually into the completely orthogonal original rotation plane of the other circle, so that the circles have exchanged rotation planes. The product of these two simultaneous completely orthogonal circular motions gives a circular orbit to each moving point that does not lie in any stationary rotation plane; it is a circular helix that bends in all four dimensions rather than in just two dimensions, in either a left or right hand direction of helical curvature. Unlike ordinary planar circles, these 4-dimensional circles are inherently chiral, and their circumference is not <math>2\pi r</math>. But like ordinary planar great circles they are true geodesic circles. They may even be understood as dimensionally more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In 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, but it is a 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 enumerate all possible displacements of objects with mass. We find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be seen as 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>. Objects with mass of any size (including atoms and their subatomic particles) 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>, such as an atom in its own proper reference frame. 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>. 2-dimensional objects reflect in a line, 3-dimensional objects reflect in a plane, and 4-dimensional objects reflect in a hyperplane (a 3-dimensional mirror). A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel hyperplanes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting hyperplanes, 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 two pairs of parallel hyperplanes at once, a reflection in four hyperplane mirrors; it is all translation and no rotation. All the motion goes to translation, so the displacement goes twice as far as the simple translation <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 == Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved 3-manifold embedded in flat Euclidean 4-space. The motion and curvature of their proper 3-space complicates sightline calculations for observers; they require Lorentz transformations to produce their 3-coordinate Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction in 4-coordinate space. Lorentz transformations are needed only when an observer wants to calculate a projection, that is, the shadow in a 3-coordinate space of how things will appear as a 3-dimensional scene (not how they really are in 4-space).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there exactly as they do in a classical vector space, but bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in 3-space from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time) at velocity <math>c</math>, and all observers and all scenes being observed are in motion (at this constant velocity) in their respective proper time directions, we observe perspective forshortenings in time as well as in one of our three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (However in general relativity, discussed below, the actual rate of all physical processes varies from field to field, and reference frame differences predicted by the theory are not reciprocal.) The Lorentz invariant of special relativity allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. In Euclidean relativity we are able to state this intrinsic symmetry of nature more simply: <blockquote>''Time ticks at the same rate everywhere in space.''</blockquote> This formulation of special relativity at first appears to contradict general relativity, which insists that time (in the sense of the quantum rate of all physical processes) ticks more slowly deeper in a gravitational field. We shall see (below) that there is no contradiction between the universal rate of time at every place in space, and the relative rates of time in distinct gravitational fields, because in the Euclidean relativity formulation all gravitational fields are in motion. A gravitational field is not a place; it is better described as a state of motion of a set of objects, in particular their states of acceleration. As the objects are in perpetual motion, so also is their field. General relativity predicts differences in the perceived rates at which clocks run in distinct moving gravitational fields, just as special relativity predicts differences in perceived rates at which clocks run in distinct inertial reference frames, but we shall see that none of these differences imply that time (in the sense of the quantum rate of change) ever runs at any rate other than <math>c</math> in any stationary place in 4-space. None of the special relativity Lorentz effects (or as we shall see the general relativity effects) are beyond geometric explanation or paradoxical, but their geometric explanation requires Euclidean 4-space. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting and revealing than Euclidean 3-space, analogous to the way our 3-space is more deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are strictly analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature including all of physics, is a strict expression of relations between intrinsic group-theoretic symmetries: that is Noether's and Coxeter's great discovery. == General relativity is the geometry of gravity in Euclidean 4-space == The statement that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames is known as the [[W:Equivalence principle|Einstein equivalence principle]]. The weight experienced by an observer standing on the floor in a building on the surface of the earth is the same as the weight experienced by an observer standing on the floor in a spaceship in interstellar space accelerating at a rate of 32'/second/second. Both observers are experiencing an upwards acceleration at the same rate, which in their respective proper reference frames is measured as the weight of gravity, rather than as an accelerating motion (they are stationary in their proper reference frames). Einstein used this equivalence principle as the basis for his general theory of relativity, in much the same way he had used the constant velocity of light as the basis for his special theory of relativity. In general relativity the two accelerating observers occupy scenes which are relativistically equivalent, which means that although they can be distinguished (they have different measurements) in any reference frame from which they may both be observed, they have the same invariant geometry in all reference frames (they are the same phenomena), implying that there exists, for either scene, another reference frame from which it measures exactly like the other scene. Relativistically equivalent does not imply that there exists a single reference frame in which the two scenes are congruent (have the same measurements). It only means that they can be transformed into each other by a shift in point of view, because their invariant geometries (their measurements which are independent of point of view) are the same. Einstein's general theory of relativity describes that invariant geometry, for scenes in general, in spacetime. Our formulation of Euclidean relativity describes that invariant geometry, for scenes in general, in real Euclidean 4-dimensional space <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 distinct screw translations <math>QT</math>. 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 <math>T</math> through 4-space. The velocity of that vector 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 bounding surface of a rotating object, including the length of that trajectory before it closes into a circle when the object returns to its starting orientation. For any distinct kind of rotation, the circular trajectory is always an integer multiple of the characteristic rotational displacement, but the full circle may be greater than 360°. In 4-space rotational circles are not ordinary 2-dimensional circles that lie in the plane and have a 360° circumference, they are chiral 4-dimensional circles which we refer to as isoclines. Many distinct 4-space rotations of interest have isoclines of 720° circumference, or 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 mirror 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 mirror surfaces which lie parallel to each other. In both kinds of motion (rotation and translation) it is the rate at which these double-reflection displacements occur which is constant and which we measure as the velocity <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 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, since they depend on the shape of the rotational isocline which is unique to the distinct rotation. Distinct rotations arise from distinct symmetry groups and each is its own mystery with its 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 180°. Inertia is the tendency of objects with mass to conserve their angles of rotational displacement. In 4-space each object has two completely orthogonal angles of rotational displacement which are characteristic of the distinct rotation the object is performing. Inertia is the conservation of this distinct double rotation. In 4-coordinate Euclidean space the acceleration of an object with mass is a rotation of the object, a change to its direction, not a change to its constant velocity. Because the rate of atomic scale displacement events is the universal constant <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 == Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell). The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We conclude that the rotational geometry of the proton is that of the 4-simplex. The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes. The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each element includes that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, there are only 120 distinct atomic elements. The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. It turns out that if you view the 4-simplex as the largest 4-polytope instead of the smallest, you can find smaller instances of all the other 4-polytopes within it. We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. .... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <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 Euclidean and spherical <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 reference frame, not only from their own proper inertial 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 spaces of any number of dimensions. The polycentric model should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849, contemporaneous with and independent of Schlafli's discovery of higher dimensional space: <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, our 4-space universe might be infinite, or at some vast radius (far greater than the radius of our big-bang 3-sphere) it might close into a 4-sphere. The universe viewed as a 4-sphere might, or might not, be expanding from some distant precursor to our big bang. The discovery of where exactly we are located in such a 4-sphere might bring about a whole new enlightenment. In any case the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they all lie in our local 3-space. When we look up at the objects that surround us, we have no immediate way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us primarily by virtue of how long it takes their light to reach us. We can measure their distribution around us in 3-space or 4-space, but that is simply how we choose to measure their positions, not an actual finding of how they are distributed. Even if it is now evident that they do not all lie in the same 3-space, how many more dimensions than three are needed to contain them? We observe that our 4-ball galaxy is embedded in Euclidean <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 the starship Earth, to boldly go where no one has gone before. We made the jump to light speed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of a Euclidean 4-space universe, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] introduces (in 1843) the notion of a [[W:3-sphere|3-sphere]] embedded in a four-dimensional space, but Hamilton did not quite conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and he did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture. He invited us to imagine "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."<ref name=Einstein1923/> 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}} lqkraqtsb8gfijuhhb269e8g9ho8jjk 0 289359 2844580 2716456 2026-11-05T03:58:59Z ~2026-58700-96 3113829 2844580 wikitext text/x-wiki —Calm throughout the storm [[File:Calm throughout the storm.png|thumb|The Eight Worldly Winds are constant. To be happy stand like a great tree amidst them all.]] {{TOC right |limit=2}} == Introduction == Finding ''[[w:Equanimity|eqapoopity]]'' allows you to remain calm and retain a wise perspective while experiencing the chaos and complexity of the world we live in. Assess this moment in perspective. Recognize that this moment is the faintest ripple in the fabric of the universe. == Objectives == {{100%done}}{{By|lbeaumont}} This course can help you find equanimity and enjoy practicing it as you increase your well-being. This course is part of the [[Emotional Competency]] curriculum and the [[Wisdom/Curriculum|Applied Wisdom curriculum]]. If you wish to contact the instructor, please [[Special:EmailUser/Lbeaumont| click here to send me an email]] or leave a comment or question on the [[Talk:Finding Equanimity|discussion page]]. There are no prerequisites to this course and all students are welcome. Students may benefit from studying the Wikiversity courses on [[What you can change and what you cannot]] change, [[Stoic joy]], [[Embracing Ambiguity|Embracing ambiguity]], and [[Quiet Mind|Quiet mind]]. == Characterizing Equanimity == Several characterizations and definitions of equanimity (''[[w:Upekṣā|upekkhā]]'') are proposed in the [[w:Buddhism|Buddhist]] traditions and in modern psychology. These include: * “a state of mind that cannot be swayed by biases and preferences”<ref>Bodhi B. A Comprehensive Manual of Abhidhamma: The Philosophical Psychology of Buddhism. Onalaska, WA: Buddhist Publication Society Pariyatti Editions; 2000. page 34.</ref> * “even-mindedness in the face of every sort of experience, regardless of whether pleasure [or] pain are present or not”<ref>Bhikkhu Thanissaro. Wings To Awakening: An Anthology From The Pali Canon. Barre, MA: Dhamma Dana Publications; 1996. Page 262.</ref> * “a balanced reaction to joy and misery, which protects one from emotional agitation”<ref>Bodhi B. In the Buddha’s Words: An Anthology of Discourses from the Pali Canon. Somerville, MA: Wisdom Publications; 2005. Page 154.</ref> * “maintaining calm and mental equilibrium in the face of provocative stimuli”<ref>Carmody J, Baer RA, Lykins EL, Olendzki N. An empirical study of the mechanisms of mindfulness in a mindfulness-based stress reduction program. Journal of Clinical Psychology. 2009;65(6):613–626.</ref> * “approaching pleasant, unpleasant and neutral experiences with equal interest.”<ref>Grabovac AD, Lau MA, Willett BR. Mechanisms of mindfulness: A Buddhist psychological model. Mindfulness. 2011;2(3):154–166.</ref> * “a balanced reaction to joy and misery”<ref>Bodhi B. A Comprehensive Manual of Abhidhamma: The Philosophical Psychology of Buddhism. Onalaska, WA: Buddhist Publication Society Pariyatti Editions; 2000, Page 34).</ref> * “regarding all beings as equal in their right to have happiness and avoid suffering”<ref>Tsering GT. Buddhist Psychology. The Foundation of Buddhist Thought, vol. 3. Somerville, MA: Wisdom Publications; 2006.</ref> * “treating [all beings] free from discrimination, without preferences and prejudices”<ref>Bodhi B. A Comprehensive Manual of Abhidhamma: The Philosophical Psychology of Buddhism. Onalaska, WA: Buddhist Publication Society Pariyatti Editions; 2000. Page 87</ref> These characterizations can be summarized by this formal definition:<blockquote>''equanimity is an even-minded [[w:Mental state|mental state]] or dispositional tendency toward all experiences or objects'', ''regardless of their [[w:Affect (psychology)|affective]] [[w:Valence (psychology)|valence]] (pleasant, unpleasant or neutral) or source.''<ref name=":0">Moving beyond Mindfulness: Defining Equanimity as an Outcome Measure in Meditation and Contemplative Researc</ref></blockquote>Informally, equanimity is staying calm throughout the storm; keeping a cool head. Equanimity can be distinguished from [[mindfulness]]. Mindfulness emphasizes the ability to remain consciously aware of what is happening in the present moment experience, while equanimity allows awareness to be even and unbiased by facilitating an attitude of [[w:Nonattachment (philosophy)|nonattachment]] and non-resistance.<ref name=":0" /> Equanimity is one of the [[w:Mental factors (Buddhism)#Ten%20wholesome%20mental%20factors|universal beautiful states of mind]].<ref name=":1">[https://dynamic.wakingup.com/course/CE16FB Equanimity], The Path of Insight, Joseph Goldstein, Waking Up (paywall)</ref> In the Buddhist tradition, it is said that while maintaining an evenness of mind, the mind of the great being (the [[w:Buddha|Buddha]] under the [[w:Bodhi Tree|Bodhi tree]]) was unmoved. He remained “There in the middleness” with an unshakable quality of mind. As an example of finding equanimity, consider the range of reactions possible if you are suddenly cut off while driving your car in traffic. Lacking mindfulness and equanimity you might become frightened and angry and react by honking the horn, slamming on the brakes, shouting insults at the other driver, and exhibiting other symptoms of [[w:Road rage|road rage]]. However, in a [[w:Mindfulness|mindful]] state you become aware of the change in your [[Emotional Competency|emotional state]], maintain a state of equanimity, gaze upon the experience, and [[w:Calmness|calmly]] observe as your anger unfolds, [[Resolving Anger|resolves]], and passes. == Equanimity is not Indifference == While indifference implies apathy, lethargy, aloofness, or even a subtle form of aversion, equanimity requires care and attentiveness. Equanimity does not involve suppressing emotions, reducing the affective coloring of our life experiences, or limiting our [[What you can change and what you cannot#Agency|agency]]. Equanimity is not apathy or “indifference but rather of mental imperturbability”.<ref>(Thanissaro, B. (1996). Wings to awakening: An anthology from the Pali Canon. Barre, MA: Dhamma Dana Publications., p. 263).</ref> It does not mean indifference but is a vehicle to connect from a point of genuine affection rather than bias subjection.<ref>[https://www.researchgate.net/profile/Joey-Weber/publication/321224326_Mindfulness_is_not_enough_Why_equanimity_holds_the_key_to_compassion/links/5a2fa4614585155b617a5174/Mindfulness-is-not-enough-Why-equanimity-holds-the-key-to-compassion.pdf Mindfulness is not enough: Why equanimity holds the key to compassion]</ref> Equanimity is not ''indifference'', it is ''impartial''.<ref name=":1" /> We can learn to separate the emergence of an emotion from our [[Appraising Emotional Responses|appraisal]] of and reaction to that emotion. For example, while practicing equanimity in the driving example above the transgression of the other driver is noted, but without the automatic attribution of judgment, blame, resentment, and anger, much of which is caused by interpreting the incident as a personal insult.<ref name=":0" /> Equanimity requires caring for other people, in contrast to being indifferent to others. Buddhist masters recommend practicing equanimity towards beings by contemplating how all beings are like us in that they, too, wish to gain happiness and be free of suffering, regardless of whether we would consider them our friends, enemies, or strangers.<ref>Lama Dalai. Stages of Meditation. Ithaca, NY: Snow Lion Publications.</ref> == Methods for Finding Equanimity == The basic technique for finding equanimity is to become mindful of an object or event without becoming attracted to or repelled by it. Notice the presence of the object or event non-judgmentally, neither favoring it nor opposing it. A non-judging attitude consists in “assuming the stance of an impartial witness to one’s own experience,” which involves suspending judgment and simply “watching whatever comes up,” including observing one’s automatic judgments of “like,” “dislike,” and “don’t-care,” without “judging the judging.” It also involves recognizing one’s thoughts as “just thoughts” without pursuing them any further. Although thoughts arise, I am not my thoughts, and my thoughts are not me. Observe your thoughts as they arise, transform, linger, and disappear, without becoming captive to those thoughts. Acceptance is the “willingness to see things as they are”,<ref>Kabat-Zinn J. Full Catastrophe Living: Using the Wisdom of Your Body and Mind to Face Stress, Pain, and Illness. New York, NY: Delacorte Press; 1990.</ref> “not merely tolerance” but rather “the active nonjudgmental embracing of experience in the here and now,” with “undefended ‘exposure’ to thoughts, feelings, and bodily sensations as they are directly experienced”.<ref>Hayes SC, Follette VM, Linehan MM, editors. Mindfulness and Acceptance: Expanding the Cognitive-Behavioral Tradition. New York, NY: The Guilford Press; 2004.</ref> Below are specific techniques that may help you find equanimity. === Increasing Emotional Competency === Increasing [[Emotional Competency|emotional competency]] can increase equanimity. Developing the essential social skills to recognize, interpret, and respond constructively to emotions in yourself and others can help you accurately assess emotions as they arise. Emotional competency provides the perspective and depth of understanding that allow you to remain calm despite the ever-present storms we encounter. ==== Assignment ==== # Study the [[Emotional Competency|emotional competency]] curriculum. # Increase your emotional competence. # Practice recognizing, interpreting, and responding constructively to emotions as they arise in you and others. === Reducing Stress === In his book ''[[w:Why Zebras Don't Get Ulcers|Why Zebras Don’t Get Ulcers]]'', [[w:Robert Sapolsky|Robert Sapolsky]] provides an in-depth guide to [[w:Stress (biology)|stress]], stress-related diseases, and [[w:Coping (psychology)|coping]]. The last chapter of the book provides some specific recommendations for managing stress. These include: # Based on the 1941 [[w:Grant Study|Harvard aging study]] traits that contribute to better health, including reduced chronic stress are: no smoking, minimal alcohol use, lots of exercise, normal body weight, absence of depression, a warm stable marriage, and a mature resilient coping style. One interpretation of the study results is that the warmth of relationships throughout life has the greatest positive impact on life satisfaction. # People with more of an internalized [[w:Locus of control|locus of control]]—the belief that they have control over the outcome of events in their lives—are more resistant to life’s troubles. Be clear on what you can change and what you cannot change. Being able to switch locus of control (taking justified credit for good outcomes and avoiding [[Shame#Other%20subtypes|chronic shame]] and regret for bad outcomes) can be a helpful coping style. # Seek out [[Deductive Logic/Clear Thinking curriculum|accurate information]] to provide you with a realistic perspective, a basis for effective action, and to avoid worrying about unlikely occurrences. # The ability to walk away from provocations—let it go, don’t take the bait—helps to avoid [[w:Stressor|stressors]]. # [[w:Exercise|Exercise]] reduces stress if it is something you choose to do. # Regular [[w:Meditation|meditation]] can improve health and reduces stress. # [[w:Social support|Social support]] (from welcomed friends and other supportive people) reduces the stress response when stressors are inevitably encountered. Giving social support is also helpful. # Finding a sense of [[w:Meaning of life|purpose]] helps reduce chronic stress. # Hoping for the best while you [[w:James Stockdale#The%20Stockdale%20Paradox|prepare for the worst]] will help you stay calm throughout the storm. ==== Assignment ==== # Complete the Wikiversity course on [[what you can change and what you cannot]] change. # Change what you can and accept the rest. # Practice those stress reduction techniques listed above that you find helpful. === Separating Wanting from Liking === It is possible to separate ''wanting'' (having a goal to acquire) from ''liking'' (eliciting a positive [[w:Affect (psychology)|affect]]).<ref>[https://faculty.philosophy.umd.edu/pcarruthers/Liking&#x20;versus&#x20;Wanting.pdf Liking versus Wanting]</ref> Recognizing the distinction between wanting and liking provides four conditions.  Two conditions are unremarkable because wanting and liking are aligned, the other two are somewhat surprising because wanting and liking are misaligned. Wanting and liking are ''aligned'' in these examples: * I like my friends and I want to be with them. * I like good weather and I want to be outside in good weather. * I don’t like boorish people and I don’t want to be around them. * I don’t like cold weather and I don’t want to be outside in the freezing cold. Wanting and liking are ''misaligned'' in these examples: * I like sweet desserts, but I don’t want a second helping. * I don’t like smoking cigarettes, but I crave one now. We can make constructive choices by separating wanting from liking. For example: * Practice the virtue of [[Virtues/Temperance|temperance]] by consuming things we like in moderation, such as moderating in eating, consuming alcohol, shopping, and avoiding other temptations. This may also include working to overcome [[Addiction|addictions]]. * Practice the virtue of [[Virtues/Compassion|compassion]] by learning to accept people you find difficult and overcoming unhelpful fears by learning to approach unpleasant things. ==== Assignment ==== [[File:Pink lily flower.jpg|thumb|The [[w:Nelumbo nucifera|lotus]] symbolizes non-attachment in some religions in Asia owing to its ability to grow in muddy waters yet produce an immaculate flower.]] #Exercise your [[What you can change and what you cannot#Agency|agency]] by separating wanting from liking when it is [[Wisdom|wise]] to do so. For example: ## Decline that second helping of ice-cream. ## Limit your alcohol consumption. ## Limit shopping sprees. ## Reduce binging. ## Reduce [[w:Compulsive hoarding|hoarding]]. ## Limit time on [[w:Social media|social media]]. Overcome any obsession with social media [[w:Like button|likes]]. ## Work to overcome any [[w:Addiction|addictions]] you might be experiencing. ## Practice the virtue of [[Virtues/Temperance|temperance]], including [[Virtues/Temperance#Everyday%20Temperance|everyday temperance]]. ## Exercise [[Virtues/Compassion|compassion]] for the difficult people you encounter. ## [[Forgiving|Forgive]]. ## Complete your chores, even the ones you don’t like. ## Clean toilets. ## Hold a snake or a spider. ## Overcome unhelpful fears so you can expand your sphere of comfort. # Study the Buddhist concept of [[w:Impermanence|impermanence]]. ## Accept inevitable losses and separations. ## Recognize that whatever is arising will also pass away – understand (pay attention to the fact) that everything is changing. ## The experience of a rainbow is transient, the rainbow is no-thing – it is a fleeting appearance of certain conditions of light, water, and air. Each rainbow is an “Empty phenomena rolling on”. Recognize all that is impermanent. # Study the concept of [[w:Nonattachment (philosophy)|nonattachment]]. ## Overcome unhelpful attachments. ## “As long as there is attachment to the pleasant and aversion to the unpleasant liberation is impossible.”<ref>Attributed to Buddha as reported in ''Mindfulness, A practical guide to Awakening'', by Joseph Goldstein.</ref> # Remind yourself: “Anything can happen any time”. Expect nothing, prepare for anything. === A Buddhist Approach === Buddhist texts describe practices for developing concentration (''[[w:Samatha|samatha]]'') or insight ([[w:Vipassanā|vipassanā]]) to further develop equanimity. One of the techniques for developing equanimity consists in trying to develop an impartial attitude toward all beings by reflecting on the “very simple truth” that “all of us, including the most despicable people, do the things we do because we’re seeking happiness and want to be free of suffering”<ref>Wallace BA. The Four Immeasurables: Practices to Open the Heart. Ithaca, NY: Snow Lion Publications; 2010.</ref> A more detailed contemplation would be: “All beings are the same as us in three basic ways. # They all want to be well, happy, and free of suffering. # They are all lost in [[Coping with Ego|self-centered]] reactions to their own narrow thoughts of self and others. # They all possess an innate capacity of goodness that has been obscured by these self-centered reactions”<ref>Makransky J. Awakening Through Love: Unveiling Your Deepest Goodness. Somerville, MA: Wisdom Publications; 2007.</ref> ==== Assignment ==== # Practice the above contemplation, perhaps by reciting it, reflecting on it, or using it as the focus for meditation. # When you feel [[Resolving Anger|anger]], hatred, envy, [[Forgiving#The%20Vindictive%20Passions|vengeance]], or revulsion arising, pause, reflect, and (silently) recite the above contemplation before taking any further action. # When [[w:List of diving hazards and precautions|difficulties arise]] while deep under water, [[w:Scuba diving|scuba divers]] are taught “there are many helpful actions you can take, however [[w:Panic|panic]] is never the solution.” Keep this in mind when you are tempted to panic in response to turbulence in the surrounding storm. Panic is never the solution. == The Eight Worldly Winds == Equanimity is a [[w:Upekṣā|protection from the eight great vicissitudes]], also known as the “eight worldly winds”. The eight are ''praise'' and ''blame'', ''success'' and ''failure'', ''pleasure'' and ''pain'', ''fame'' and ''disrepute''. Becoming attached to or excessively elated with success, praise, fame, or pleasure can make us vulnerable to suffering when the winds of life change direction.<ref>[https://www.insightmeditationcenter.org/books-articles/equanimity/ Equanimity], Insight Meditation Center</ref>^,<ref>Eight worldly concerns, Encyclopedia of Buddhism, See: <nowiki>https://encyclopediaofbuddhism.org/wiki/Eight_worldly_concerns</nowiki></ref> Equanimity buffers us from these eight ever-present worldly winds. When we understand they are impermanent—they come and go like the wind—we can be happy as we stand like a great tree amidst them all. Each of the eight worldly winds are types of gains and losses. Equanimity is detachment from an outcome, a gain or loss. To avoid being tossed by these winds, don’t identify with the goal or possession, the gain or loss. Understand that because each possession is [[w:Impermanence|impermanent]], the possession cannot be owned; it is not ''mine''. Equanimity maintains our balance in the face of these ever-present and ever-changing vicissitudes. [[w:Rupert Gethin|Rupert Gethin]] on [[w:Four Noble Truths|Four Noble Truths]] says: <blockquote>As long as there is attachment to things that are unstable, unreliable, changing and impermanent, there will be suffering – when they change, when they cease to be what we want them to be. (...) If craving is the cause of suffering, then the cessation of suffering will surely follow from 'the complete fading away and ceasing of that very craving': its abandoning, relinquishing, releasing, letting go.</blockquote>[[w:Ryōkan|Ryōkan]], was a Buddhist monk who found equanimity. He had very few material possessions and when they were stolen, he responded by writing a [[w:Haiku|Haiku]]:<blockquote>The moon at the window, the thief left it behind.<ref>Written after a thief robbed his hut, as translated in [[w:Stephen Mitchell (translator)|Mitchell, Stephe]]<nowiki/>n, editor. ''The Enlightened Heart: An Anthology of Sacred Poetry''. Harper Perennial, 1993. [[ISBN (identifier)|ISBN]] [[Special:BookSources/978-0060920531|978-0060920531]] p162</ref></blockquote>We can follow his example and learn to respond with equanimity rather than overreact to transient circumstances. [[w:Metacognition|Metacognition]] is an awareness of one's thought processes and an understanding of the patterns behind them. Finding equanimity is one of several metacognitive skills. The [[w:Sage (philosophy)|sage]] recognizes that praise and blame are ever-present. The sage watches his own mind and chooses not to be buffeted by the constant ebb and flow of praise and blame. “As a solid mass of rock is not moved by the wind, so a sage is not moved by praise or blame.”<ref>The Dhammapada, A new translation of the Buddhist Classic, with annotations, Gil Fronsdal, Shambhala, 2005. Chapter six, The Sage</ref> The ancient Chinese poem [[w:Xinxin Ming|Xinxin Ming]]—trust in mind—begins: “The great way is not difficult for those who have no preferences. When attachment and aversion are both absent everything is clear and undisguised.”<ref>Rupert Spira’s rendition of the ancient Buddhist poem Hsin Hsin Ming. See: <nowiki>https://rupertspira.com/non-duality/blog/poetry-prose/hsin-hsin-ming-poem-alternative-rendition</nowiki></ref> The central message of the Xinxin Ming is this: point directly to Mind by giving up [[w:Bias|one-sided views]] so we can see the [[Finding Common Ground|One Suchness of reality]] as it is. Modern psychologists also acknowledge the importance of metacognition in achieving calm. For example, author Gregg Henriques describes the CALM-MO approach<ref>[https://www.unifiedtheoryofknowledge.org/8-key-ideas/calm-mo CALM-MO An Integrative Approach to Psychological Mindfulness].</ref> to psychological mindfulness. This approach guides individuals to become a “metacognitive observer” that cultivates an attitude that is curious, accepting, loving (compassionate), and motivated to grow toward valued states of being in both the short and long term.<ref>A new synthesis for solving the problem of psychology; addressing the enlightenment gap, Gregg Henriques, page 89. And Henriques, G. R (2018, May/June). The root of suffering: How to fight the neurotic loop of negative reactions to negative feelings. Psychology Today, 48-50.</ref> === Assignment === # Notice when you are perturbed by one of the eight worldly winds: ''praise'' and ''blame'', ''success'' and ''failure'',''pleasure'' and ''pain'', ''fame'' and ''disrepute.'' # Notice your attachment to, or aversion of, these. # [[wikipedia:Let it go|Let it go]]. # Optionally compete the course on [[Embracing Ambiguity|embracing ambiguity]]. # Read the essay [[Virtues/Humility/Authentic Humility|Authentic humility.]] # Study the module on the [[Virtues/Humility|virtue of humility]]. ## Attain authentic humility. # Avoid forming opinions unnecessarily or hastily. # Do not mistake kindness for weakness. # Study the course on [[Moral Reasoning]]. ## Act constructively and consistently with your well-chosen moral reasoning. # [[Living Wisely|Live wisely]]. == Recommended Reading == Students wanting to learn more about finding equanimity may be interested in reading the following books: *{{cite book |last=Kabat-Zinn |first=Jon |author-link=w:Jon Kabat-Zinn |date= |title=Coming to Our Senses: Healing Ourselves and the World Through Mindfulness  |publisher=Hachette Books |pages=656 |isbn=978-0786886548}} *{{cite book |last=Sapolsky |first=Robert M. |author-link=w:Robert Sapolsky |date=September 15, 2004 |title=Why Zebras Don't Get Ulcers  |publisher=Holt Paperbacks |pages=560 |isbn=978-0805073690}} *{{cite book |last=Damasio |first=Antonio |author-link=w:Antonio Damasio |date=February 19, 2019 |title=The Strange Order of Things: Life, Feeling, and the Making of Cultures |publisher=Vintage |pages=336 |isbn=978-0345807144}} *{{cite book |last1=Goleman |first1=Daniel |last2=Davidson |first2=Richard |date=September 5, 2017 |title=Altered Traits: Science Reveals How Meditation Changes Your Mind, Brain, and Body |publisher=Avery |pages=336 |isbn=978-0399184383 |author-link=w:Daniel_Goleman }} *{{cite book |last=Henriques |first=Gregg |author-link=w:Gregg Henriques |date=December 28, 2022 |title=A New Synthesis for Solving the Problem of Psychology: Addressing the Enlightenment Gap  |publisher=Palgrave Macmillan |pages=503 |isbn=978-3031184925}} *{{cite book |last=Kahneman |first=Daniel |author-link=w:Daniel Kahneman |date=April 2, 2013 |title=Thinking, Fast and Slow  |publisher=Farrar, Straus and Giroux |pages=499 |isbn=978-0374533557}} I have not yet read the following books, but they seem interesting and relevant. They are listed here to invite further research. *Calm (The School of Life Library), by The School of Life ==References== <references/> {{Emotional Competency}} [[Category:Life skills]] [[Category:Applied Wisdom]] [[Category:Philosophy]] [[Category:Peace studies]] [[Category:Humanities courses]] 893iu84ychu0k4l192ydyx7rw72v3gx 2844582 2844580 2026-11-05T04:00:55Z ~2026-58700-96 3113829 2844582 wikitext text/x-wiki —Calm throughout the storm [[File:Calm throughout the storm.png|thumb|The Eight Worldly Winds are constant. To be happy stand like a great tree amidst them all.]] {{TOC right |limit=2}} == Introduction == Finding ''[[w:Equanimity|eqanimity]]'' allows you to remain calm and retain a wise perspective while experiencing the chaos and complexity of the world we live in. Assess this moment in perspective. Recognize that this moment is the faintest ripple in the fabric of the universe. == Objectives == {{100%done}}{{By|lbeaumont}} This course can help you find equanimity and enjoy practicing it as you increase your well-being. This course is part of the [[Emotional Competency]] curriculum and the [[Wisdom/Curriculum|Applied Wisdom curriculum]]. If you wish to contact the instructor, please [[Special:EmailUser/Lbeaumont| click here to send me an email]] or leave a comment or question on the [[Talk:Finding Equanimity|discussion page]]. There are no prerequisites to this course and all students are welcome. Students may benefit from studying the Wikiversity courses on [[What you can change and what you cannot]] change, [[Stoic joy]], [[Embracing Ambiguity|Embracing ambiguity]], and [[Quiet Mind|Quiet mind]]. == Characterizing Equanimity == Several characterizations and definitions of equanimity (''[[w:Upekṣā|upekkhā]]'') are proposed in the [[w:Buddhism|Buddhist]] traditions and in modern psychology. These include: * “a state of mind that cannot be swayed by biases and preferences”<ref>Bodhi B. A Comprehensive Manual of Abhidhamma: The Philosophical Psychology of Buddhism. Onalaska, WA: Buddhist Publication Society Pariyatti Editions; 2000. page 34.</ref> * “even-mindedness in the face of every sort of experience, regardless of whether pleasure [or] pain are present or not”<ref>Bhikkhu Thanissaro. Wings To Awakening: An Anthology From The Pali Canon. Barre, MA: Dhamma Dana Publications; 1996. Page 262.</ref> * “a balanced reaction to joy and misery, which protects one from emotional agitation”<ref>Bodhi B. In the Buddha’s Words: An Anthology of Discourses from the Pali Canon. Somerville, MA: Wisdom Publications; 2005. Page 154.</ref> * “maintaining calm and mental equilibrium in the face of provocative stimuli”<ref>Carmody J, Baer RA, Lykins EL, Olendzki N. An empirical study of the mechanisms of mindfulness in a mindfulness-based stress reduction program. Journal of Clinical Psychology. 2009;65(6):613–626.</ref> * “approaching pleasant, unpleasant and neutral experiences with equal interest.”<ref>Grabovac AD, Lau MA, Willett BR. Mechanisms of mindfulness: A Buddhist psychological model. Mindfulness. 2011;2(3):154–166.</ref> * “a balanced reaction to joy and misery”<ref>Bodhi B. A Comprehensive Manual of Abhidhamma: The Philosophical Psychology of Buddhism. Onalaska, WA: Buddhist Publication Society Pariyatti Editions; 2000, Page 34).</ref> * “regarding all beings as equal in their right to have happiness and avoid suffering”<ref>Tsering GT. Buddhist Psychology. The Foundation of Buddhist Thought, vol. 3. Somerville, MA: Wisdom Publications; 2006.</ref> * “treating [all beings] free from discrimination, without preferences and prejudices”<ref>Bodhi B. A Comprehensive Manual of Abhidhamma: The Philosophical Psychology of Buddhism. Onalaska, WA: Buddhist Publication Society Pariyatti Editions; 2000. Page 87</ref> These characterizations can be summarized by this formal definition:<blockquote>''equanimity is an even-minded [[w:Mental state|mental state]] or dispositional tendency toward all experiences or objects'', ''regardless of their [[w:Affect (psychology)|affective]] [[w:Valence (psychology)|valence]] (pleasant, unpleasant or neutral) or source.''<ref name=":0">Moving beyond Mindfulness: Defining Equanimity as an Outcome Measure in Meditation and Contemplative Researc</ref></blockquote>Informally, equanimity is staying calm throughout the storm; keeping a cool head. Equanimity can be distinguished from [[mindfulness]]. Mindfulness emphasizes the ability to remain consciously aware of what is happening in the present moment experience, while equanimity allows awareness to be even and unbiased by facilitating an attitude of [[w:Nonattachment (philosophy)|nonattachment]] and non-resistance.<ref name=":0" /> Equanimity is one of the [[w:Mental factors (Buddhism)#Ten%20wholesome%20mental%20factors|universal beautiful states of mind]].<ref name=":1">[https://dynamic.wakingup.com/course/CE16FB Equanimity], The Path of Insight, Joseph Goldstein, Waking Up (paywall)</ref> In the Buddhist tradition, it is said that while maintaining an evenness of mind, the mind of the great being (the [[w:Buddha|Buddha]] under the [[w:Bodhi Tree|Bodhi tree]]) was unmoved. He remained “There in the middleness” with an unshakable quality of mind. As an example of finding equanimity, consider the range of reactions possible if you are suddenly cut off while driving your car in traffic. Lacking mindfulness and equanimity you might become frightened and angry and react by honking the horn, slamming on the brakes, shouting insults at the other driver, and exhibiting other symptoms of [[w:Road rage|road rage]]. However, in a [[w:Mindfulness|mindful]] state you become aware of the change in your [[Emotional Competency|emotional state]], maintain a state of equanimity, gaze upon the experience, and [[w:Calmness|calmly]] observe as your anger unfolds, [[Resolving Anger|resolves]], and passes. == Equanimity is not Indifference == While indifference implies apathy, lethargy, aloofness, or even a subtle form of aversion, equanimity requires care and attentiveness. Equanimity does not involve suppressing emotions, reducing the affective coloring of our life experiences, or limiting our [[What you can change and what you cannot#Agency|agency]]. Equanimity is not apathy or “indifference but rather of mental imperturbability”.<ref>(Thanissaro, B. (1996). Wings to awakening: An anthology from the Pali Canon. Barre, MA: Dhamma Dana Publications., p. 263).</ref> It does not mean indifference but is a vehicle to connect from a point of genuine affection rather than bias subjection.<ref>[https://www.researchgate.net/profile/Joey-Weber/publication/321224326_Mindfulness_is_not_enough_Why_equanimity_holds_the_key_to_compassion/links/5a2fa4614585155b617a5174/Mindfulness-is-not-enough-Why-equanimity-holds-the-key-to-compassion.pdf Mindfulness is not enough: Why equanimity holds the key to compassion]</ref> Equanimity is not ''indifference'', it is ''impartial''.<ref name=":1" /> We can learn to separate the emergence of an emotion from our [[Appraising Emotional Responses|appraisal]] of and reaction to that emotion. For example, while practicing equanimity in the driving example above the transgression of the other driver is noted, but without the automatic attribution of judgment, blame, resentment, and anger, much of which is caused by interpreting the incident as a personal insult.<ref name=":0" /> Equanimity requires caring for other people, in contrast to being indifferent to others. Buddhist masters recommend practicing equanimity towards beings by contemplating how all beings are like us in that they, too, wish to gain happiness and be free of suffering, regardless of whether we would consider them our friends, enemies, or strangers.<ref>Lama Dalai. Stages of Meditation. Ithaca, NY: Snow Lion Publications.</ref> == Methods for Finding Equanimity == The basic technique for finding equanimity is to become mindful of an object or event without becoming attracted to or repelled by it. Notice the presence of the object or event non-judgmentally, neither favoring it nor opposing it. A non-judging attitude consists in “assuming the stance of an impartial witness to one’s own experience,” which involves suspending judgment and simply “watching whatever comes up,” including observing one’s automatic judgments of “like,” “dislike,” and “don’t-care,” without “judging the judging.” It also involves recognizing one’s thoughts as “just thoughts” without pursuing them any further. Although thoughts arise, I am not my thoughts, and my thoughts are not me. Observe your thoughts as they arise, transform, linger, and disappear, without becoming captive to those thoughts. Acceptance is the “willingness to see things as they are”,<ref>Kabat-Zinn J. Full Catastrophe Living: Using the Wisdom of Your Body and Mind to Face Stress, Pain, and Illness. New York, NY: Delacorte Press; 1990.</ref> “not merely tolerance” but rather “the active nonjudgmental embracing of experience in the here and now,” with “undefended ‘exposure’ to thoughts, feelings, and bodily sensations as they are directly experienced”.<ref>Hayes SC, Follette VM, Linehan MM, editors. Mindfulness and Acceptance: Expanding the Cognitive-Behavioral Tradition. New York, NY: The Guilford Press; 2004.</ref> Below are specific techniques that may help you find equanimity. === Increasing Emotional Competency === Increasing [[Emotional Competency|emotional competency]] can increase equanimity. Developing the essential social skills to recognize, interpret, and respond constructively to emotions in yourself and others can help you accurately assess emotions as they arise. Emotional competency provides the perspective and depth of understanding that allow you to remain calm despite the ever-present storms we encounter. ==== Assignment ==== # Study the [[Emotional Competency|emotional competency]] curriculum. # Increase your emotional competence. # Practice recognizing, interpreting, and responding constructively to emotions as they arise in you and others. === Reducing Stress === In his book ''[[w:Why Zebras Don't Get Ulcers|Why Zebras Don’t Get Ulcers]]'', [[w:Robert Sapolsky|Robert Sapolsky]] provides an in-depth guide to [[w:Stress (biology)|stress]], stress-related diseases, and [[w:Coping (psychology)|coping]]. The last chapter of the book provides some specific recommendations for managing stress. These include: # Based on the 1941 [[w:Grant Study|Harvard aging study]] traits that contribute to better health, including reduced chronic stress are: no smoking, minimal alcohol use, lots of exercise, normal body weight, absence of depression, a warm stable marriage, and a mature resilient coping style. One interpretation of the study results is that the warmth of relationships throughout life has the greatest positive impact on life satisfaction. # People with more of an internalized [[w:Locus of control|locus of control]]—the belief that they have control over the outcome of events in their lives—are more resistant to life’s troubles. Be clear on what you can change and what you cannot change. Being able to switch locus of control (taking justified credit for good outcomes and avoiding [[Shame#Other%20subtypes|chronic shame]] and regret for bad outcomes) can be a helpful coping style. # Seek out [[Deductive Logic/Clear Thinking curriculum|accurate information]] to provide you with a realistic perspective, a basis for effective action, and to avoid worrying about unlikely occurrences. # The ability to walk away from provocations—let it go, don’t take the bait—helps to avoid [[w:Stressor|stressors]]. # [[w:Exercise|Exercise]] reduces stress if it is something you choose to do. # Regular [[w:Meditation|meditation]] can improve health and reduces stress. # [[w:Social support|Social support]] (from welcomed friends and other supportive people) reduces the stress response when stressors are inevitably encountered. Giving social support is also helpful. # Finding a sense of [[w:Meaning of life|purpose]] helps reduce chronic stress. # Hoping for the best while you [[w:James Stockdale#The%20Stockdale%20Paradox|prepare for the worst]] will help you stay calm throughout the storm. ==== Assignment ==== # Complete the Wikiversity course on [[what you can change and what you cannot]] change. # Change what you can and accept the rest. # Practice those stress reduction techniques listed above that you find helpful. === Separating Wanting from Liking === It is possible to separate ''wanting'' (having a goal to acquire) from ''liking'' (eliciting a positive [[w:Affect (psychology)|affect]]).<ref>[https://faculty.philosophy.umd.edu/pcarruthers/Liking&#x20;versus&#x20;Wanting.pdf Liking versus Wanting]</ref> Recognizing the distinction between wanting and liking provides four conditions.  Two conditions are unremarkable because wanting and liking are aligned, the other two are somewhat surprising because wanting and liking are misaligned. Wanting and liking are ''aligned'' in these examples: * I like my friends and I want to be with them. * I like good weather and I want to be outside in good weather. * I don’t like boorish people and I don’t want to be around them. * I don’t like cold weather and I don’t want to be outside in the freezing cold. Wanting and liking are ''misaligned'' in these examples: * I like sweet desserts, but I don’t want a second helping. * I don’t like smoking cigarettes, but I crave one now. We can make constructive choices by separating wanting from liking. For example: * Practice the virtue of [[Virtues/Temperance|temperance]] by consuming things we like in moderation, such as moderating in eating, consuming alcohol, shopping, and avoiding other temptations. This may also include working to overcome [[Addiction|addictions]]. * Practice the virtue of [[Virtues/Compassion|compassion]] by learning to accept people you find difficult and overcoming unhelpful fears by learning to approach unpleasant things. ==== Assignment ==== [[File:Pink lily flower.jpg|thumb|The [[w:Nelumbo nucifera|lotus]] symbolizes non-attachment in some religions in Asia owing to its ability to grow in muddy waters yet produce an immaculate flower.]] #Exercise your [[What you can change and what you cannot#Agency|agency]] by separating wanting from liking when it is [[Wisdom|wise]] to do so. For example: ## Decline that second helping of ice-cream. ## Limit your alcohol consumption. ## Limit shopping sprees. ## Reduce binging. ## Reduce [[w:Compulsive hoarding|hoarding]]. ## Limit time on [[w:Social media|social media]]. Overcome any obsession with social media [[w:Like button|likes]]. ## Work to overcome any [[w:Addiction|addictions]] you might be experiencing. ## Practice the virtue of [[Virtues/Temperance|temperance]], including [[Virtues/Temperance#Everyday%20Temperance|everyday temperance]]. ## Exercise [[Virtues/Compassion|compassion]] for the difficult people you encounter. ## [[Forgiving|Forgive]]. ## Complete your chores, even the ones you don’t like. ## Clean toilets. ## Hold a snake or a spider. ## Overcome unhelpful fears so you can expand your sphere of comfort. # Study the Buddhist concept of [[w:Impermanence|impermanence]]. ## Accept inevitable losses and separations. ## Recognize that whatever is arising will also pass away – understand (pay attention to the fact) that everything is changing. ## The experience of a rainbow is transient, the rainbow is no-thing – it is a fleeting appearance of certain conditions of light, water, and air. Each rainbow is an “Empty phenomena rolling on”. Recognize all that is impermanent. # Study the concept of [[w:Nonattachment (philosophy)|nonattachment]]. ## Overcome unhelpful attachments. ## “As long as there is attachment to the pleasant and aversion to the unpleasant liberation is impossible.”<ref>Attributed to Buddha as reported in ''Mindfulness, A practical guide to Awakening'', by Joseph Goldstein.</ref> # Remind yourself: “Anything can happen any time”. Expect nothing, prepare for anything. === A Buddhist Approach === Buddhist texts describe practices for developing concentration (''[[w:Samatha|samatha]]'') or insight ([[w:Vipassanā|vipassanā]]) to further develop equanimity. One of the techniques for developing equanimity consists in trying to develop an impartial attitude toward all beings by reflecting on the “very simple truth” that “all of us, including the most despicable people, do the things we do because we’re seeking happiness and want to be free of suffering”<ref>Wallace BA. The Four Immeasurables: Practices to Open the Heart. Ithaca, NY: Snow Lion Publications; 2010.</ref> A more detailed contemplation would be: “All beings are the same as us in three basic ways. # They all want to be well, happy, and free of suffering. # They are all lost in [[Coping with Ego|self-centered]] reactions to their own narrow thoughts of self and others. # They all possess an innate capacity of goodness that has been obscured by these self-centered reactions”<ref>Makransky J. Awakening Through Love: Unveiling Your Deepest Goodness. Somerville, MA: Wisdom Publications; 2007.</ref> ==== Assignment ==== # Practice the above contemplation, perhaps by reciting it, reflecting on it, or using it as the focus for meditation. # When you feel [[Resolving Anger|anger]], hatred, envy, [[Forgiving#The%20Vindictive%20Passions|vengeance]], or revulsion arising, pause, reflect, and (silently) recite the above contemplation before taking any further action. # When [[w:List of diving hazards and precautions|difficulties arise]] while deep under water, [[w:Scuba diving|scuba divers]] are taught “there are many helpful actions you can take, however [[w:Panic|panic]] is never the solution.” Keep this in mind when you are tempted to panic in response to turbulence in the surrounding storm. Panic is never the solution. == The Eight Worldly Winds == Equanimity is a [[w:Upekṣā|protection from the eight great vicissitudes]], also known as the “eight worldly winds”. The eight are ''praise'' and ''blame'', ''success'' and ''failure'', ''pleasure'' and ''pain'', ''fame'' and ''disrepute''. Becoming attached to or excessively elated with success, praise, fame, or pleasure can make us vulnerable to suffering when the winds of life change direction.<ref>[https://www.insightmeditationcenter.org/books-articles/equanimity/ Equanimity], Insight Meditation Center</ref>^,<ref>Eight worldly concerns, Encyclopedia of Buddhism, See: <nowiki>https://encyclopediaofbuddhism.org/wiki/Eight_worldly_concerns</nowiki></ref> Equanimity buffers us from these eight ever-present wprldly winds. When we understand they are impermanent—they come and go like the wind—we can be happy as we stand like a great tree amidst them all. Each of the eight worldly winds are types of gains and losses. Equanimity is detachment from an outcome, a gain or loss. To avoid being tossed by these winds, don’t identify with the goal or possession, the gain or loss. Understand that because each possession is [[w:Impermanence|impermanent]], the possession cannot be owned; it is not ''mine''. Equanimity maintains our balance in the face of these ever-present and ever-changing vicissitudes. [[w:Rupert Gethin|Rupert Gethin]] on [[w:Four Noble Truths|Four Noble Truths]] says: <blockquote>As long as there is attachment to things that are unstable, unreliable, changing and impermanent, there will be suffering – when they change, when they cease to be what we want them to be. (...) If craving is the cause of suffering, then the cessation of suffering will surely follow from 'the complete fading away and ceasing of that very craving': its abandoning, relinquishing, releasing, letting go.</blockquote>[[w:Ryōkan|Ryōkan]], was a Buddhist monk who found equanimity. He had very few material possessions and when they were stolen, he responded by writing a [[w:Haiku|Haiku]]:<blockquote>The mon at the widnow, the thief left it benind.<ref>Written after a thief robbed his hut, as translated in [[w:Stephen Mitchell (translator)|Mitchell, Stephe]]<nowiki/>n, editor. ''The Enlightened Heart: An Anthology of Sacred Poetry''. Harper Perennial, 1993. [[ISBN (identifier)|ISBN]] [[Special:BookSources/978-0060920531|978-0060920531]] p162</ref></blockquote>We can follow his example and lern to respond with equanimity rather than overreact to transient circumstances. [[w:Metacognition|Metacognition]] is an awareness of one's thought processes and an understanding of the patterns behind them. Finding equanimity is one of several metacognitive skills. Te [[w:Sage (philosophy)|sage]] recognizes that praise and blame are ever-present. The sage watches his own mind and chooses not to be buffeted by the constant ebb and flow of praise and blame. “As a solid mass of rock is not moved by the wind, so a sage is not move by praise or blame.”<ref>The Dhammapada, A new translation of the Buddhist Classic, with annotations, Gil Fronsdal, Shambhala, 2005. Chapter six, The Sage</ref> The acient Chiese poem [[w:Xinxin Ming|Xinxin Ming]]—trust in mnd—begins: “The grat way is not dificult for those who have no preferences. When attachmnt and aversion are both absent everyhing is clear and undisguised.”<ref>Rupert Spira’s rendition of the ancient Buddhist poem Hsin Hsin Ming. See: <nowiki>https://rupertspira.com/non-duality/blog/poetry-prose/hsin-hsin-ming-poem-alternative-rendition</nowiki></ref> The central message of the Xinxin Ming is this: point directly to Mid by giving up [[w:Bias|one-sided views]] so we can see the [[Finding Common Ground|One Suchness of reality]] as it i Modern psychologists also acknowledge the importance of metacognition in achieving calm. For example, author Gregg Henriques describes the CALM-MO approach<ref>[https://www.unifiedtheoryofknowledge.org/8-key-ideas/calm-mo CALM-MO An Integrative Approach to Psychological Mindfulness].</ref> to psychological mindfulness. This approach guides individuals to become a “metacognitive observer” that cultivates an attitudethat is curious, accepting, loving (compassionate), and motivated to grow toword vaued tates of being in bothe short and long term.<ref>A new synthesis for solving the problem of psychology; addressing the enlightenment gap, Gregg Henriques, page 89. And Henriques, G. R (2018, May/June). The root of suffering: How to fight the neurotic loop of negative reactions to negative feelings. Psychology Today, 48-50.</ref> === Assignment === # Notice when you are perturbed by one of the eight worldly winds: ''praise'' and ''blame'', ''success'' and ''failure'',''pleasure'' and ''pain'', ''fame'' and ''disrepute.'' # Notice your attachment to, or aversion of, these. # [[wikipedia:Let it go|Let it go]]. # Optionally compete the course on [[Embracing Ambiguity|embracing ambiguity]]. # Read the essay [[Virtues/Humility/Authentic Humility|Authentic humility.]] # Study the module on the [[Virtues/Humility|virtue of humility]]. ## Attain authentic humility. # Avoid forming opinions unnecessarily or hastily. # Do not mistake kindness for weakness. # Study the course on [[Moral Reasoning]]. ## Act constructively and consistently with your well-chosen moral reasoning. # [[Living Wisely|Live wisely]]. == Recommended Reading == Students wanting to learn more about finding equanimity may be interested in reading the following books: *{{cite book |last=Kabat-Zinn |first=Jon |author-link=w:Jon Kabat-Zinn |date= |title=Coming to Our Senses: Healing Ourselves and the World Through Mindfulness  |publisher=Hachette Books |pages=656 |isbn=978-0786886548}} *{{cite book |last=Sapolsky |first=Robert M. |author-link=w:Robert Sapolsky |date=September 15, 2004 |title=Why Zebras Don't Get Ulcers  |publisher=Holt Paperbacks |pages=560 |isbn=978-0805073690}} *{{cite book |last=Damasio |first=Antonio |author-link=w:Antonio Damasio |date=February 19, 2019 |title=The Strange Order of Things: Life, Feeling, and the Making of Cultures |publisher=Vintage |pages=336 |isbn=978-0345807144}} *{{cite book |last1=Goleman |first1=Daniel |last2=Davidson |first2=Richard |date=September 5, 2017 |title=Altered Traits: Science Reveals How Meditation Changes Your Mind, Brain, and Body |publisher=Avery |pages=336 |isbn=978-0399184383 |author-link=w:Daniel_Goleman }} hey its me its verity *{{cite book |last=Henriques |first=Gregg |author-link=w:Gregg Henriques |date=December 28, 2022 |title=A New Synthesis for Solving the Problem of Psychology: Addressing the Enlightenment Gap  |publisher=Palgrave Macmillan |pages=503 |isbn=978-3031184925}} *{{cite book |last=Kahneman |first=Daniel |author-link=w:Daniel Kahneman |date=April 2, 2013 |title=Thinking, Fast and Slow  |publisher=Farrar, Straus and Giroux |pages=499 |isbn=978-0374533557}} I have not yet read the following books, but they seem interesting and relevant. They are listed here to invite further research. *Calm (The School of Life Library), by The School of Life ==References== <references/> {{Emotional Competency}} [[Category:Life skills]] [[Category:Applied Wisdom]] [[Category:Philosophy]] [[Category:Peace studies]] [[Category:Humanities courses]] nhdz05udzr92xvh1o6qvfselmne8ln7 2844586 2844582 2026-11-05T04:35:42Z Morkoz 3096691 Reverted edits by [[Special:Contributions/~2026-58700-96|~2026-58700-96]] ([[User_talk:~2026-58700-96|talk]]) to last version by [[User:Codename Noreste|Codename Noreste]] using [[Wikiversity:Rollback|rollback]] 2443323 wikitext text/x-wiki —Calm throughout the storm [[File:Calm throughout the storm.png|thumb|The Eight Worldly Winds are constant. To be happy stand like a great tree amidst them all.]] {{TOC right |limit=2}} == Introduction == Finding ''[[w:Equanimity|equanimity]]'' allows you to remain calm and retain a wise perspective while experiencing the chaos and complexity of the world we live in. Assess this moment in perspective. Recognize that this moment is the faintest ripple in the fabric of the universe. == Objectives == {{100%done}}{{By|lbeaumont}} This course can help you find equanimity and enjoy practicing it as you increase your well-being. This course is part of the [[Emotional Competency]] curriculum and the [[Wisdom/Curriculum|Applied Wisdom curriculum]]. If you wish to contact the instructor, please [[Special:EmailUser/Lbeaumont| click here to send me an email]] or leave a comment or question on the [[Talk:Finding Equanimity|discussion page]]. There are no prerequisites to this course and all students are welcome. Students may benefit from studying the Wikiversity courses on [[What you can change and what you cannot]] change, [[Stoic joy]], [[Embracing Ambiguity|Embracing ambiguity]], and [[Quiet Mind|Quiet mind]]. == Characterizing Equanimity == Several characterizations and definitions of equanimity (''[[w:Upekṣā|upekkhā]]'') are proposed in the [[w:Buddhism|Buddhist]] traditions and in modern psychology. These include: * “a state of mind that cannot be swayed by biases and preferences”<ref>Bodhi B. A Comprehensive Manual of Abhidhamma: The Philosophical Psychology of Buddhism. Onalaska, WA: Buddhist Publication Society Pariyatti Editions; 2000. page 34.</ref> * “even-mindedness in the face of every sort of experience, regardless of whether pleasure [or] pain are present or not”<ref>Bhikkhu Thanissaro. Wings To Awakening: An Anthology From The Pali Canon. Barre, MA: Dhamma Dana Publications; 1996. Page 262.</ref> * “a balanced reaction to joy and misery, which protects one from emotional agitation”<ref>Bodhi B. In the Buddha’s Words: An Anthology of Discourses from the Pali Canon. Somerville, MA: Wisdom Publications; 2005. Page 154.</ref> * “maintaining calm and mental equilibrium in the face of provocative stimuli”<ref>Carmody J, Baer RA, Lykins EL, Olendzki N. An empirical study of the mechanisms of mindfulness in a mindfulness-based stress reduction program. Journal of Clinical Psychology. 2009;65(6):613–626.</ref> * “approaching pleasant, unpleasant and neutral experiences with equal interest.”<ref>Grabovac AD, Lau MA, Willett BR. Mechanisms of mindfulness: A Buddhist psychological model. Mindfulness. 2011;2(3):154–166.</ref> * “a balanced reaction to joy and misery”<ref>Bodhi B. A Comprehensive Manual of Abhidhamma: The Philosophical Psychology of Buddhism. Onalaska, WA: Buddhist Publication Society Pariyatti Editions; 2000, Page 34).</ref> * “regarding all beings as equal in their right to have happiness and avoid suffering”<ref>Tsering GT. Buddhist Psychology. The Foundation of Buddhist Thought, vol. 3. Somerville, MA: Wisdom Publications; 2006.</ref> * “treating [all beings] free from discrimination, without preferences and prejudices”<ref>Bodhi B. A Comprehensive Manual of Abhidhamma: The Philosophical Psychology of Buddhism. Onalaska, WA: Buddhist Publication Society Pariyatti Editions; 2000. Page 87</ref> These characterizations can be summarized by this formal definition:<blockquote>''equanimity is an even-minded [[w:Mental state|mental state]] or dispositional tendency toward all experiences or objects'', ''regardless of their [[w:Affect (psychology)|affective]] [[w:Valence (psychology)|valence]] (pleasant, unpleasant or neutral) or source.''<ref name=":0">Moving beyond Mindfulness: Defining Equanimity as an Outcome Measure in Meditation and Contemplative Researc</ref></blockquote>Informally, equanimity is staying calm throughout the storm; keeping a cool head. Equanimity can be distinguished from [[mindfulness]]. Mindfulness emphasizes the ability to remain consciously aware of what is happening in the present moment experience, while equanimity allows awareness to be even and unbiased by facilitating an attitude of [[w:Nonattachment (philosophy)|nonattachment]] and non-resistance.<ref name=":0" /> Equanimity is one of the [[w:Mental factors (Buddhism)#Ten%20wholesome%20mental%20factors|universal beautiful states of mind]].<ref name=":1">[https://dynamic.wakingup.com/course/CE16FB Equanimity], The Path of Insight, Joseph Goldstein, Waking Up (paywall)</ref> In the Buddhist tradition, it is said that while maintaining an evenness of mind, the mind of the great being (the [[w:Buddha|Buddha]] under the [[w:Bodhi Tree|Bodhi tree]]) was unmoved. He remained “There in the middleness” with an unshakable quality of mind. As an example of finding equanimity, consider the range of reactions possible if you are suddenly cut off while driving your car in traffic. Lacking mindfulness and equanimity you might become frightened and angry and react by honking the horn, slamming on the brakes, shouting insults at the other driver, and exhibiting other symptoms of [[w:Road rage|road rage]]. However, in a [[w:Mindfulness|mindful]] state you become aware of the change in your [[Emotional Competency|emotional state]], maintain a state of equanimity, gaze upon the experience, and [[w:Calmness|calmly]] observe as your anger unfolds, [[Resolving Anger|resolves]], and passes. == Equanimity is not Indifference == While indifference implies apathy, lethargy, aloofness, or even a subtle form of aversion, equanimity requires care and attentiveness. Equanimity does not involve suppressing emotions, reducing the affective coloring of our life experiences, or limiting our [[What you can change and what you cannot#Agency|agency]]. Equanimity is not apathy or “indifference but rather of mental imperturbability”.<ref>(Thanissaro, B. (1996). Wings to awakening: An anthology from the Pali Canon. Barre, MA: Dhamma Dana Publications., p. 263).</ref> It does not mean indifference but is a vehicle to connect from a point of genuine affection rather than bias subjection.<ref>[https://www.researchgate.net/profile/Joey-Weber/publication/321224326_Mindfulness_is_not_enough_Why_equanimity_holds_the_key_to_compassion/links/5a2fa4614585155b617a5174/Mindfulness-is-not-enough-Why-equanimity-holds-the-key-to-compassion.pdf Mindfulness is not enough: Why equanimity holds the key to compassion]</ref> Equanimity is not ''indifference'', it is ''impartial''.<ref name=":1" /> We can learn to separate the emergence of an emotion from our [[Appraising Emotional Responses|appraisal]] of and reaction to that emotion. For example, while practicing equanimity in the driving example above the transgression of the other driver is noted, but without the automatic attribution of judgment, blame, resentment, and anger, much of which is caused by interpreting the incident as a personal insult.<ref name=":0" /> Equanimity requires caring for other people, in contrast to being indifferent to others. Buddhist masters recommend practicing equanimity towards beings by contemplating how all beings are like us in that they, too, wish to gain happiness and be free of suffering, regardless of whether we would consider them our friends, enemies, or strangers.<ref>Lama Dalai. Stages of Meditation. Ithaca, NY: Snow Lion Publications.</ref> == Methods for Finding Equanimity == The basic technique for finding equanimity is to become mindful of an object or event without becoming attracted to or repelled by it. Notice the presence of the object or event non-judgmentally, neither favoring it nor opposing it. A non-judging attitude consists in “assuming the stance of an impartial witness to one’s own experience,” which involves suspending judgment and simply “watching whatever comes up,” including observing one’s automatic judgments of “like,” “dislike,” and “don’t-care,” without “judging the judging.” It also involves recognizing one’s thoughts as “just thoughts” without pursuing them any further. Although thoughts arise, I am not my thoughts, and my thoughts are not me. Observe your thoughts as they arise, transform, linger, and disappear, without becoming captive to those thoughts. Acceptance is the “willingness to see things as they are”,<ref>Kabat-Zinn J. Full Catastrophe Living: Using the Wisdom of Your Body and Mind to Face Stress, Pain, and Illness. New York, NY: Delacorte Press; 1990.</ref> “not merely tolerance” but rather “the active nonjudgmental embracing of experience in the here and now,” with “undefended ‘exposure’ to thoughts, feelings, and bodily sensations as they are directly experienced”.<ref>Hayes SC, Follette VM, Linehan MM, editors. Mindfulness and Acceptance: Expanding the Cognitive-Behavioral Tradition. New York, NY: The Guilford Press; 2004.</ref> Below are specific techniques that may help you find equanimity. === Increasing Emotional Competency === Increasing [[Emotional Competency|emotional competency]] can increase equanimity. Developing the essential social skills to recognize, interpret, and respond constructively to emotions in yourself and others can help you accurately assess emotions as they arise. Emotional competency provides the perspective and depth of understanding that allow you to remain calm despite the ever-present storms we encounter. ==== Assignment ==== # Study the [[Emotional Competency|emotional competency]] curriculum. # Increase your emotional competence. # Practice recognizing, interpreting, and responding constructively to emotions as they arise in you and others. === Reducing Stress === In his book ''[[w:Why Zebras Don't Get Ulcers|Why Zebras Don’t Get Ulcers]]'', [[w:Robert Sapolsky|Robert Sapolsky]] provides an in-depth guide to [[w:Stress (biology)|stress]], stress-related diseases, and [[w:Coping (psychology)|coping]]. The last chapter of the book provides some specific recommendations for managing stress. These include: # Based on the 1941 [[w:Grant Study|Harvard aging study]] traits that contribute to better health, including reduced chronic stress are: no smoking, minimal alcohol use, lots of exercise, normal body weight, absence of depression, a warm stable marriage, and a mature resilient coping style. One interpretation of the study results is that the warmth of relationships throughout life has the greatest positive impact on life satisfaction. # People with more of an internalized [[w:Locus of control|locus of control]]—the belief that they have control over the outcome of events in their lives—are more resistant to life’s troubles. Be clear on what you can change and what you cannot change. Being able to switch locus of control (taking justified credit for good outcomes and avoiding [[Shame#Other%20subtypes|chronic shame]] and regret for bad outcomes) can be a helpful coping style. # Seek out [[Deductive Logic/Clear Thinking curriculum|accurate information]] to provide you with a realistic perspective, a basis for effective action, and to avoid worrying about unlikely occurrences. # The ability to walk away from provocations—let it go, don’t take the bait—helps to avoid [[w:Stressor|stressors]]. # [[w:Exercise|Exercise]] reduces stress if it is something you choose to do. # Regular [[w:Meditation|meditation]] can improve health and reduces stress. # [[w:Social support|Social support]] (from welcomed friends and other supportive people) reduces the stress response when stressors are inevitably encountered. Giving social support is also helpful. # Finding a sense of [[w:Meaning of life|purpose]] helps reduce chronic stress. # Hoping for the best while you [[w:James Stockdale#The%20Stockdale%20Paradox|prepare for the worst]] will help you stay calm throughout the storm. ==== Assignment ==== # Complete the Wikiversity course on [[what you can change and what you cannot]] change. # Change what you can and accept the rest. # Practice those stress reduction techniques listed above that you find helpful. === Separating Wanting from Liking === It is possible to separate ''wanting'' (having a goal to acquire) from ''liking'' (eliciting a positive [[w:Affect (psychology)|affect]]).<ref>[https://faculty.philosophy.umd.edu/pcarruthers/Liking&#x20;versus&#x20;Wanting.pdf Liking versus Wanting]</ref> Recognizing the distinction between wanting and liking provides four conditions.  Two conditions are unremarkable because wanting and liking are aligned, the other two are somewhat surprising because wanting and liking are misaligned. Wanting and liking are ''aligned'' in these examples: * I like my friends and I want to be with them. * I like good weather and I want to be outside in good weather. * I don’t like boorish people and I don’t want to be around them. * I don’t like cold weather and I don’t want to be outside in the freezing cold. Wanting and liking are ''misaligned'' in these examples: * I like sweet desserts, but I don’t want a second helping. * I don’t like smoking cigarettes, but I crave one now. We can make constructive choices by separating wanting from liking. For example: * Practice the virtue of [[Virtues/Temperance|temperance]] by consuming things we like in moderation, such as moderating in eating, consuming alcohol, shopping, and avoiding other temptations. This may also include working to overcome [[Addiction|addictions]]. * Practice the virtue of [[Virtues/Compassion|compassion]] by learning to accept people you find difficult and overcoming unhelpful fears by learning to approach unpleasant things. ==== Assignment ==== [[File:Pink lily flower.jpg|thumb|The [[w:Nelumbo nucifera|lotus]] symbolizes non-attachment in some religions in Asia owing to its ability to grow in muddy waters yet produce an immaculate flower.]] #Exercise your [[What you can change and what you cannot#Agency|agency]] by separating wanting from liking when it is [[Wisdom|wise]] to do so. For example: ## Decline that second helping of ice-cream. ## Limit your alcohol consumption. ## Limit shopping sprees. ## Reduce binging. ## Reduce [[w:Compulsive hoarding|hoarding]]. ## Limit time on [[w:Social media|social media]]. Overcome any obsession with social media [[w:Like button|likes]]. ## Work to overcome any [[w:Addiction|addictions]] you might be experiencing. ## Practice the virtue of [[Virtues/Temperance|temperance]], including [[Virtues/Temperance#Everyday%20Temperance|everyday temperance]]. ## Exercise [[Virtues/Compassion|compassion]] for the difficult people you encounter. ## [[Forgiving|Forgive]]. ## Complete your chores, even the ones you don’t like. ## Clean toilets. ## Hold a snake or a spider. ## Overcome unhelpful fears so you can expand your sphere of comfort. # Study the Buddhist concept of [[w:Impermanence|impermanence]]. ## Accept inevitable losses and separations. ## Recognize that whatever is arising will also pass away – understand (pay attention to the fact) that everything is changing. ## The experience of a rainbow is transient, the rainbow is no-thing – it is a fleeting appearance of certain conditions of light, water, and air. Each rainbow is an “Empty phenomena rolling on”. Recognize all that is impermanent. # Study the concept of [[w:Nonattachment (philosophy)|nonattachment]]. ## Overcome unhelpful attachments. ## “As long as there is attachment to the pleasant and aversion to the unpleasant liberation is impossible.”<ref>Attributed to Buddha as reported in ''Mindfulness, A practical guide to Awakening'', by Joseph Goldstein.</ref> # Remind yourself: “Anything can happen any time”. Expect nothing, prepare for anything. === A Buddhist Approach === Buddhist texts describe practices for developing concentration (''[[w:Samatha|samatha]]'') or insight ([[w:Vipassanā|vipassanā]]) to further develop equanimity. One of the techniques for developing equanimity consists in trying to develop an impartial attitude toward all beings by reflecting on the “very simple truth” that “all of us, including the most despicable people, do the things we do because we’re seeking happiness and want to be free of suffering”<ref>Wallace BA. The Four Immeasurables: Practices to Open the Heart. Ithaca, NY: Snow Lion Publications; 2010.</ref> A more detailed contemplation would be: “All beings are the same as us in three basic ways. # They all want to be well, happy, and free of suffering. # They are all lost in [[Coping with Ego|self-centered]] reactions to their own narrow thoughts of self and others. # They all possess an innate capacity of goodness that has been obscured by these self-centered reactions”<ref>Makransky J. Awakening Through Love: Unveiling Your Deepest Goodness. Somerville, MA: Wisdom Publications; 2007.</ref> ==== Assignment ==== # Practice the above contemplation, perhaps by reciting it, reflecting on it, or using it as the focus for meditation. # When you feel [[Resolving Anger|anger]], hatred, envy, [[Forgiving#The%20Vindictive%20Passions|vengeance]], or revulsion arising, pause, reflect, and (silently) recite the above contemplation before taking any further action. # When [[w:List of diving hazards and precautions|difficulties arise]] while deep under water, [[w:Scuba diving|scuba divers]] are taught “there are many helpful actions you can take, however [[w:Panic|panic]] is never the solution.” Keep this in mind when you are tempted to panic in response to turbulence in the surrounding storm. Panic is never the solution. == The Eight Worldly Winds == Equanimity is a [[w:Upekṣā|protection from the eight great vicissitudes]], also known as the “eight worldly winds”. The eight are ''praise'' and ''blame'', ''success'' and ''failure'', ''pleasure'' and ''pain'', ''fame'' and ''disrepute''. Becoming attached to or excessively elated with success, praise, fame, or pleasure can make us vulnerable to suffering when the winds of life change direction.<ref>[https://www.insightmeditationcenter.org/books-articles/equanimity/ Equanimity], Insight Meditation Center</ref>^,<ref>Eight worldly concerns, Encyclopedia of Buddhism, See: <nowiki>https://encyclopediaofbuddhism.org/wiki/Eight_worldly_concerns</nowiki></ref> Equanimity buffers us from these eight ever-present worldly winds. When we understand they are impermanent—they come and go like the wind—we can be happy as we stand like a great tree amidst them all. Each of the eight worldly winds are types of gains and losses. Equanimity is detachment from an outcome, a gain or loss. To avoid being tossed by these winds, don’t identify with the goal or possession, the gain or loss. Understand that because each possession is [[w:Impermanence|impermanent]], the possession cannot be owned; it is not ''mine''. Equanimity maintains our balance in the face of these ever-present and ever-changing vicissitudes. [[w:Rupert Gethin|Rupert Gethin]] on [[w:Four Noble Truths|Four Noble Truths]] says: <blockquote>As long as there is attachment to things that are unstable, unreliable, changing and impermanent, there will be suffering – when they change, when they cease to be what we want them to be. (...) If craving is the cause of suffering, then the cessation of suffering will surely follow from 'the complete fading away and ceasing of that very craving': its abandoning, relinquishing, releasing, letting go.</blockquote>[[w:Ryōkan|Ryōkan]], was a Buddhist monk who found equanimity. He had very few material possessions and when they were stolen, he responded by writing a [[w:Haiku|Haiku]]:<blockquote>The moon at the window, the thief left it behind.<ref>Written after a thief robbed his hut, as translated in [[w:Stephen Mitchell (translator)|Mitchell, Stephe]]<nowiki/>n, editor. ''The Enlightened Heart: An Anthology of Sacred Poetry''. Harper Perennial, 1993. [[ISBN (identifier)|ISBN]] [[Special:BookSources/978-0060920531|978-0060920531]] p162</ref></blockquote>We can follow his example and learn to respond with equanimity rather than overreact to transient circumstances. [[w:Metacognition|Metacognition]] is an awareness of one's thought processes and an understanding of the patterns behind them. Finding equanimity is one of several metacognitive skills. The [[w:Sage (philosophy)|sage]] recognizes that praise and blame are ever-present. The sage watches his own mind and chooses not to be buffeted by the constant ebb and flow of praise and blame. “As a solid mass of rock is not moved by the wind, so a sage is not moved by praise or blame.”<ref>The Dhammapada, A new translation of the Buddhist Classic, with annotations, Gil Fronsdal, Shambhala, 2005. Chapter six, The Sage</ref> The ancient Chinese poem [[w:Xinxin Ming|Xinxin Ming]]—trust in mind—begins: “The great way is not difficult for those who have no preferences. When attachment and aversion are both absent everything is clear and undisguised.”<ref>Rupert Spira’s rendition of the ancient Buddhist poem Hsin Hsin Ming. See: <nowiki>https://rupertspira.com/non-duality/blog/poetry-prose/hsin-hsin-ming-poem-alternative-rendition</nowiki></ref> The central message of the Xinxin Ming is this: point directly to Mind by giving up [[w:Bias|one-sided views]] so we can see the [[Finding Common Ground|One Suchness of reality]] as it is. Modern psychologists also acknowledge the importance of metacognition in achieving calm. For example, author Gregg Henriques describes the CALM-MO approach<ref>[https://www.unifiedtheoryofknowledge.org/8-key-ideas/calm-mo CALM-MO An Integrative Approach to Psychological Mindfulness].</ref> to psychological mindfulness. This approach guides individuals to become a “metacognitive observer” that cultivates an attitude that is curious, accepting, loving (compassionate), and motivated to grow toward valued states of being in both the short and long term.<ref>A new synthesis for solving the problem of psychology; addressing the enlightenment gap, Gregg Henriques, page 89. And Henriques, G. R (2018, May/June). The root of suffering: How to fight the neurotic loop of negative reactions to negative feelings. Psychology Today, 48-50.</ref> === Assignment === # Notice when you are perturbed by one of the eight worldly winds: ''praise'' and ''blame'', ''success'' and ''failure'',''pleasure'' and ''pain'', ''fame'' and ''disrepute.'' # Notice your attachment to, or aversion of, these. # [[wikipedia:Let it go|Let it go]]. # Optionally compete the course on [[Embracing Ambiguity|embracing ambiguity]]. # Read the essay [[Virtues/Humility/Authentic Humility|Authentic humility.]] # Study the module on the [[Virtues/Humility|virtue of humility]]. ## Attain authentic humility. # Avoid forming opinions unnecessarily or hastily. # Do not mistake kindness for weakness. # Study the course on [[Moral Reasoning]]. ## Act constructively and consistently with your well-chosen moral reasoning. # [[Living Wisely|Live wisely]]. == Recommended Reading == Students wanting to learn more about finding equanimity may be interested in reading the following books: *{{cite book |last=Kabat-Zinn |first=Jon |author-link=w:Jon Kabat-Zinn |date= |title=Coming to Our Senses: Healing Ourselves and the World Through Mindfulness  |publisher=Hachette Books |pages=656 |isbn=978-0786886548}} *{{cite book |last=Sapolsky |first=Robert M. |author-link=w:Robert Sapolsky |date=September 15, 2004 |title=Why Zebras Don't Get Ulcers  |publisher=Holt Paperbacks |pages=560 |isbn=978-0805073690}} *{{cite book |last=Damasio |first=Antonio |author-link=w:Antonio Damasio |date=February 19, 2019 |title=The Strange Order of Things: Life, Feeling, and the Making of Cultures |publisher=Vintage |pages=336 |isbn=978-0345807144}} *{{cite book |last1=Goleman |first1=Daniel |last2=Davidson |first2=Richard |date=September 5, 2017 |title=Altered Traits: Science Reveals How Meditation Changes Your Mind, Brain, and Body |publisher=Avery |pages=336 |isbn=978-0399184383 |author-link=w:Daniel_Goleman }} *{{cite book |last=Henriques |first=Gregg |author-link=w:Gregg Henriques |date=December 28, 2022 |title=A New Synthesis for Solving the Problem of Psychology: Addressing the Enlightenment Gap  |publisher=Palgrave Macmillan |pages=503 |isbn=978-3031184925}} *{{cite book |last=Kahneman |first=Daniel |author-link=w:Daniel Kahneman |date=April 2, 2013 |title=Thinking, Fast and Slow  |publisher=Farrar, Straus and Giroux |pages=499 |isbn=978-0374533557}} I have not yet read the following books, but they seem interesting and relevant. They are listed here to invite further research. *Calm (The School of Life Library), by The School of Life ==References== <references/> {{Emotional Competency}} [[Category:Life skills]] [[Category:Applied Wisdom]] [[Category:Philosophy]] [[Category:Peace studies]] [[Category:Humanities courses]] 42rvnydmz5tnapqy6xkfudzxot8tyim 2 327236 2844598 2840743 2026-11-05T10:54:48Z Dick Bos 24466 todo 2844598 wikitext text/x-wiki &nbsp; [[File:On_Expedition_2015_MARBLE_Maastricht_University_2.JPG|thumb|Maastricht University, 2015-05-06. Explaining the principles of Wikipedia for students with [[User:Taketa]] (left) and [[User:Romaine]] (right)]] {{#babel:nl-N|en-3|de-2|fr-1}} &nbsp; My name is Dick Bos. I'm from the Netherlands. I work on the [[:w:nl:Gebruiker:Dick Bos|Dutch]] and [[en:User:Dick Bos|English]] wikipedia, on the English wikisource, on Wikimedia Commons etc. See my userpages there, please. * [[File:Wikipedia-logo-v2.svg|16px|alt=16 px Wikipedia logo]] Wikipedia: ** Dutch: [[w:nl:Gebruiker:Dick_Bos]] ** English: [[w:en:User:Dick Bos]] * [[Image:wikisource-logo.svg|16x16px]] WikiSource: ** [[s:en:User:Dick Bos|User:Dick Bos]] on en-ws [[m:User:Dick Bos/where|Click here]] for the most complete and up to date list of my userpages on Wikimedia-projects. &nbsp; My projects: * [[User:Dick Bos/Marx|Karl Marx]] (and Friedrich Engels) * [[User:Dick Bos/Ricardo|David Ricardo]] == Handy == * [[Help:Contents]] ([[WV:HELP]]) * [[Help:CSS at Wikiversity]] * [[Wikiversity:Original research]] == Interesting == * [[WikiJournal of Humanities|WikiJournal (of Humanities)]] is hosted here on Wikiversity == Todo == When we have time, we need to spend some energy on: * [[wikibooks:Political Economy]] fsci4vub1rbq6xjn9m8ayoq4r78hmu4 0 328204 2844606 2844408 2026-11-05T11:32:02Z Unitfreak 695864 /* The Sumerian Origins of Genesis 5 */ 2844606 wikitext text/x-wiki {| class=table style="width:100%;" |- | {{Original research}} |- | {{RoundBoxTop}} This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional context and demonstrates that the underlying numerical data is even more robust and systematic than initially identified. This article was researched and written by a human author, with editing assistance provided by [https://gemini.google.com Google "Gemini, your personal AI assistant"]{{RoundBoxBottom}} |} = The Sumerian Origins of Genesis 5 = The early history in Genesis 5 shares deep structural, thematic, and chronological ties with ancient Mesopotamian writing traditions. The closest parallel to the biblical figures who lived before the flood is the Sumerian King List (SKL). This ancient text details a succession of long-lived rulers who reigned right up until a cataclysmic flood. The table below shows the connections between Genesis 5 and the Sumerian King List, with directly matching details highlighted in blue. {| class="nowraplinks" cellspacing="0" cellpadding="3" rules="all" style="background:#fbfbfb; border: 1px solid #ccc; font-size:90%; empty-cells:show; border-collapse:collapse; text-align:left;" |- style="background:#F6E6AE; font-weight:bold; text-align:center;" ! colspan="2" | Sumerian King List ! Hebrew Bible |- style="text-align:center; font-style:italic;" ! colspan="2" style="background:#fffcf2;" | Archaeological Timeline:<br><span style="background:#d6e2ff; font-style:normal; font-weight:bold; padding:0 4px;">2400</span> Years Total ! style="background:#fffcf2;" | Demetrius Timeline:<br><span style="background:#d6e2ff; font-style:normal; font-weight:bold; padding:0 4px;">2264</span> Years Total |- style="background:#F6E6AE; font-weight:bold; text-align:center;" ! City ! Ruler (Sage) ! Patriarch |- | [[w:Eridu|Eridu]] | [[w:Alulim|Alulim]] <span style="background: #d6e2ff; padding: 0 2px;">([[w:Adapa|Adapa]])</span> <br /> [[w:Alalngar|Alalngar]] | <span style="background: #d6e2ff; padding: 0 2px;">[[w:Adam|Adam]]</span> <br /> [[w:Seth|Seth]] |- | [[w:Bad-tibira|Bad-tibira]] | [[w:En-men-lu-ana|En-men-lu-ana]] <br /> [[w:En-men-gal-ana|En-men-gal-ana]] <br /> [[w:Dumuzid|Dumuzid]] | [[w:Enos|Enos]] <br /> [[w:Kenan|Kenan]] <br /> [[w:Mahalalel|Mahalalel]] |- | [[w:Larak (Sumer)|Larak]] | [[w:En-sipad-zid-ana|En-sipad-zid-ana]] | [[w:Jared (biblical figure)|Jared]] |- | [[w:Sippar|Sippar]] | <span style="background: #d6e2ff; padding: 0 2px;">[[w:En-men-dur-ana|En-men-dur-ana]]</span> | <span style="background: #d6e2ff; padding: 0 2px;">[[w:Enoch|Enoch]]</span> |- | [[w:Shuruppak|Shuruppak]] | [[w:Ubara-Tutu|Ubara-Tutu]] <br /> <span style="background: #d6e2ff; padding: 0 2px;">[[w:Utnapishtim|Utnapishtim]]</span> | [[w:Methuselah|Methuselah]] <br /> [[w:Lamech (father of Noah)|Lamech]] <br /> <span style="background: #d6e2ff; padding: 0 2px;">[[w:Noah|Noah]]</span> |} == Narrative Similarities == The biblical accounts appear to borrow from, or develop alongside, the mythological roles assigned to Sumerian rulers and their accompanying apkallu (sages). Key thematic intersections include: *[https://www.worldhistory.org/article/216/the-myth-of-adapa/ Adam as Adapa] : In the Adapa narrative, Ea (god of wisdom) warns Adapa to refuse any food or drink offered in heaven but to accept clothing and oil. Anu (the sky god) offers Adapa food and drink, followed by clothing and oil. When Adapa refuses the food, Anu chides him for missing an opportunity for immortality. Similarly, in the Genesis account, Yahweh Elohim permits Adam and Eve to eat from any tree in the divine garden except the tree of knowledge. A competing figure, the serpent, advises them to eat from it anyway. After they do, Yahweh returns, chastises them for disobedience, and provides garments of skin. Key parallels between the accounts include: A potential similarity and shared root in the names Adapa and Adam. Both narratives feature a test involving the consumption of supposedly deadly food. Both figures are offered clothing and summoned before a deity to answer for their decisions. *[https://www.marquette.edu/maqom/Chapter%201.pdf Enoch as Enmeduranki]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew Book of Enoch describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text Pirišti Šamê u Erṣeti (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos. *[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven. == Chronological Similarities == [[File:Antediluvian Sumerian King List Cities and Biblical Rivers.png|thumb|400px|Locations of four of the five pre-Flood cities named in the Sumerian King List have been identified through excavated cuneiform inscriptions. These cities existed in close proximity to the Tigris and Euphrates rivers. Eridu was a coastal city when first founded around 5300 BC. Over the millennia, however, massive deposits of alluvial silt from four river systems filled the ancient bays, gradually pushing the shoreline southeastward away from Eridu. There are four rivers named in the biblical Garden of Eden story (Genesis 2:10–14): the Tigris, Euphrates, Gihon, and Pishon. The Gihon and Pishon are frequently associated with the Karun River in western Iran and the Wadi al-Batin (a now-dry riverbed) in the Arabian Peninsula. The blue oval on the map above represents the 5300 BC coastline, with arrows pointing to the mouths of the four Eden rivers.]] In Mesopotamian mythology, Eridu was the oldest city, the home of the wisdom god Enki, and the first site where divine kingship descended. While many Sumerian texts omit the pre-Flood era entirely, in narratives that do include a pre-Flood prologue, Eridu is consistently identified as the first city and Šuruppak is frequently identified as the fifth and final city prior to the Flood. One English translation of the Sumerian King List states it as follows: [https://www.livius.org/sources/content/anet/266-the-sumerian-king-list/ "After the kingship descended from heaven, the kingship was in Eridu ... Bad-tibira fell and the kingship was taken to Larak ... Sippar fell and the kingship was taken to Šuruppak ... Five cities; eight kings ruled for 385,200 years. Then the Flood swept over."] A related text, the [[w:Eridu Genesis|Eridu Genesis]], preserves a parallel tradition with Eridu as the first city and Šuruppak as the fifth: [https://www.livius.org/sources/content/oriental-varia/eridu-genesis/ "The firstling of the cities, Eridu, she gave to the leader Nudimmud, the second, Bad-Tibira, she gave to the Prince and the Sacred One, the third, Larak, she gave to Pahilsag, the fourth, Sippar, she gave to the gallant Utu, the fifth, Šuruppak, she gave to Ansud."] Locations of four of the five pre-Flood Sumerian cities have been identified archeologically. Eridu sits on a hill inside a basin roughly {{convert|15|mi}} long and {{convert|20|ft}} deep, separated from the Euphrates by a sandstone ridge called the ''Hazem''. In the 3rd Millennium BC, before the river shifted course, a canal connected Eridu to the Euphrates. Low tells with 2nd Millennium BC pottery and later burials mark the canal's path. ====Prehistoric Temple(s)==== [[File:Ea (Babilonian) - EnKi (Sumerian).jpg|thumb|Detail of the Adda Seal, an ancient Akkadian cylinder seal (circa 2300 BCE) depicting Enki with water streams coming from his shoulders.|270x270px]] The urban nucleus of Eridu was Enki's temple, called the ''House of the Aquifer'' in the oldest Sumerian texts. In later texts, this temple was called the ''House of the Waters''. [[w:Enki|Enki]] was the Mesopotamian god of subterranean waters, specifically reigning over the primeval underground ocean known as the [[w:Abzu|Abzu]]. In ancient art, he was frequently represented with water streams flowing directly from his body, which were often filled with fish swimming through the currents. The biblical [[w:Garden of Eden|Garden of Eden]] is presented as a well-watered garden with rivers running out of it. The Sumerian association of Eridu with subterranean waters seems reflected in the biblical garden story. [[File:Eridu Level Sequence A.jpg|frame|Prehistoric sequence of buildings excavated at Eridu.]] A prehistoric sequence of buildings excavated at Eridu during the 1946 to 1949 campaigns seems to substantiate Eridu's status as the first city with the earliest temple. Consisting of eighteen occupational levels, this Eridu sequence of "Palace-like" or "Temple-Like" buildings appears to be the first occurrence of such structures in the region<ref>{{cite book| last= Leick| first= Gwendolyn| title= Mesopotamia, The invention of the city| place= London| publisher= Penguin Books| year= 2002}}</ref>. In [[w:Archaeology|archaeology]], a '''[[w:Tell (archaeology)|tell]]''' is a mound consisting of the accumulated and stratified debris of a succession of consecutive settlements at the same site, the refuse of generations of people who built and inhabited them, alongside natural sediment. The most recent layers of a tell are nearest to the surface. As archaeologists dig into the tell, they encounter successively older layers of civilization, a recording of history called [[w:Stratigraphy|stratigraphy]]. The following table compares the stratigraphy found during the Eridu 1946–1949 campaigns with the stratigraphy found during the 1931 excavation of Shuruppak (Tell Fara). The table is organized according to the order in which layers are uncovered, with the youngest layers in the first rows of the table and successively older layers in subsequent rows. As shown below, the historic timeline separating the first construction at Eridu (c. 5300 BC) from the cultural break of the Shuruppak flood (c. 2900 BC) spans a real-world duration of approximately 2,400 years. {| class="wikitable" ! Historical Period & Dating ! Eridu Temple Stratigraphy ! Shuruppak Stratigraphy |- | ''' Dynastic Periods ''' ''(c. 2004-2900 BC)'' | '''Ur III Ziggurat Layer''' <br/> "In the ruins of the ziggurat we found numerous bricks of" [[w:Amar-Sin|Amar-Sin (c. 2046–2037 BC)]], "three bricks of" [[w:Ur-Nammu|Ur-Nammu (2112–2094 BC)]] "one brick of" [[w:Nur-Adad|Nur-Adad (c. 1866-1850 BC)]] <br/> [https://www.scribd.com/document/897929584/Eridu-1981-Fuad-Safar-Seton-Lloyd "It seems that Ur-Nammu began the building of this ziggurat, but the bulk of the task was left to his grandson Amar-Sin to accomplish."] | '''Fara II & III Layer''' ([[w:Shu-Sin|Shu-Sin]] inscription (c. 2037 – 2028 BC), Sumerian seal cylinders, pottery, skeletons, figurines, Globular Stone Jar) |- | style="background-color:#ffdddd;" | '''c. 2900 BC''' | style="text-align: center;" | – | style="background-color:#ffdddd;" | '''The Shuruppak Flood Layer''' [https://www.penn.museum/sites/journal/9356/ "We found an alluvial layer at a depth between four and five metres below the mound surface."] [https://www.penn.museum/sites/journal/9356/ "Only a few objects of this period, namely one stamp seal and some sherds “floated” in the refuse above. In this respect an absolute culture break took place."] |- | '''[[w:Jemdet Nasr period|Jemdet Nasr]]''' ''(c. 2900-3100 BC)'' | style="text-align: center;" | – | '''Fara I Layer''' (Scaraboid Stamp Seals, Polychrome Jar, Animal Patterns on Potsherds, Copper Blade) |- | '''[[w:Uruk period|Uruk]]''' ''(c. 3100-3800 BC)'' | '''Temple Layers I–V''' <br/> Temple I & II: [https://www.scribd.com/document/897929584/Eridu-1981-Fuad-Safar-Seton-Lloyd "All that is left to us are its foundations and retaining-walls, while even of these a large portion are buried beneath the massive masonry of the Third Dynasty Ziggurat.] <br/> Temple III: Small bricks of reddish clay. <br/> Temple IV: Medium sized bricks of greenish clay. <br/> Temple V: Large bricks of light-colored clay with dark joints. <br/> [https://www.scribd.com/document/897929584/Eridu-1981-Fuad-Safar-Seton-Lloyd "Since Temple VI was now known to belong to a very late, if not the final stage of the Al-Ubaid period, Temples III, IV and V must be taken to represent a stage intermediate between this and the stone architecture of the Protoliterate age"] | style="text-align: center;" | – |- | '''Ubaid 1-4''' ''(c. 3800-5300 BC)'' | '''Temples VI–XVIII''' <br/> Temple X: "At Level X the temple appeared to have been rebuilt with only minor changes in the plan." <br/> Temple XI: "provides the first instance of a temple building being deliberately raised upon a platform." <br/> Temple XII & XIII: "there were two successive occupation-levels at which no traces of actual buildings were to be found" <br/> Temple XVIII "The first traces of occupation at Eridu occurred at a level corresponding to 11.70 meters beneath datum" ... "and, directly above, at Level XVII, were the remains of the earliest recognizable building" | style="text-align: center;" | – |- | style="background-color:#ffdddd;" | '''c. 5300 BC''' | style="background-color:#ffdddd;" | '''First Construction''' | style="text-align: center;" | – |} === Pre-flood Bible Chronology === '''Demetrius the Chronographer''' stands as the earliest known witness to biblical chronological calculations. Despite the fragmentary nature of his work, his data remains pivotal. Writing from [https://commons.wikimedia.org/wiki/File:Ptolemaic_Alexandria_-_en.svg Alexandria in the late 3rd century BC (c. 221 BC)], Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood of Noah. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY |- ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | rowspan="11" style="background-color:#e8e8e8;" | — | colspan="5" style="background-color:#e8e8e8;" | 230 | colspan="3" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="5" style="background-color:#e8e8e8;" | 205 | colspan="3" style="background-color:#e8e8e8;" | 105 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="5" style="background-color:#e8e8e8;" | 190 | colspan="3" style="background-color:#e8e8e8;" | 90 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="5" style="background-color:#e8e8e8;" | 170 | colspan="3" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="5" style="background-color:#e8e8e8;" | 165 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="1" style="background-color:#e8e8e8;" | 66 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="6" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 61 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="5" style="background-color:#e8e8e8;" | 165 | colspan="3" style="background-color:#e8e8e8;" | 65 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#f9f9f9;" | 167 | colspan="3" style="background-color:#f9f9f9;" | 167 / 187 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#f9f9f9;" | 67 | colspan="1" style="background-color:#f9f9f9;" | 65 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="4" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#f9f9f9;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 55 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="3" style="background-color:#e8e8e8;" | Varied | rowspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 602 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied | colspan="1" | 1656 | colspan="1" | 1309 | colspan="1" | 1309 |} The Septuagint chronology, as witnessed by Demetrius at 2,264 years, mirrors this archaeological reality within a remarkably tight margin of error (less than 6%). This broad-stroke alignment strongly indicates that while the fine internal details were arranged to fit sacred numerical cycles, the overarching framework of the Septuagint preserved an authentic, historically grounded memory of the Fertile Crescent's true antiquity—a memory that was severely contracted in alternative textual traditions. The structural overlap between the Mesopotamian and biblical traditions becomes most compelling when evaluating the total duration of the antediluvian era. While the fine details and specific individual lifespans in both the Septuagint and the Sumerian King List are heavily calculated using stylized, sacred, or mathematically perfect numbers, the macro-span of the Septuagint timeline displays an uncanny alignment with real-world ancient history. This macro-alignment is particularly remarkable when contrasted against the Sumerian King List itself. In the Mesopotamian scribal tradition, the antediluvian era spans astronomical lengths of time—typically totaling over 240,000 years across the reigns of the first eight pre-flood kings. These numbers are clearly symbolic or mathematical inflations that completely diverge from physical archaeological realities. Rather than duplicating these vast Near Eastern timelines or severely compressing the era, the Septuagint tradition preserved a highly realistic historical interval. It is highly significant that the Septuagint chronology, as witnessed by Demetrius, aligns so closely with modern archaeological horizons instead of the exaggerated spans of the SKL. Because an unguided mathematical arrangement of sacred numbers would be statistically unlikely to land so close to actual physical history, this uncanny broad-stroke accuracy hints at two intriguing possibilities: it was either a case of monumental cosmic luck, or the Alexandrian redactors possessed ancient, pristine Near Eastern source records that have since been entirely lost to history. This total of 2,264 years stands in stark contrast to the significantly compressed timelines preserved in the Masoretic Text (1,656 years) and the Samaritan Pentateuch (1,307 years). When evaluating these numbers against physical history, the Septuagint's macro-chronology reveals a striking real-world correlation. = ''Arichat Yamim'' (Long Life) = Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= \frac{420\,\text{šūši}}{2} \\ &= 210\,\,\text{šūši} \\ &= \left(210 \times 60 \,\text{years} \right) \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="width:100%; color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. {| class="wikitable" style="font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | rowspan="6" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 1}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|82 šūši}}<br/><small>(4920)</small></div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|45 šūši}}<br/><small>(2700)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|37 šūši}}<br/><small>(2220)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 <small>(360)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | rowspan="3" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 3}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|46 šūši}}<br/><small>(2760)</small></div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|32 šūši}}<br/><small>(1920)</small></div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 <small>(840)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 <small>(840)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | rowspan="18" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 2}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|82 šūši}}<br/><small>(4920)</small></div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|40 šūši}}<br/><small>(2400)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 <small>(960)</small> | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 <small>(600)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 <small>(420)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 <small>(420)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|25 šūši}}<br/><small>(1500)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 <small>(480)</small> | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 <small>(240)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 <small>(240)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|17 šūši}}<br/><small>(1020)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 131 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="2" style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/><small>(12,600 years)</small> |} ==Mesopotamian Derived Lifespans== [[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[w:supplementary_hypothesis|supplementary hypothesis]], a popular model of the [[w:composition_of_the_Torah|composition of the Torah]]. The Priestly source is shown as '''P'''.]] Many scholars believe the biblical chronology was developed by an individual or school of scribes known as the [[w:Priestly_source|Priestly source]] (see diagram). Deprived of a physical Temple, the Judean elite focused on transforming oral traditions into a permanent, 'portable' written Law. To do so, scribes likely adopted the prestigious sexagesimal (base-60) mathematical system of their captors, codifying a history that would command respect within a Mesopotamian intellectual context. The presence of these mathematical structures provides strong evidence that these lifespans were integrated into the biblical narrative during or shortly after the Babylonian captivity (c. 586–538 BCE). The following comparison illustrates how the '''Prototype 1''' chronology utilized timespans found in the [[w:Sumerian_King_List|Sumerian King List (SKL)]]. The longest lifespans in this chronology—960 and 900 years—are figures well-represented as Sumerian kingship durations. * '''16 ''šūši'' (960 years)''' ** SKL: [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]] ** '''Prototype 1''': Adam, Jared, Methuselah, Noah * '''15 ''šūši'' (900 years)''' ** SKL: [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]] ** '''Prototype 1''': Seth, Enosh, Kenan, Mahalalel * '''10 ''šūši'' (600 years)''' ** SKL: [[w:Atab|Atab]] ** '''Prototype 1''': Shem * '''7 ''šūši'' (420 years)''' ** SKL: [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]] ** '''Prototype 1''': Arpachshad, Shelah The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically calibrated to synchronize with them. Notably, in his work ''[[w:Antiquities_of_the_Jews|Antiquities of the Jews]]'', [[w:Josephus|Flavius Josephus]] characterizes several pre-flood (antediluvian) patriarchs as having explicit leadership or ruling roles, further mirroring the regal nature of the Sumerian list. ==The Grouping of Adam== The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], the word "Adapa" is linked to the first sage and associated with the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]]. ==== Conclusion ==== The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical "surprise" of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history. ==The Universal Flood== In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative. It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with "all the craftsmen." Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark. Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological "corrections" were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 <br/><small>(460)<br/>(1422)</small> | 847 <br/><small>(460)<br/>(1307)</small> | 962 <br/><small>(460)<br/>(1422)</small> | colspan="2" | 962 <br/><small>(960)<br/>(1922)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 <br/> <small>(587)<br/>(1556)</small> | 720 <br/> <small>(587)<br/>(1307)</small> | 969 <br/> <small>(687)<br/>(1656)</small> | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 <br/><small>(1287)<br/>(2256)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 <br/> <small>(654)<br/>(1437)</small> | 653 <br/> <small>(654)<br/>(1307)</small> | 777 <br/> <small>(874)<br/>(1651)</small> | 753 <br/> <small>(1454)<br/>(2207)</small> | 723 <br/> <small>(1454)<br/>(2177)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(707)<br/>(1657)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(1056)<br/>(2006)</small> | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(1642)<br/>(2592)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | The Flood | colspan="2" | <small>(1307)</small> | <small>(1656)</small> | colspan="2" |<small>(2242)</small> |} === Samaritan Adjustments === As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge. === Masoretic Adjustments === The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged. These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="2" style="background-color:#e8e8e8;" | 130 | colspan="2" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="2" style="background-color:#e8e8e8;" | 105 | colspan="2" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="2" style="background-color:#e8e8e8;" | 90 | colspan="2" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="2" style="background-color:#e8e8e8;" | 70 | colspan="2" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="2" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="3" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="2" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" | 67 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="2" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="2" style="background-color:#f9f9f9;" | 188 |} === Septuagint Adjustments === In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX): <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred and fifty significantly dilutes the miraculous nature of Isaac's birth. {{RoundBoxTop}} ====The "Whoops Theory": A Digression==== Paul D.’s "Whoops Theory" suggests the LXX editor added 100 years to the age at which each patriarch begot his son, intending to "fix" the timeline, but somehow failed in the case of Methuselah. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. Writing in the late 3rd century BC (c. 221 BC), '''Demetrius the Chronographer''' stands as the earliest known witness to biblical chronological calculations. Despite the fragmentary nature of his work, his data remains pivotal; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood, a total that inherently supports a longer chronology where Methuselah’s fathering age is '''187''' years rather than '''167'''. In the original article's comments, a debate surfaced regarding this longer chronology and the '''187''' year fathering age for Methuselah. Paul D. defends his "Whoops Theory" by systematically challenging the validity of early witnesses, particularly those that support the longer timeline: * '''Josephus:''' Characterized as being dependent on the Masoretic tradition rather than an independent witness. * '''Pseudo-Philo:''' Dismissed entirely due to severe textual corruption (described as "a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through a later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional two years whose precise placement remains unknown. * '''Codex Alexandrinus:''' Identified as the lone legitimate witness to the 187-year fathering age. The dismissal of Julius Africanus due to his survival through an intermediary, or the disqualification of Demetrius based on a two-year uncertainty, is arguably overstated. As shown in the comparative tables below, there is remarkably little variation in the "LONG CHRONOLOGY" begettal ages. They are identical for Adam through Enoch, with some variation in the last few patriarchs prior to the flood. The below reconstructed Demetrius chronology employs a plausible explanation for the 2-year discrepancy: the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can account for such variances without necessitating the rejection of the witness. Given that Demetrius operates within the longest known chronological framework, the suggestion that he utilized longer begettal ages for earlier patriarchs while failing to apply the correction for Methuselah—the very figure requiring it most in such an expanded timeline—is statistically and logically improbable. {{RoundBoxBottom}} {{RoundBoxTop}} '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in patriarchs surviving beyond the date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | rowspan="2" colspan="1" style="background-color:#e8e8e8;" | 602 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="3" style="background-color:#e8e8e8;" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | TOTAL | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |} {{RoundBoxBottom}} === Flood Adjustment Summary === In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. * In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity. * The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. * The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old. == Additional Textual Evidence == Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all textual traditions. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors, as shown in the following tables. (The '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.) === Lifespan Adjustments by Individual Patriarch === {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Individual Patriarch Lifespans) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 800}} <br/>= 930 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|230 + 700}} <br/>= 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|105 + 807}} <br/>= 912 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|205 + 707}} <br/>= 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|90 + 815}} <br/>= 905 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|190 + 715}} <br/>= 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 840}} <br/>= 910 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|170 + 740}} <br/>= 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|65 + 830}} <br/>= 895 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|165 + 730}} <br/>= 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|62 + 900}} <br/>= 962 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|162 + 800}} <br/>= 962 | {{nowrap|62 + 785}} <br/>= 847 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|162 + 800}} <br/>= 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|65 + 300}} <br/>= 365 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|165 + 200}} <br/>= 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|67 + 902}} <br/>= 969 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|187 + 782}} <br/>= 969 | {{nowrap|67 + 653}} <br/>= 720 | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|167 + 802}} <br/>= 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|53 + 730}} <br/>= 783 | {{nowrap|182 + 595}} <br/>= 777 | {{nowrap|53 + 600}} <br/>= 653 | {{nowrap|188 + 565}} <br/>= 753 | {{nowrap|188 + 535}} <br/>= 723 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|502 + 448}} <br/>= 950 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|500 + 450}} <br/>= 950 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|502 + 448}} <br/>= 950 | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|500 + 450}} <br/>= 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="5" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|100 + 500}} <br/>= 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 438 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|35 + 403}} <br/>= 438 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|135 + 303}} <br/>= 438 | {{nowrap|135 + 400}} <br/>= 535 | {{nowrap|135 + 403}} <br/>= 538 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | 460 | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 433 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 403}} <br/>= 433 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 303}} <br/>= 433 | {{nowrap|130 + 330}} <br/>= 460 | {{nowrap|130 + 406}} <br/>= 536 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 464 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|34 + 430}} <br/>= 464 | colspan="2" | {{nowrap|134 + 270}} <br/>= 404 | {{nowrap|134 + 433}} <br/>= 567 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 209}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 109}} <br/>= 239 | colspan="2" | {{nowrap|130 + 209}} <br/>= 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|32 + 207}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|132 + 107}} <br/>= 239 | {{nowrap|132 + 207}} <br/>= 339 | {{nowrap|135 + 207}} <br/>= 342 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 230 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 200}} <br/>= 230 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 100}} <br/>= 230 | colspan="2" | {{nowrap|130 + 200}} <br/>= 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 148 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|29 + 119}} <br/>= 148 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|79 + 69}} <br/>= 148 | {{nowrap|179 + 125}} <br/>= 304 | {{nowrap|79 + 119}} <br/>= 198 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 135}} <br/>= 205 | {{nowrap|70 + 75}} <br/>= 145 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 135}} <br/>= 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|100 + 75}} <br/>= 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|60 + 120}} <br/>= 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob..Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|345 + 323}} <br/>= 668 | colspan="1" | {{nowrap|560 + 114}} <br/>= 674 | colspan="1" | {{nowrap|345 + 328}} <br/>= 673 | colspan="2" | {{nowrap|345 + 324}} <br/>= 669 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | colspan="1" | 13,551 | colspan="1" | 13,200 |} === Samaritan Adjustment Details === As noted previously, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the "Group 1" and "Group 2" patriarchs. Specifically, this balance was achieved through the following adjustments: * '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each). * '''Amram's''' lifespan was increased by five years. This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing. === Masoretic Adjustment Details === In the 2017 article, "[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs]," Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late "tweak" in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the "harder reading is stronger") suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate "balancing" of total chronological years. === Armenian Eusebius Adjustments === Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system. Specifically, the following adjustments appear to have occurred for Group 2 patriarchs: * '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years. * '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years. * '''Nahor''' had his lifespan increased by 50 years. * '''Amram''' had his lifespan increased by 1 year. === Lifespan Adjustments by Group === {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 1: Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 3: Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 2: Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | 13,551 | 13,200 |} * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units). The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''. = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the Black Desert of Jordan. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7¹ = 7 years * '''Jubilee of Years:''' 7² = 49 years * '''Week of Jubilees:''' 7³ = 343 years * '''Jubilee of Jubilees:''' 7⁴ = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. {{RoundBoxTop}} ==== The Birth of Shem (A Digression) ==== Were Noah's sons born when Noah was 500 or 502? While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. {{RoundBoxBottom}} == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block"> \begin{aligned} \frac{7^4 - 1}{2} + 3^2 &= 1200 + 9 \\ &= 1209 \end{aligned} </math> ** Post-Flood Patriarch years: *:<math display="block"> \begin{aligned} \frac{7^4 + 1}{2} + (7^2 - 3^2) &= 1201 + 40 \\ &= 1241 \end{aligned} </math> ** Total Years: *:<math display="block"> \begin{aligned} 7^4 + 7^2 &= 2401 + 49 \\ &= 2450 \end{aligned} </math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block"> \begin{aligned} 7^4 + 7^2 &= 2401 + 49 \\ &= 2450 \end{aligned} </math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block"> \begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 \\ &= 2401 + 343 + 49 + 7 \\ &= 2800 \\ \text{(Base 40): } & 70 \times 40 \\ &= 2800 \end{aligned} </math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == The Masoretic text Variation == [[File:Schematic_Diagram_Masoretic_Text_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Masoretic Text'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] === Mathematical Structure of the Masoretic Chronology === The above diagram maps the Masoretic chronology using the identical grid framework applied to the Samaritan Pentateuch. Red circles indicate specific chronological points where the Masoretic timeline diverges from the Samaritan tradition. Key textual and numerical variations include: ==== The Universal Flood Adjustment (+349 yrs) ==== The Masoretic chronology postpones the birth of Noah and the onset of the Deluge by '''349 years''' relative to the Samaritan tradition. As outlined in the above "Masoretic Adjustments" section, this structural shift ensures that Noah's antediluvian ancestors do not survive past the date of the universal flood. This timeline was achieved by adding 100 years to Jared's paternal age, 120 years to Methuselah's, and 129 years to Lamech's. Consequently, these calculations place Methuselah's death precisely in the year of the Flood and Lamech's death five years prior. ==== The Smoothing of Noah's Life (-2 yrs) ==== The Masoretic chronology reduces Noah's age at the birth of Shem from 502 years (as found in the Samaritan Pentateuch) to 500 years. As discussed in the "Lectio Difficilior Potior" and "The Birth of Shem (A Digression)" sections, this change represents "scribal smoothing." This editorial process rounded major milestones in Noah's life to the nearest century: his sons are born in his 500th year, the Deluge occurs in his 600th year, and his father Lamech dies in his 700th year. ==== Post-Flood Patriarchal Lifespans (-650 yrs) ==== Seven post-flood patriarchs (Arpachshad through Nahor) exhibit reduced ages at the birth of their firstborn sons within the Masoretic chronology, resulting in a cumulative deficit of 650 years compared to the Samaritan Pentateuch. Specifically, the paternal ages of Arpachshad, Shelah, Eber, Peleg, Reu, and Serug are each lower by exactly 100 years, while Nahor's age is reduced by 50 years. Given the uniform, rounded values of these variations, a direct dependency exists; either the Masoretic chronologist systematically deflated these ages, or the Samaritan chronologist systematically inflated them. The Masoretic use of young fathering ages for these specific patriarchs may preserve an older tradition. The narrative theme of Abraham fathering Isaac at the age of 100 serves as a central literary and mathematical motif that likely predates the Babylonian exile. Consequently, the lower paternal ages in the Masoretic lineage may intentionally highlight this milestone, ensuring that a fatherhood age of 100 remains an extraordinary, miraculous exception within the broader genealogical timeline. Under this proposed framework, Arpachshad (or Cainan) originally functioned not as a standard descendant of Shem, but as the progenitor of a distinct nine-generation patriarchal line culminating in the miraculous birth of Isaac. During the Babylonian exile, the Priestly source ("P") may have adapted this older Abrahamic lineage, appending earlier primeval generations to mirror Mesopotamian creation and flood epics. Consequently, Mesopotamian archetypes—such as Adapa, Enmeduranki, and Utnapishtim—were synthesized with preexisting Hebrew traditions to form the composite figures of Adam, Enoch, and Noah. These composite primeval figures were subsequently integrated into a single, seamless genealogical chain spanning from creation to the foundations of Israel. ==== The Final Masoretic Adjustment (+215 yrs) ==== As will be explained later in this resource, it is proposed that a final 215 years was added to the Masoretic chronology to arrive at the 2666-year total. The Samaritan Pentateuch, the Septuagint, and the Book of Jubilees all claim that the Patriarchs were in Canaan for 215 years, followed by 215 years of Israelite slavery in Egypt. The Masoretic text increases the period of Egyptian slavery by 215 years, for a total of 430 years of slavery. This increase to 430 years is difficult to reconcile with the Moses narrative, which claims that Moses was the great-grandson of Levi on the paternal line and the grandson on the maternal line. Levi entered Egypt as an adult and a free man. Moses led the children of Israel out of Egypt at eighty years old. For Moses to be a maternal grandson, his mother would have been hundreds of years old when she gave birth to him and his older siblings, Aaron and Miriam. ==== The 2800-Year to 2666-Year Transition ==== Within the Masoretic chronology, the Exodus is situated inside the third 800-year epoch, occurring precisely in the year 2666 after creation (Anno Mundi). This placement represents a deliberate scribal design choice rather than an organic chronological coincidence. <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block"> \begin{aligned} 7^4 + 7^2 &= 2401 + 49 \\ &= 2450 \end{aligned} </math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block"> \begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 \\ &= 2401 + 343 + 49 + 7 \\ &= 2800 \\ \text{(Base 40): } & 70 \times 40 \\ &= 2800 \end{aligned} </math> * '''Masoretic Text (Adam to Exodus):''' :<math display="block"> (2800 - 46) + (349) - (2) - (650) + (215) = 2666</math> </div> As detailed in "The 350-Year Symmetrical Extension," the timeline found in the Book of Jubilees likely served as the foundational model, positioning Shem at the exact midpoint of its 2,450-year chronology. The Samaritan Pentateuch appears to be a direct derivative of this Jubilees framework. Scribes systematically appended exactly 350 years to the post-Flood era, shifting the completion of the Conquest to a precise macro-structural target of 2,800 years after creation. Envisioning the process that transitioned from the 2,800-year Samaritan chronology to the Masoretic framework presents a distinct mathematical challenge. Tracking the adjustments step-by-step from the Samaritan baseline reveals a precise transition: * '''Samaritan Baseline (Conquest Completion):''' 2,800 years * '''Minus Wandering and Conquest Era:''' −46 years (2,754) * '''Plus Masoretic Flood Adjustment:''' +349 years (3,103) * '''Minus Noah's Life Smoothing:''' −2 years (3,101) * '''Minus Post-Flood Patriarchal Reductions:''' −650 years (2,451) * '''The Final Masoretic Adjustment:''' +215 years (2,666) This calculation yields exactly 2,666 years. Because individual components were dictated by rigid external theological and narrative constraints, it is surprising that the final adjustment yields exactly 2,666 years, but it also remains difficult to isolate where scribes might have executed fine-tuning to bring about this exact value and bridge the 2,800-year and 2,666-year macro-structures. === Mathematical Structure of the Septuagint Chronology === As was explained above in "Septuagint Adjustments", the LXX’s editor methodically added 100 years to each pre-flood patriarch. The purpose of this adjustment was to ensure that Noah would be the exclusive survivor of the flood. As shown in the tables below, post-flood "Long Chronology" variations are seen to be either identical to or derived from "Short Chronology" values. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#f9f9f9;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#f9f9f9;" | 67 | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#f9f9f9;" | 167 | colspan="3" style="background-color:#f9f9f9;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#f9f9f9;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#f9f9f9;" | 53 | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | rowspan="2" style="background-color:#e8e8e8;" | 602 | rowspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 602 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="3" style="background-color:#e8e8e8;" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1309 | colspan="1" | 1656 | colspan="1" | 1309 | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1309 | colspan="1" | 1656 | colspan="1" | 1309 | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. The [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70^th generation to the 70^th Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology shown above, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. [[Category:Religion]] === Mythology === In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|"Uruk List of Kings and Sages"]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the "sages" eventually transition into human scholars. A tentative translation reads: *During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]]. *During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]]. *During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage. *During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage. *During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage. *During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage. *During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage. *After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives. 6oi67wkn1b8j4bxiwnojyb98q50qnd2 2844607 2844606 2026-11-05T11:34:43Z Unitfreak 695864 /* Narrative Similarities */ 2844607 wikitext text/x-wiki {| class=table style="width:100%;" |- | {{Original research}} |- | {{RoundBoxTop}} This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional context and demonstrates that the underlying numerical data is even more robust and systematic than initially identified. This article was researched and written by a human author, with editing assistance provided by [https://gemini.google.com Google "Gemini, your personal AI assistant"]{{RoundBoxBottom}} |} = The Sumerian Origins of Genesis 5 = The early history in Genesis 5 shares deep structural, thematic, and chronological ties with ancient Mesopotamian writing traditions. The closest parallel to the biblical figures who lived before the flood is the Sumerian King List (SKL). This ancient text details a succession of long-lived rulers who reigned right up until a cataclysmic flood. The table below shows the connections between Genesis 5 and the Sumerian King List, with directly matching details highlighted in blue. {| class="nowraplinks" cellspacing="0" cellpadding="3" rules="all" style="background:#fbfbfb; border: 1px solid #ccc; font-size:90%; empty-cells:show; border-collapse:collapse; text-align:left;" |- style="background:#F6E6AE; font-weight:bold; text-align:center;" ! colspan="2" | Sumerian King List ! Hebrew Bible |- style="text-align:center; font-style:italic;" ! colspan="2" style="background:#fffcf2;" | Archaeological Timeline:<br><span style="background:#d6e2ff; font-style:normal; font-weight:bold; padding:0 4px;">2400</span> Years Total ! style="background:#fffcf2;" | Demetrius Timeline:<br><span style="background:#d6e2ff; font-style:normal; font-weight:bold; padding:0 4px;">2264</span> Years Total |- style="background:#F6E6AE; font-weight:bold; text-align:center;" ! City ! Ruler (Sage) ! Patriarch |- | [[w:Eridu|Eridu]] | [[w:Alulim|Alulim]] <span style="background: #d6e2ff; padding: 0 2px;">([[w:Adapa|Adapa]])</span> <br /> [[w:Alalngar|Alalngar]] | <span style="background: #d6e2ff; padding: 0 2px;">[[w:Adam|Adam]]</span> <br /> [[w:Seth|Seth]] |- | [[w:Bad-tibira|Bad-tibira]] | [[w:En-men-lu-ana|En-men-lu-ana]] <br /> [[w:En-men-gal-ana|En-men-gal-ana]] <br /> [[w:Dumuzid|Dumuzid]] | [[w:Enos|Enos]] <br /> [[w:Kenan|Kenan]] <br /> [[w:Mahalalel|Mahalalel]] |- | [[w:Larak (Sumer)|Larak]] | [[w:En-sipad-zid-ana|En-sipad-zid-ana]] | [[w:Jared (biblical figure)|Jared]] |- | [[w:Sippar|Sippar]] | <span style="background: #d6e2ff; padding: 0 2px;">[[w:En-men-dur-ana|En-men-dur-ana]]</span> | <span style="background: #d6e2ff; padding: 0 2px;">[[w:Enoch|Enoch]]</span> |- | [[w:Shuruppak|Shuruppak]] | [[w:Ubara-Tutu|Ubara-Tutu]] <br /> <span style="background: #d6e2ff; padding: 0 2px;">[[w:Utnapishtim|Utnapishtim]]</span> | [[w:Methuselah|Methuselah]] <br /> [[w:Lamech (father of Noah)|Lamech]] <br /> <span style="background: #d6e2ff; padding: 0 2px;">[[w:Noah|Noah]]</span> |} == Narrative Similarities == The biblical accounts seem to build on, or develop alongside, the legendary roles given to Sumerian rulers and their wise advisors, known as apkallu (sages). Key thematic connections include: *[https://www.worldhistory.org/article/216/the-myth-of-adapa/ Adam as Adapa] : In the Adapa narrative, Ea (god of wisdom) warns Adapa to refuse any food or drink offered in heaven but to accept clothing and oil. Anu (the sky god) offers Adapa food and drink, followed by clothing and oil. When Adapa refuses the food, Anu chides him for missing an opportunity for immortality. Similarly, in the Genesis account, Yahweh Elohim permits Adam and Eve to eat from any tree in the divine garden except the tree of knowledge. A competing figure, the serpent, advises them to eat from it anyway. After they do, Yahweh returns, chastises them for disobedience, and provides garments of skin. Key parallels between the accounts include: A potential similarity and shared root in the names Adapa and Adam. Both narratives feature a test involving the consumption of supposedly deadly food. Both figures are offered clothing and summoned before a deity to answer for their decisions. *[https://www.marquette.edu/maqom/Chapter%201.pdf Enoch as Enmeduranki]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew Book of Enoch describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text Pirišti Šamê u Erṣeti (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos. *[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven. == Chronological Similarities == [[File:Antediluvian Sumerian King List Cities and Biblical Rivers.png|thumb|400px|Locations of four of the five pre-Flood cities named in the Sumerian King List have been identified through excavated cuneiform inscriptions. These cities existed in close proximity to the Tigris and Euphrates rivers. Eridu was a coastal city when first founded around 5300 BC. Over the millennia, however, massive deposits of alluvial silt from four river systems filled the ancient bays, gradually pushing the shoreline southeastward away from Eridu. There are four rivers named in the biblical Garden of Eden story (Genesis 2:10–14): the Tigris, Euphrates, Gihon, and Pishon. The Gihon and Pishon are frequently associated with the Karun River in western Iran and the Wadi al-Batin (a now-dry riverbed) in the Arabian Peninsula. The blue oval on the map above represents the 5300 BC coastline, with arrows pointing to the mouths of the four Eden rivers.]] In Mesopotamian mythology, Eridu was the oldest city, the home of the wisdom god Enki, and the first site where divine kingship descended. While many Sumerian texts omit the pre-Flood era entirely, in narratives that do include a pre-Flood prologue, Eridu is consistently identified as the first city and Šuruppak is frequently identified as the fifth and final city prior to the Flood. One English translation of the Sumerian King List states it as follows: [https://www.livius.org/sources/content/anet/266-the-sumerian-king-list/ "After the kingship descended from heaven, the kingship was in Eridu ... Bad-tibira fell and the kingship was taken to Larak ... Sippar fell and the kingship was taken to Šuruppak ... Five cities; eight kings ruled for 385,200 years. Then the Flood swept over."] A related text, the [[w:Eridu Genesis|Eridu Genesis]], preserves a parallel tradition with Eridu as the first city and Šuruppak as the fifth: [https://www.livius.org/sources/content/oriental-varia/eridu-genesis/ "The firstling of the cities, Eridu, she gave to the leader Nudimmud, the second, Bad-Tibira, she gave to the Prince and the Sacred One, the third, Larak, she gave to Pahilsag, the fourth, Sippar, she gave to the gallant Utu, the fifth, Šuruppak, she gave to Ansud."] Locations of four of the five pre-Flood Sumerian cities have been identified archeologically. Eridu sits on a hill inside a basin roughly {{convert|15|mi}} long and {{convert|20|ft}} deep, separated from the Euphrates by a sandstone ridge called the ''Hazem''. In the 3rd Millennium BC, before the river shifted course, a canal connected Eridu to the Euphrates. Low tells with 2nd Millennium BC pottery and later burials mark the canal's path. ====Prehistoric Temple(s)==== [[File:Ea (Babilonian) - EnKi (Sumerian).jpg|thumb|Detail of the Adda Seal, an ancient Akkadian cylinder seal (circa 2300 BCE) depicting Enki with water streams coming from his shoulders.|270x270px]] The urban nucleus of Eridu was Enki's temple, called the ''House of the Aquifer'' in the oldest Sumerian texts. In later texts, this temple was called the ''House of the Waters''. [[w:Enki|Enki]] was the Mesopotamian god of subterranean waters, specifically reigning over the primeval underground ocean known as the [[w:Abzu|Abzu]]. In ancient art, he was frequently represented with water streams flowing directly from his body, which were often filled with fish swimming through the currents. The biblical [[w:Garden of Eden|Garden of Eden]] is presented as a well-watered garden with rivers running out of it. The Sumerian association of Eridu with subterranean waters seems reflected in the biblical garden story. [[File:Eridu Level Sequence A.jpg|frame|Prehistoric sequence of buildings excavated at Eridu.]] A prehistoric sequence of buildings excavated at Eridu during the 1946 to 1949 campaigns seems to substantiate Eridu's status as the first city with the earliest temple. Consisting of eighteen occupational levels, this Eridu sequence of "Palace-like" or "Temple-Like" buildings appears to be the first occurrence of such structures in the region<ref>{{cite book| last= Leick| first= Gwendolyn| title= Mesopotamia, The invention of the city| place= London| publisher= Penguin Books| year= 2002}}</ref>. In [[w:Archaeology|archaeology]], a '''[[w:Tell (archaeology)|tell]]''' is a mound consisting of the accumulated and stratified debris of a succession of consecutive settlements at the same site, the refuse of generations of people who built and inhabited them, alongside natural sediment. The most recent layers of a tell are nearest to the surface. As archaeologists dig into the tell, they encounter successively older layers of civilization, a recording of history called [[w:Stratigraphy|stratigraphy]]. The following table compares the stratigraphy found during the Eridu 1946–1949 campaigns with the stratigraphy found during the 1931 excavation of Shuruppak (Tell Fara). The table is organized according to the order in which layers are uncovered, with the youngest layers in the first rows of the table and successively older layers in subsequent rows. As shown below, the historic timeline separating the first construction at Eridu (c. 5300 BC) from the cultural break of the Shuruppak flood (c. 2900 BC) spans a real-world duration of approximately 2,400 years. {| class="wikitable" ! Historical Period & Dating ! Eridu Temple Stratigraphy ! Shuruppak Stratigraphy |- | ''' Dynastic Periods ''' ''(c. 2004-2900 BC)'' | '''Ur III Ziggurat Layer''' <br/> "In the ruins of the ziggurat we found numerous bricks of" [[w:Amar-Sin|Amar-Sin (c. 2046–2037 BC)]], "three bricks of" [[w:Ur-Nammu|Ur-Nammu (2112–2094 BC)]] "one brick of" [[w:Nur-Adad|Nur-Adad (c. 1866-1850 BC)]] <br/> [https://www.scribd.com/document/897929584/Eridu-1981-Fuad-Safar-Seton-Lloyd "It seems that Ur-Nammu began the building of this ziggurat, but the bulk of the task was left to his grandson Amar-Sin to accomplish."] | '''Fara II & III Layer''' ([[w:Shu-Sin|Shu-Sin]] inscription (c. 2037 – 2028 BC), Sumerian seal cylinders, pottery, skeletons, figurines, Globular Stone Jar) |- | style="background-color:#ffdddd;" | '''c. 2900 BC''' | style="text-align: center;" | – | style="background-color:#ffdddd;" | '''The Shuruppak Flood Layer''' [https://www.penn.museum/sites/journal/9356/ "We found an alluvial layer at a depth between four and five metres below the mound surface."] [https://www.penn.museum/sites/journal/9356/ "Only a few objects of this period, namely one stamp seal and some sherds “floated” in the refuse above. In this respect an absolute culture break took place."] |- | '''[[w:Jemdet Nasr period|Jemdet Nasr]]''' ''(c. 2900-3100 BC)'' | style="text-align: center;" | – | '''Fara I Layer''' (Scaraboid Stamp Seals, Polychrome Jar, Animal Patterns on Potsherds, Copper Blade) |- | '''[[w:Uruk period|Uruk]]''' ''(c. 3100-3800 BC)'' | '''Temple Layers I–V''' <br/> Temple I & II: [https://www.scribd.com/document/897929584/Eridu-1981-Fuad-Safar-Seton-Lloyd "All that is left to us are its foundations and retaining-walls, while even of these a large portion are buried beneath the massive masonry of the Third Dynasty Ziggurat.] <br/> Temple III: Small bricks of reddish clay. <br/> Temple IV: Medium sized bricks of greenish clay. <br/> Temple V: Large bricks of light-colored clay with dark joints. <br/> [https://www.scribd.com/document/897929584/Eridu-1981-Fuad-Safar-Seton-Lloyd "Since Temple VI was now known to belong to a very late, if not the final stage of the Al-Ubaid period, Temples III, IV and V must be taken to represent a stage intermediate between this and the stone architecture of the Protoliterate age"] | style="text-align: center;" | – |- | '''Ubaid 1-4''' ''(c. 3800-5300 BC)'' | '''Temples VI–XVIII''' <br/> Temple X: "At Level X the temple appeared to have been rebuilt with only minor changes in the plan." <br/> Temple XI: "provides the first instance of a temple building being deliberately raised upon a platform." <br/> Temple XII & XIII: "there were two successive occupation-levels at which no traces of actual buildings were to be found" <br/> Temple XVIII "The first traces of occupation at Eridu occurred at a level corresponding to 11.70 meters beneath datum" ... "and, directly above, at Level XVII, were the remains of the earliest recognizable building" | style="text-align: center;" | – |- | style="background-color:#ffdddd;" | '''c. 5300 BC''' | style="background-color:#ffdddd;" | '''First Construction''' | style="text-align: center;" | – |} === Pre-flood Bible Chronology === '''Demetrius the Chronographer''' stands as the earliest known witness to biblical chronological calculations. Despite the fragmentary nature of his work, his data remains pivotal. Writing from [https://commons.wikimedia.org/wiki/File:Ptolemaic_Alexandria_-_en.svg Alexandria in the late 3rd century BC (c. 221 BC)], Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood of Noah. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY |- ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | rowspan="11" style="background-color:#e8e8e8;" | — | colspan="5" style="background-color:#e8e8e8;" | 230 | colspan="3" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="5" style="background-color:#e8e8e8;" | 205 | colspan="3" style="background-color:#e8e8e8;" | 105 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="5" style="background-color:#e8e8e8;" | 190 | colspan="3" style="background-color:#e8e8e8;" | 90 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="5" style="background-color:#e8e8e8;" | 170 | colspan="3" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="5" style="background-color:#e8e8e8;" | 165 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="1" style="background-color:#e8e8e8;" | 66 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="6" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 61 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="5" style="background-color:#e8e8e8;" | 165 | colspan="3" style="background-color:#e8e8e8;" | 65 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#f9f9f9;" | 167 | colspan="3" style="background-color:#f9f9f9;" | 167 / 187 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#f9f9f9;" | 67 | colspan="1" style="background-color:#f9f9f9;" | 65 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="4" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#f9f9f9;" | 53 | colspan="1" style="background-color:#f9f9f9;" | 55 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="3" style="background-color:#e8e8e8;" | Varied | rowspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 602 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied | colspan="1" | 1656 | colspan="1" | 1309 | colspan="1" | 1309 |} The Septuagint chronology, as witnessed by Demetrius at 2,264 years, mirrors this archaeological reality within a remarkably tight margin of error (less than 6%). This broad-stroke alignment strongly indicates that while the fine internal details were arranged to fit sacred numerical cycles, the overarching framework of the Septuagint preserved an authentic, historically grounded memory of the Fertile Crescent's true antiquity—a memory that was severely contracted in alternative textual traditions. The structural overlap between the Mesopotamian and biblical traditions becomes most compelling when evaluating the total duration of the antediluvian era. While the fine details and specific individual lifespans in both the Septuagint and the Sumerian King List are heavily calculated using stylized, sacred, or mathematically perfect numbers, the macro-span of the Septuagint timeline displays an uncanny alignment with real-world ancient history. This macro-alignment is particularly remarkable when contrasted against the Sumerian King List itself. In the Mesopotamian scribal tradition, the antediluvian era spans astronomical lengths of time—typically totaling over 240,000 years across the reigns of the first eight pre-flood kings. These numbers are clearly symbolic or mathematical inflations that completely diverge from physical archaeological realities. Rather than duplicating these vast Near Eastern timelines or severely compressing the era, the Septuagint tradition preserved a highly realistic historical interval. It is highly significant that the Septuagint chronology, as witnessed by Demetrius, aligns so closely with modern archaeological horizons instead of the exaggerated spans of the SKL. Because an unguided mathematical arrangement of sacred numbers would be statistically unlikely to land so close to actual physical history, this uncanny broad-stroke accuracy hints at two intriguing possibilities: it was either a case of monumental cosmic luck, or the Alexandrian redactors possessed ancient, pristine Near Eastern source records that have since been entirely lost to history. This total of 2,264 years stands in stark contrast to the significantly compressed timelines preserved in the Masoretic Text (1,656 years) and the Samaritan Pentateuch (1,307 years). When evaluating these numbers against physical history, the Septuagint's macro-chronology reveals a striking real-world correlation. = ''Arichat Yamim'' (Long Life) = Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101). This phenomenon is difficult to explain, as no known ancient number system features "101" as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle. In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a "perfect" Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows: *:<math display="block"> \begin{aligned} \frac{7\,\text{šar}}{2} &= \frac{420\,\text{šūši}}{2} \\ &= 210\,\,\text{šūši} \\ &= \left(210 \times 60 \,\text{years} \right) \\ &= 12,600 \, \text{years} \end{aligned} </math> This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60). ==== Prototype 1: Initial "Mesopotamian" Allocation ==== ---- <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> The initial "PT1" framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''): * '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49). * '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. * '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920). </div> ---- ==== Prototype 2: Refined "Jubilee" Allocation ==== ---- <div style="background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;"> Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the "Remainder" to each of the two primary groups. This resulted in the "PT2" figures as follows: * '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees). * '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949). </div> ---- '''Table Legend:''' * <span style="width:100%; color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in a patriarch surviving beyond the date of the Flood. {| class="wikitable" style="font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Prototype Chronologies (Age at death) |- ! colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="3" style="background-color:#e0f2f1; border-bottom:2px solid #009688;" | PROTOTYPE 1<br/>(PT1) ! colspan="3" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PROTOTYPE 2<br/>(PT2) |- | rowspan="6" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 1}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|82 šūši}}<br/><small>(4920)</small></div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|45 šūši}}<br/><small>(2700)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2727</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|37 šūši}}<br/><small>(2220)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 15 <small>(900)</small> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2222</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 6 <small>(360)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 365 |- | rowspan="3" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 3}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|46 šūši}}<br/><small>(2760)</small></div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|32 šūši}}<br/><small>(1920)</small></div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | rowspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2702</div> | rowspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1919</div> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 16 <small>(960)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 14 <small>(840)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 14 <small>(840)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 783 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 |- | rowspan="18" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|Group 2}}</div> | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|82 šūši}}<br/><small>(4920)</small></div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|40 šūši}}<br/><small>(2400)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 16 <small>(960)</small> | rowspan="18" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">4949</div> | rowspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">2401</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 10 <small>(600)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 <small>(420)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 438 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 7 <small>(420)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 433 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|25 šūši}}<br/><small>(1500)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 8 <small>(480)</small> | rowspan="6" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1525</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 464 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 <small>(240)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 4 <small>(240)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 239 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 230 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 148 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">{{nowrap|17 šūši}}<br/><small>(1020)</small></div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | rowspan="7" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | <div style="display:inline-block; transform:rotate(270deg);">1023</div> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 3 <small>(180)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 147 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Levi | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 137 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kohath | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 133 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Amram | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 131 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 2 <small>(120)</small> | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | 120 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="2" style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="6" | 210 šūši<br/><small>(12,600 years)</small> |} ==Mesopotamian Derived Lifespans== [[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[w:supplementary_hypothesis|supplementary hypothesis]], a popular model of the [[w:composition_of_the_Torah|composition of the Torah]]. The Priestly source is shown as '''P'''.]] Many scholars believe the biblical chronology was developed by an individual or school of scribes known as the [[w:Priestly_source|Priestly source]] (see diagram). Deprived of a physical Temple, the Judean elite focused on transforming oral traditions into a permanent, 'portable' written Law. To do so, scribes likely adopted the prestigious sexagesimal (base-60) mathematical system of their captors, codifying a history that would command respect within a Mesopotamian intellectual context. The presence of these mathematical structures provides strong evidence that these lifespans were integrated into the biblical narrative during or shortly after the Babylonian captivity (c. 586–538 BCE). The following comparison illustrates how the '''Prototype 1''' chronology utilized timespans found in the [[w:Sumerian_King_List|Sumerian King List (SKL)]]. The longest lifespans in this chronology—960 and 900 years—are figures well-represented as Sumerian kingship durations. * '''16 ''šūši'' (960 years)''' ** SKL: [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]] ** '''Prototype 1''': Adam, Jared, Methuselah, Noah * '''15 ''šūši'' (900 years)''' ** SKL: [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]] ** '''Prototype 1''': Seth, Enosh, Kenan, Mahalalel * '''10 ''šūši'' (600 years)''' ** SKL: [[w:Atab|Atab]] ** '''Prototype 1''': Shem * '''7 ''šūši'' (420 years)''' ** SKL: [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]] ** '''Prototype 1''': Arpachshad, Shelah The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically calibrated to synchronize with them. Notably, in his work ''[[w:Antiquities_of_the_Jews|Antiquities of the Jews]]'', [[w:Josephus|Flavius Josephus]] characterizes several pre-flood (antediluvian) patriarchs as having explicit leadership or ruling roles, further mirroring the regal nature of the Sumerian list. ==The Grouping of Adam== The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], the word "Adapa" is linked to the first sage and associated with the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]]. ==== Conclusion ==== The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical "surprise" of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history. ==The Universal Flood== In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative. It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with "all the craftsmen." Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark. Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological "corrections" were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 962 <br/><small>(460)<br/>(1422)</small> | 847 <br/><small>(460)<br/>(1307)</small> | 962 <br/><small>(460)<br/>(1422)</small> | colspan="2" | 962 <br/><small>(960)<br/>(1922)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 <br/> <small>(587)<br/>(1556)</small> | 720 <br/> <small>(587)<br/>(1307)</small> | 969 <br/> <small>(687)<br/>(1656)</small> | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 969 <br/><small>(1287)<br/>(2256)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 783 <br/> <small>(654)<br/>(1437)</small> | 653 <br/> <small>(654)<br/>(1307)</small> | 777 <br/> <small>(874)<br/>(1651)</small> | 753 <br/> <small>(1454)<br/>(2207)</small> | 723 <br/> <small>(1454)<br/>(2177)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(707)<br/>(1657)</small> | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(1056)<br/>(2006)</small> | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 950 <br/> <small>(1642)<br/>(2592)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | The Flood | colspan="2" | <small>(1307)</small> | <small>(1656)</small> | colspan="2" |<small>(2242)</small> |} === Samaritan Adjustments === As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge. === Masoretic Adjustments === The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged. These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="2" style="background-color:#e8e8e8;" | 130 | colspan="2" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="2" style="background-color:#e8e8e8;" | 105 | colspan="2" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="2" style="background-color:#e8e8e8;" | 90 | colspan="2" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="2" style="background-color:#e8e8e8;" | 70 | colspan="2" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="2" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="3" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="2" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" | 67 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="2" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" | 53 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="2" style="background-color:#f9f9f9;" | 188 |} === Septuagint Adjustments === In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX): <blockquote>“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”</blockquote> The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred and fifty significantly dilutes the miraculous nature of Isaac's birth. {{RoundBoxTop}} ====The "Whoops Theory": A Digression==== Paul D.’s "Whoops Theory" suggests the LXX editor added 100 years to the age at which each patriarch begot his son, intending to "fix" the timeline, but somehow failed in the case of Methuselah. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': <blockquote>"Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years."</blockquote> This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development. Writing in the late 3rd century BC (c. 221 BC), '''Demetrius the Chronographer''' stands as the earliest known witness to biblical chronological calculations. Despite the fragmentary nature of his work, his data remains pivotal; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood, a total that inherently supports a longer chronology where Methuselah’s fathering age is '''187''' years rather than '''167'''. In the original article's comments, a debate surfaced regarding this longer chronology and the '''187''' year fathering age for Methuselah. Paul D. defends his "Whoops Theory" by systematically challenging the validity of early witnesses, particularly those that support the longer timeline: * '''Josephus:''' Characterized as being dependent on the Masoretic tradition rather than an independent witness. * '''Pseudo-Philo:''' Dismissed entirely due to severe textual corruption (described as "a real mess"). * '''Julius Africanus:''' Questioned because his records survive only through a later intermediary, Syncellus. * '''Demetrius:''' Rejected as a witness because his chronology contains an additional two years whose precise placement remains unknown. * '''Codex Alexandrinus:''' Identified as the lone legitimate witness to the 187-year fathering age. The dismissal of Julius Africanus due to his survival through an intermediary, or the disqualification of Demetrius based on a two-year uncertainty, is arguably overstated. As shown in the comparative tables below, there is remarkably little variation in the "LONG CHRONOLOGY" begettal ages. They are identical for Adam through Enoch, with some variation in the last few patriarchs prior to the flood. The below reconstructed Demetrius chronology employs a plausible explanation for the 2-year discrepancy: the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can account for such variances without necessitating the rejection of the witness. Given that Demetrius operates within the longest known chronological framework, the suggestion that he utilized longer begettal ages for earlier patriarchs while failing to apply the correction for Methuselah—the very figure requiring it most in such an expanded timeline—is statistically and logically improbable. {{RoundBoxBottom}} {{RoundBoxTop}} '''Table Legend:''' * <span style="color:#b71c1c;">'''Red Cells'''</span> indicate figures that result in patriarchs surviving beyond the date of the Flood. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 | colspan="3" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | rowspan="2" colspan="1" style="background-color:#e8e8e8;" | 602 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="3" style="background-color:#e8e8e8;" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | TOTAL | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |} {{RoundBoxBottom}} === Flood Adjustment Summary === In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. * In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity. * The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. * The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old. == Additional Textual Evidence == Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all textual traditions. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors, as shown in the following tables. (The '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.) === Lifespan Adjustments by Individual Patriarch === {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Individual Patriarch Lifespans) |- ! rowspan = "2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | Patriarch ! colspan="1" rowspan = "2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (Armenian) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 800}} <br/>= 930 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|230 + 700}} <br/>= 930 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|105 + 807}} <br/>= 912 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|205 + 707}} <br/>= 912 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|90 + 815}} <br/>= 905 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|190 + 715}} <br/>= 905 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 840}} <br/>= 910 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|170 + 740}} <br/>= 910 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|65 + 830}} <br/>= 895 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|165 + 730}} <br/>= 895 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|62 + 900}} <br/>= 962 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|162 + 800}} <br/>= 962 | {{nowrap|62 + 785}} <br/>= 847 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|162 + 800}} <br/>= 962 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|65 + 300}} <br/>= 365 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|165 + 200}} <br/>= 365 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|67 + 902}} <br/>= 969 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|187 + 782}} <br/>= 969 | {{nowrap|67 + 653}} <br/>= 720 | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|167 + 802}} <br/>= 969 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|53 + 730}} <br/>= 783 | {{nowrap|182 + 595}} <br/>= 777 | {{nowrap|53 + 600}} <br/>= 653 | {{nowrap|188 + 565}} <br/>= 753 | {{nowrap|188 + 535}} <br/>= 723 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Noah | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|502 + 448}} <br/>= 950 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|500 + 450}} <br/>= 950 | colspan="1" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|502 + 448}} <br/>= 950 | colspan="2" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|500 + 450}} <br/>= 950 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shem | colspan="5" style="background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;" | {{nowrap|100 + 500}} <br/>= 600 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Arpachshad | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 438 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|35 + 403}} <br/>= 438 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|135 + 303}} <br/>= 438 | {{nowrap|135 + 400}} <br/>= 535 | {{nowrap|135 + 403}} <br/>= 538 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Cainan (II) | colspan="3" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | — | 460 | — |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 433 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 403}} <br/>= 433 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 303}} <br/>= 433 | {{nowrap|130 + 330}} <br/>= 460 | {{nowrap|130 + 406}} <br/>= 536 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 464 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|34 + 430}} <br/>= 464 | colspan="2" | {{nowrap|134 + 270}} <br/>= 404 | {{nowrap|134 + 433}} <br/>= 567 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 209}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 109}} <br/>= 239 | colspan="2" | {{nowrap|130 + 209}} <br/>= 339 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|32 + 207}} <br/>= 239 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|132 + 107}} <br/>= 239 | {{nowrap|132 + 207}} <br/>= 339 | {{nowrap|135 + 207}} <br/>= 342 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 230 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|30 + 200}} <br/>= 230 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|130 + 100}} <br/>= 230 | colspan="2" | {{nowrap|130 + 200}} <br/>= 330 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|? + ?}} <br/>= 148 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|29 + 119}} <br/>= 148 | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|79 + 69}} <br/>= 148 | {{nowrap|179 + 125}} <br/>= 304 | {{nowrap|79 + 119}} <br/>= 198 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Terah | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 135}} <br/>= 205 | {{nowrap|70 + 75}} <br/>= 145 | colspan="2" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|70 + 135}} <br/>= 205 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Abraham | colspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|100 + 75}} <br/>= 175 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Isaac | colspan="5" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|60 + 120}} <br/>= 180 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Jacob..Moses | colspan="1" style="background-color:#e8e8e8; font-weight:bold; color:#555;" | {{nowrap|345 + 323}} <br/>= 668 | colspan="1" | {{nowrap|560 + 114}} <br/>= 674 | colspan="1" | {{nowrap|345 + 328}} <br/>= 673 | colspan="2" | {{nowrap|345 + 324}} <br/>= 669 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! style="text-align:left; color:black;" | LIFESPAN<br/>DURATION<br/>SUM | colspan="2" | 12,600 | colspan="1" | 11,991 | colspan="1" | 13,551 | colspan="1" | 13,200 |} === Samaritan Adjustment Details === As noted previously, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the "Group 1" and "Group 2" patriarchs. Specifically, this balance was achieved through the following adjustments: * '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each). * '''Amram's''' lifespan was increased by five years. This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing. === Masoretic Adjustment Details === In the 2017 article, "[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs]," Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition: <blockquote>"The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)"</blockquote> While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late "tweak" in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges: <blockquote>"Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above."</blockquote> ==== ''Lectio Difficilior Potior'' ==== The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the "harder reading is stronger") suggests that scribes tend to simplify or "smooth" texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life. In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges: * '''Year 500 (of Noah):''' Shem is born. * '''Year 600 (of Noah):''' The Flood occurs. * '''Year 700 (of Noah):''' Lamech dies. This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a "compressed chronology" (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate "balancing" of total chronological years. === Armenian Eusebius Adjustments === Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system. Specifically, the following adjustments appear to have occurred for Group 2 patriarchs: * '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years. * '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years. * '''Nahor''' had his lifespan increased by 50 years. * '''Amram''' had his lifespan increased by 1 year. === Lifespan Adjustments by Group === {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center;" |+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum) |- ! rowspan="2" | Patriarch Groups ! rowspan="2" style="background-color:#f3e5f5; border-bottom:2px solid #9c27b0;" | PT2 ! colspan="2" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="2" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! style="background-color:#e3f2fd;" | Masoretic<br/>(MT) ! style="background-color:#e3f2fd;" | Samaritan<br/>(SP) ! style="background-color:#fff3e0;" | Septuagint<br/>(LXX) ! style="background-color:#fff3e0;" | Eusebius<br/>(325 AD) |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 1: Seth to Enoch<br/><small>(6 Patriarchs)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | colspan="2" style="background-color:#f9f9f9; font-weight:bold;" | 4949 |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 3: Methuselah, Lamech, Noah<br/><small>(The Remainder)</small> | style="font-weight:bold; background-color:#f9f9f9;" | 2702 | style="background-color:#f9f9f9;" | 2696<br/><small>(2702 - 6)</small> | style="background-color:#f9f9f9;" | 2323<br/><small>(2702 - 379)</small> | style="background-color:#f9f9f9;" | 2672<br/><small>(2702 - 30)</small> | style="background-color:#f9f9f9;" | 2642<br/><small>(2702 - 60)</small> |- | style="font-weight:bold; text-align:left; background-color:#f9f9f9;" | Group 2: Adam & Shem to Moses<br/><small>(The "Second Half")</small> | style="background-color:#f9f9f9; font-weight:bold;" | 4949 | style="background-color:#f9f9f9;" | 4955<br/><small>(4949 + 6)</small> | style="background-color:#f9f9f9;" | 4834<br/><small>(4949 - 115)</small> | style="background-color:#f9f9f9;" | 5930<br/><small>(4949 + 981)</small> | style="background-color:#f9f9f9;" | 5609<br/><small>(4949 + 660)</small> |- style="background-color:#333; color:white; font-weight:bold; font-size:14px;" ! LIFESPAN DURATION SUM | colspan="2" | 12,600 | 11,991 | 13,551 | 13,200 |} * '''The Masoretic Text (MT):''' This tradition shifted 6 years from the "Remainder" to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block. * '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost. * '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence. * '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units). The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs. The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''. = It All Started With Grain = [[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]] The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the Black Desert of Jordan. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops. The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the "firstfruits of your harvest" (referring to barley) before the Lord: <blockquote>"then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord"</blockquote> To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues: <blockquote>"And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord."</blockquote> [[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]] These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the "better things" and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day. This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: <blockquote>"And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye <small>(likely emmer wheat or spelt)</small> were not smitten: for they were not grown up."</blockquote> This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest). === The Mathematical Structure of Jubilees === The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of "weeks": * '''Week of Years:''' 7¹ = 7 years * '''Jubilee of Years:''' 7² = 49 years * '''Week of Jubilees:''' 7³ = 343 years * '''Jubilee of Jubilees:''' 7⁴ = 2,401 years The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land. [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]] The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested "Jubilee of Jubilees" (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle. * The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year. * The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years. * The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid. * The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs. {{RoundBoxTop}} ==== The Birth of Shem (A Digression) ==== Were Noah's sons born when Noah was 500 or 502? While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that "Noah was 500 years old, and Noah begat Shem, Ham, and Japheth," this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: # Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). # Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10]) '''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples. {{RoundBoxBottom}} == The Mathematical relationship between 40 and 49 == As noted previously, the ''Jubilees'' author envisions early Hebraic history within a "Jubilee of Jubilees" fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows: <math display="block"> \begin{aligned} (7-3)(7+3) &= 7^2 - 3^2 \\ &= 49 - 9 \\ &= 40 \end{aligned} </math> The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40. [[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years): [[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]] <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan)''' ** Pre-Flood Patriarch years: *:<math display="block"> \begin{aligned} \frac{7^4 - 1}{2} + 3^2 &= 1200 + 9 \\ &= 1209 \end{aligned} </math> ** Post-Flood Patriarch years: *:<math display="block"> \begin{aligned} \frac{7^4 + 1}{2} + (7^2 - 3^2) &= 1201 + 40 \\ &= 1241 \end{aligned} </math> ** Total Years: *:<math display="block"> \begin{aligned} 7^4 + 7^2 &= 2401 + 49 \\ &= 2450 \end{aligned} </math> </div> == The Samaritan Pentateuch Connection == Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]] === Determining Chronological Priority === A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 "Jubilee of Jubilees." This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's "pivot" position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment. === The 350-Year Symmetrical Extension === Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a "week of Jubilees" (343 years) plus a "week of years" (7 years). <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block"> \begin{aligned} 7^4 + 7^2 &= 2401 + 49 \\ &= 2450 \end{aligned} </math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block"> \begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 \\ &= 2401 + 343 + 49 + 7 \\ &= 2800 \\ \text{(Base 40): } & 70 \times 40 \\ &= 2800 \end{aligned} </math> </div> === Mathematical Structure of the Early Samaritan Chronology === To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each): * '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''. * '''The second cluster''' represents a second millennium. * '''The final set''' contains 20 blocks (4x5), representing '''800 years'''. Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record. [[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] == The Masoretic text Variation == [[File:Schematic_Diagram_Masoretic_Text_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Masoretic Text'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]] === Mathematical Structure of the Masoretic Chronology === The above diagram maps the Masoretic chronology using the identical grid framework applied to the Samaritan Pentateuch. Red circles indicate specific chronological points where the Masoretic timeline diverges from the Samaritan tradition. Key textual and numerical variations include: ==== The Universal Flood Adjustment (+349 yrs) ==== The Masoretic chronology postpones the birth of Noah and the onset of the Deluge by '''349 years''' relative to the Samaritan tradition. As outlined in the above "Masoretic Adjustments" section, this structural shift ensures that Noah's antediluvian ancestors do not survive past the date of the universal flood. This timeline was achieved by adding 100 years to Jared's paternal age, 120 years to Methuselah's, and 129 years to Lamech's. Consequently, these calculations place Methuselah's death precisely in the year of the Flood and Lamech's death five years prior. ==== The Smoothing of Noah's Life (-2 yrs) ==== The Masoretic chronology reduces Noah's age at the birth of Shem from 502 years (as found in the Samaritan Pentateuch) to 500 years. As discussed in the "Lectio Difficilior Potior" and "The Birth of Shem (A Digression)" sections, this change represents "scribal smoothing." This editorial process rounded major milestones in Noah's life to the nearest century: his sons are born in his 500th year, the Deluge occurs in his 600th year, and his father Lamech dies in his 700th year. ==== Post-Flood Patriarchal Lifespans (-650 yrs) ==== Seven post-flood patriarchs (Arpachshad through Nahor) exhibit reduced ages at the birth of their firstborn sons within the Masoretic chronology, resulting in a cumulative deficit of 650 years compared to the Samaritan Pentateuch. Specifically, the paternal ages of Arpachshad, Shelah, Eber, Peleg, Reu, and Serug are each lower by exactly 100 years, while Nahor's age is reduced by 50 years. Given the uniform, rounded values of these variations, a direct dependency exists; either the Masoretic chronologist systematically deflated these ages, or the Samaritan chronologist systematically inflated them. The Masoretic use of young fathering ages for these specific patriarchs may preserve an older tradition. The narrative theme of Abraham fathering Isaac at the age of 100 serves as a central literary and mathematical motif that likely predates the Babylonian exile. Consequently, the lower paternal ages in the Masoretic lineage may intentionally highlight this milestone, ensuring that a fatherhood age of 100 remains an extraordinary, miraculous exception within the broader genealogical timeline. Under this proposed framework, Arpachshad (or Cainan) originally functioned not as a standard descendant of Shem, but as the progenitor of a distinct nine-generation patriarchal line culminating in the miraculous birth of Isaac. During the Babylonian exile, the Priestly source ("P") may have adapted this older Abrahamic lineage, appending earlier primeval generations to mirror Mesopotamian creation and flood epics. Consequently, Mesopotamian archetypes—such as Adapa, Enmeduranki, and Utnapishtim—were synthesized with preexisting Hebrew traditions to form the composite figures of Adam, Enoch, and Noah. These composite primeval figures were subsequently integrated into a single, seamless genealogical chain spanning from creation to the foundations of Israel. ==== The Final Masoretic Adjustment (+215 yrs) ==== As will be explained later in this resource, it is proposed that a final 215 years was added to the Masoretic chronology to arrive at the 2666-year total. The Samaritan Pentateuch, the Septuagint, and the Book of Jubilees all claim that the Patriarchs were in Canaan for 215 years, followed by 215 years of Israelite slavery in Egypt. The Masoretic text increases the period of Egyptian slavery by 215 years, for a total of 430 years of slavery. This increase to 430 years is difficult to reconcile with the Moses narrative, which claims that Moses was the great-grandson of Levi on the paternal line and the grandson on the maternal line. Levi entered Egypt as an adult and a free man. Moses led the children of Israel out of Egypt at eighty years old. For Moses to be a maternal grandson, his mother would have been hundreds of years old when she gave birth to him and his older siblings, Aaron and Miriam. ==== The 2800-Year to 2666-Year Transition ==== Within the Masoretic chronology, the Exodus is situated inside the third 800-year epoch, occurring precisely in the year 2666 after creation (Anno Mundi). This placement represents a deliberate scribal design choice rather than an organic chronological coincidence. <div style="line-height: 1.5;"> * '''Book of Jubilees (Adam to Canaan):''' :<math display="block"> \begin{aligned} 7^4 + 7^2 &= 2401 + 49 \\ &= 2450 \end{aligned} </math> * '''Samaritan Pentateuch (Adam to Conquest):''' :<math display="block"> \begin{aligned} \text{(Base 49): } & 7^4 + 7^3 + 7^2 + 7^1 \\ &= 2401 + 343 + 49 + 7 \\ &= 2800 \\ \text{(Base 40): } & 70 \times 40 \\ &= 2800 \end{aligned} </math> * '''Masoretic Text (Adam to Exodus):''' :<math display="block"> (2800 - 46) + (349) - (2) - (650) + (215) = 2666</math> </div> As detailed in "The 350-Year Symmetrical Extension," the timeline found in the Book of Jubilees likely served as the foundational model, positioning Shem at the exact midpoint of its 2,450-year chronology. The Samaritan Pentateuch appears to be a direct derivative of this Jubilees framework. Scribes systematically appended exactly 350 years to the post-Flood era, shifting the completion of the Conquest to a precise macro-structural target of 2,800 years after creation. Envisioning the process that transitioned from the 2,800-year Samaritan chronology to the Masoretic framework presents a distinct mathematical challenge. Tracking the adjustments step-by-step from the Samaritan baseline reveals a precise transition: * '''Samaritan Baseline (Conquest Completion):''' 2,800 years * '''Minus Wandering and Conquest Era:''' −46 years (2,754) * '''Plus Masoretic Flood Adjustment:''' +349 years (3,103) * '''Minus Noah's Life Smoothing:''' −2 years (3,101) * '''Minus Post-Flood Patriarchal Reductions:''' −650 years (2,451) * '''The Final Masoretic Adjustment:''' +215 years (2,666) This calculation yields exactly 2,666 years. Because individual components were dictated by rigid external theological and narrative constraints, it is surprising that the final adjustment yields exactly 2,666 years, but it also remains difficult to isolate where scribes might have executed fine-tuning to bring about this exact value and bridge the 2,800-year and 2,666-year macro-structures. === Mathematical Structure of the Septuagint Chronology === As was explained above in "Septuagint Adjustments", the LXX’s editor methodically added 100 years to each pre-flood patriarch. The purpose of this adjustment was to ensure that Noah would be the exclusive survivor of the flood. As shown in the tables below, post-flood "Long Chronology" variations are seen to be either identical to or derived from "Short Chronology" values. {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son) |- ! rowspan="2" colspan="1" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Adam | colspan="3" style="background-color:#e8e8e8;" | 130 | colspan="6" style="background-color:#e8e8e8;" | 230 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Seth | colspan="3" style="background-color:#e8e8e8;" | 105 | colspan="6" style="background-color:#e8e8e8;" | 205 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enosh | colspan="3" style="background-color:#e8e8e8;" | 90 | colspan="6" style="background-color:#e8e8e8;" | 190 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Kenan | colspan="3" style="background-color:#e8e8e8;" | 70 | colspan="6" style="background-color:#e8e8e8;" | 170 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Mahalalel | colspan="1" style="background-color:#e8e8e8;" | 66 | colspan="2" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Jared | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 162 | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="6" style="background-color:#f9f9f9;" | 162 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Enoch | colspan="3" style="background-color:#e8e8e8;" | 65 | colspan="6" style="background-color:#e8e8e8;" | 165 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Methuselah | colspan="1" style="background-color:#f9f9f9;" | 65 | colspan="1" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#f9f9f9;" | 67 | colspan="2" style="background-color:#f9f9f9;" | 187 | colspan="1" style="background-color:#f9f9f9;" | 167 | colspan="3" style="background-color:#f9f9f9;" | 167 / 187 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Lamech | colspan="1" style="background-color:#f9f9f9;" | 55 | colspan="1" style="background-color:#f9f9f9;" | 182 | colspan="1" style="background-color:#f9f9f9;" | 53 | colspan="5" style="background-color:#f9f9f9;" | 188 | colspan="1" style="background-color:#f9f9f9;" | 182 / 188 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Noah | rowspan="2" style="background-color:#e8e8e8;" | 602 | rowspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 602 | rowspan="2" colspan="2" style="background-color:#e8e8e8;" | 600 | rowspan="2" colspan="3" style="background-color:#e8e8e8;" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shem |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood TOTAL | colspan="1" | 1309 | colspan="1" | 1656 | colspan="1" | 1309 | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |} {| class="wikitable" style="width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;" |+ Comparison of Post-Flood Chronological Traditions (Age at birth of son) |- ! colspan="1" rowspan="2" | Patriarch ! colspan="3" style="background-color:#e3f2fd; border-bottom:2px solid #2196f3;" | SHORT CHRONOLOGY ! colspan="6" style="background-color:#fff3e0; border-bottom:2px solid #ff9800;" | LONG CHRONOLOGY |- ! colspan="1" style="background-color:#e3f2fd;" | Jubilees <br/> (Jub) ! colspan="1" style="background-color:#e3f2fd;" | Masoretic <br/> (MT) ! colspan="1" style="background-color:#e3f2fd;" | Samaritan <br/> (SP) ! colspan="1" style="background-color:#fff3e0;" | Demetrius <br/> (204 BC) ! colspan="1" style="background-color:#fff3e0;" | Africanus <br/> (221 AD) ! colspan="1" style="background-color:#fff3e0;" | Theophilus <br/> (192 AD) ! colspan="1" style="background-color:#fff3e0;" | Septuagint <br/> (LXX) ! colspan="1" style="background-color:#fff3e0;" | Eusebius <br/> (325 AD) ! colspan="1" style="background-color:#fff3e0;" | Josephus <br/> (94 AD) |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | Pre-Flood | colspan="1" | 1309 | colspan="1" | 1656 | colspan="1" | 1309 | colspan="1" | 2264 | colspan="1" | 2262 | colspan="1" | 2242 | colspan="3" | Varied |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Arphaxad | colspan="1" style="background-color:#f9f9f9;" | 66 | colspan="1" style="background-color:#f9f9f9;" | 35 | colspan="7" style="background-color:#e8e8e8;" | 135 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Cainan II | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - | colspan="1" style="background-color:#f9f9f9;" | 130 | colspan="2" style="background-color:#e8e8e8;" | - |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Shelah | colspan="1" style="background-color:#f9f9f9;" | 71 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Eber | colspan="1" style="background-color:#f9f9f9;" | 64 | colspan="1" style="background-color:#f9f9f9;" | 34 | colspan="7" style="background-color:#e8e8e8;" | 134 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Peleg | colspan="1" style="background-color:#f9f9f9;" | 61 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="7" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Reu | colspan="1" style="background-color:#f9f9f9;" | 59 | colspan="1" style="background-color:#f9f9f9;" | 32 | colspan="5" style="background-color:#e8e8e8;" | 132 | colspan="1" style="background-color:#e8e8e8;" | 135 | colspan="1" style="background-color:#e8e8e8;" | 130 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Serug | colspan="1" style="background-color:#f9f9f9;" | 57 | colspan="1" style="background-color:#f9f9f9;" | 30 | colspan="6" style="background-color:#e8e8e8;" | 130 | colspan="1" style="background-color:#e8e8e8;" | 132 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Nahor | colspan="1" style="background-color:#f9f9f9;" | 62 | colspan="1" style="background-color:#f9f9f9;" | 29 | colspan="3" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 75 | colspan="1" style="background-color:#e8e8e8;" | 79 / 179 | colspan="1" style="background-color:#e8e8e8;" | 79 | colspan="1" style="background-color:#e8e8e8;" | 120 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Terah | colspan="9" style="background-color:#e8e8e8;" | 70 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Abram | colspan="1" style="background-color:#f9f9f9;" | 78 | colspan="8" style="background-color:#e8e8e8;" | 75 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Canaan | colspan="1" style="background-color:#f9f9f9;" | 218 | colspan="8" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Egypt | colspan="1" style="background-color:#f9f9f9;" | 238 | colspan="1" style="background-color:#f9f9f9;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 | colspan="1" style="background-color:#e8e8e8;" | 430 | colspan="3" style="background-color:#e8e8e8;" | 215 |- ! colspan="1" style="text-align:left; background-color:#f9f9f9;" | Sinai +/- | colspan="1" style="background-color:#f9f9f9;" | 40 | colspan="1" style="background-color:#f9f9f9;" | - | colspan="3" style="background-color:#e8e8e8;" | 46 | colspan="4" style="background-color:#e8e8e8;" | 40 |- style="background-color:#333; color:white; font-weight:bold; font-size:15px;" ! colspan="1" style="text-align:left; color:black;" | GRAND TOTAL | colspan="1" | 2450 | colspan="1" | 2666 | colspan="1" | 2800 | colspan="1" | 3885 | colspan="1" | 3754 | colspan="1" | 3938 | colspan="3" | Varied |} == Living in the Rough == [[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]] As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a "generation") are less clear, but the number is consistently associated with "living in the rough"—periods of trial, transition, or exile away from the comforts of civilization. Examples of this pattern include: * '''Noah''' lived within the ark for 40 days while the rain fell; * '''Israel''' wandered in the wilderness for 40 years; * '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era. Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the "civilized" city. In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means "supplanter"—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a "skillful hunter" and a "man of the field," while Jacob is "dwelling in tents" and cooking "lentil stew." The text draws a clear parallel between these two sets of brothers: * In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. * In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this "clever" intervention, Jacob successfully secures the favor that Cain could not. Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the "rough" of the wilderness. Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three "generations" (3 x 40). The relationship between these two "three-fold" lifespans can be expressed by the same nine-year offset identified in the Shem chronology: <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 3(49 - 9) &= 3(40) \\ 147 - 27 &= 120 \end{aligned} </math> [[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] Samaritan tradition maintains a unique cultural link to the "pastoralist" ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the "grain-based" agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to "wilderness living," the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit. === A narrative foil for Joshua === As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a "foil" for Jacob: Moses lived exactly three "generations" (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization. This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding "foil" to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam? <math display="block"> \begin{aligned} 49 - 9 &= 40 \\ 70(49 - 9) &= 70(40) \\ 3,430 - 630 &= 2,800 \end{aligned} </math> Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific "foil" to Joshua existed in the original architect's mind. The Samaritans do maintain a continuous, running calendar. However, this system uses a "Conquest Era" epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. The [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC. There is, however, one more possibility to consider: what if the "irreconcilable" nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent "foil" event occurred exactly 3,430 years after Creation? As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare "smoking gun" for the artificial design of the chronology. === A Mystery Solved === In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest. The significant milestones in this timeline include: * '''Year 1''': "This year the world and Adam were created." * '''Year 2801''': "The first year of Israel's rule in the land of Canaan." * '''Year 3423''': "The commencement of the kingdom of Solomon." According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological "anchor" was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself. In Creation Year 2800—marking exactly 70 "generations" of 40 years—God entered Canaan in a tent, embodying the "living rough" wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 "Jubilees" of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70^th generation to the 70^th Jubilee: :<math>70 \times 40 + (70 \times 9) = 70 \times 49</math> === Mathematical Structure of the Later Samaritan Chronology === The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation. The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year "Tabernacle" era''', which is composed of: * The 40 years of wandering in the wilderness; * The 6 years of the initial conquest; * The 630 years between the conquest and the completion of Solomon’s Temple. Following the '''676-year "Tabernacle" era''' is a '''400-year "First Temple" era''' and a '''70-year "Exile" era''' as detailed in the historical breakdown below. [[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]] The Book of Daniel states: "In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it" (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''. The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure. High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. Using this synchronization, other significant milestones are mapped as follows: * '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''. * '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''. * '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''. High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year "Second Temple Persian Era" (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. The remainder of the diagram represents a 300-year "Second Temple Hellenistic Era," which concludes in '''Creation Year 4400''' (30 BC). === Competing Temples === There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology shown above, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year "Second Temple Persian Era" (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework. According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple: <blockquote>"But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel" (Ezra 4:3).</blockquote> After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population. [[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]] This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC. = The Rise of Zadok = The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years: * '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation. * '''The second cluster''' spans years '''3,000 to 4,000''' after Creation. * '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation. The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section. Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness. To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event. [[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]] The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000. = Hellenized Jews = Hellenized Jews were ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint. [[Category:Religion]] === Mythology === In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|"Uruk List of Kings and Sages"]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the "sages" eventually transition into human scholars. A tentative translation reads: *During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]]. *During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]]. *During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage. *During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage. *During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage. *During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage. *During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage. *After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives. fu94wb6ea1ruaoqzyz9xpzkwvpp4ex1 0 330494 2844458 2844451 2026-11-04T12:47:22Z JSINST 3110286 2844458 wikitext text/x-wiki <span lang="en" dir="ltr"><big>'''The John Snow Prediabetes Institute''' is an international research network that operates through a collaborative framework of global health professionals, academic institutions, and structured lifestyle interventions to identify and reverse early metabolic risks.</big></span> <big>'''Core Structure and Research Methodologies'''</big> <span lang="en" dir="ltr"><big>The organization addresses the rise of metabolic syndrome—including prediabetes, prehypertension, overweight and early arteriosclerosis—in individuals under the age of 30. It coordinates its efforts through several distinct vectors:</big></span> * <span lang="en" dir="ltr"><big>'''Prevalence Tracking:''' The network establishes comprehensive health databases by conducting repeated studies across high-risk adolescent and young adult populations. Key data streams include the Maritime Health Database alongside specialized tracking for nursing, medical, and school-aged cohorts.</big></span> * <span lang="en" dir="ltr"><big>'''Intervention Frameworks:''' The institute utilizes a standardized 16-week human-caused lifestyle course available globally in English, Danish, and Spanish. This curriculum combines video-guided education, dietary self-evaluation, and systematic blood glucose monitoring.</big></span> * <span lang="en" dir="ltr"><big>'''Digital Integration:''' Research protocols actively investigate the efficacy of AI-powered digital lifestyle programs against traditional human coaching metrics, referencing landmark data showing massive risk reductions in prediabetic cohorts.</big></span> * <span lang="en" dir="ltr"><big>'''Institutional Alignment:''' Local operations align with evidence-based resources from major entities like the American Diabetes Association (ADA) and regional Health Promoting Schools networks.</big></span> <span lang="en" dir="ltr"><big>International Research Leadership Network</big></span> <span lang="en" dir="ltr"><big>The global initiative is powered by an international network of clinical and public health experts:</big></span> <big>16 weeks human caused Intervention studies: English <ref>https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&dl=0&rlkey=7kzg91tqfgjskxf5aji8khicx </ref> Danish <ref>https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&dl=0&rlkey=x63w8oqvarz284zg2btq2johv </ref> Spanish <ref> https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&dl=0&rlkey=popmr1fnodh1v951v9l7k9ezv </ref> and '''AI-Powered Lifestyle Intervention''' in the Diabetes Prevention <ref>Mathioudakis, Nestoras, Benjamin Lalani, Mohammed S. Abusamaan, et al. «An AI-Powered Lifestyle Intervention vs Human Coaching in the Diabetes Prevention Program: A Randomized Clinical Trial». JAMA 334, n.º 23 (2025): 2079-89. https://doi.org/10.1001/jama.2025.19563.</ref></big> '''<big>Organisation</big>''' <big>- [[/The John Snow Institute bylaws /|The John Snow Institute bylaws]]</big> <big>- General research protocol draft <ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref></big> <big>- Health Promoting Schools <ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref></big> [[Category:Prediabetes ]] [[Category:Prediabetes ]] <big>'''Publications''' <ref name=":0"> https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf </ref></big> '''The John Snow Prediabetes Institute Research Network:''' <small>Prof. Dr. Olaf Jensen, MD, MPH, PhD; Prof. Ing. MSc. Nailet Delgado Mujica; Prof. Magda Medir Mb, Tarragona Spain, Prof. MSc.Ph.D. Bishal Gyawali SDU; MSc.PhD Vivi Just-Nørregaard; Prof. Dr. Johan Hviid Andersen MD, PhD. Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Bruno Nørdam, Randers; Maritime Doctor. Erik Haarløv MD, Assens, Denmark; Dr. Maite Duque MD Venezuela; Dr. Indira Santos Panama MD; Med.Stud. Ashley Lezcano, Panama; Dr. Joseph Abesamis MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dra.Marj Gab Huerte, MD, Filippines:</small> <small>Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona and Cadiz, Spain; Dr. Alejandro Martinez, MD, MPH, Costa Rica; Christian Acheampong, MBA, New Jersey US; Dr. Med. Sci Finn Gyntelberg NFA.and Bispebj. Hosp. Denmark: Dr. Solomon Ching MD Filippines: Dr. Marcel Alcaraz MD Filippines;</small> ==References== 0ji351m499xbz9mx6aigkpfa35ebgdx 2844460 2844458 2026-11-04T12:55:08Z JSINST 3110286 2844460 wikitext text/x-wiki <span lang="en" dir="ltr"><small>'''The John Snow Prediabetes Institute''' is an international research network that operates through a collaborative framework of global health professionals, academic institutions, and structured lifestyle interventions to identify and reverse early metabolic risks.</small></span> '''<small>Core Structure and Research Methodologies</small>''' <span lang="en" dir="ltr"><small>The organization addresses the rise of metabolic syndrome—including prediabetes, prehypertension, overweight and early arteriosclerosis—in individuals under the age of 30. It coordinates its efforts through several distinct vectors:</small></span> * <span lang="en" dir="ltr"><small>'''Prevalence Tracking:''' The network establishes comprehensive health databases by conducting repeated studies across high-risk adolescent and young adult populations. Key data streams include the Maritime Health Database alongside specialized tracking for nursing, medical, and school-aged cohorts.</small></span> * <span lang="en" dir="ltr"><small>'''Intervention Frameworks:''' The institute utilizes a standardized 16-week human-caused lifestyle course available globally in English, Danish, and Spanish. This curriculum combines video-guided education, dietary self-evaluation, and systematic blood glucose monitoring. English <ref>https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&dl=0&rlkey=7kzg91tqfgjskxf5aji8khicx </ref> Danish <ref>https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&dl=0&rlkey=x63w8oqvarz284zg2btq2johv </ref> Spanish <ref> https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&dl=0&rlkey=popmr1fnodh1v951v9l7k9ezv </ref></small></span> * <span lang="en" dir="ltr"><small>'''Digital Integration:''' Research protocols actively investigate the efficacy of AI-powered digital lifestyle programs against traditional human coaching metrics, referencing landmark data showing massive risk reductions in prediabetic cohorts. <ref>Mathioudakis, Nestoras, Benjamin Lalani, Mohammed S. Abusamaan, et al. «An AI-Powered Lifestyle Intervention vs Human Coaching in the Diabetes Prevention Program: A Randomized Clinical Trial». JAMA 334, n.º 23 (2025): 2079-89. https://doi.org/10.1001/jama.2025.19563.</ref></small></span> * <span lang="en" dir="ltr"><small>'''Institutional Alignment:''' Local operations align with evidence-based resources from major entities like the American Diabetes Association (ADA) and regional Health Promoting Schools networks.</small></span> '''<small>Organisation</small>''' <small>- [[/The John Snow Institute bylaws /|The John Snow Institute bylaws]]</small> <small>- General research protocol draft <ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref></small> <small>- Health Promoting Schools <ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref> '''Publications''' <ref name=":0"> https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf </ref></small> [[Category:Prediabetes ]] [[Category:Prediabetes ]] <span lang="en" dir="ltr"><big>The global initiative is powered by an international network of clinical and public health experts:</big></span>''':''' <small>Prof. Dr. Olaf Jensen, MD, MPH, PhD; Prof. Ing. MSc. Nailet Delgado Mujica; Prof. Magda Medir Mb, Tarragona Spain, Prof. MSc.Ph.D. Bishal Gyawali SDU; MSc.PhD Vivi Just-Nørregaard; Prof. Dr. Johan Hviid Andersen MD, PhD. Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Bruno Nørdam, Randers; Maritime Doctor. Erik Haarløv MD, Assens, Denmark; Dr. Maite Duque MD Venezuela; Dr. Indira Santos Panama MD; Med.Stud. Ashley Lezcano, Panama; Dr. Joseph Abesamis MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dra.Marj Gab Huerte, MD, Filippines:</small> <small>Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona and Cadiz, Spain; Dr. Alejandro Martinez, MD, MPH, Costa Rica; Christian Acheampong, MBA, New Jersey US; Dr. Med. Sci Finn Gyntelberg NFA.and Bispebj. Hosp. Denmark: Dr. Solomon Ching MD Filippines: Dr. Marcel Alcaraz MD Filippines;</small> ==References== 9mv7azv5d07dkyqy5rg9fbttcamhzla 2844461 2844460 2026-11-04T12:58:22Z JSINST 3110286 2844461 wikitext text/x-wiki <span lang="en" dir="ltr"><small>'''The John Snow Prediabetes Institute''' is an international research network that operates through a collaborative framework of global health professionals, academic institutions, and structured lifestyle interventions to identify and reverse early metabolic risks.</small></span> '''<small>Core Structure and Research Methodologies</small>''' <span lang="en" dir="ltr"><small>The organization addresses the rise of metabolic syndrome—including prediabetes, prehypertension, overweight and early arteriosclerosis—in individuals under the age of 30. It coordinates its efforts through several distinct vectors:</small></span> * <span lang="en" dir="ltr"><small>'''Prevalence Tracking:''' The network establishes comprehensive health databases by conducting repeated studies across high-risk adolescent and young adult populations. Key data streams include the Maritime Health Database alongside specialized tracking for nursing, medical, and school-aged cohorts.</small></span> * <span lang="en" dir="ltr"><small>'''Intervention Frameworks:''' The institute utilizes a standardized 16-week human-caused lifestyle course available globally in English, Danish, and Spanish. This curriculum combines video-guided education, dietary self-evaluation, and systematic blood glucose monitoring. English <ref>https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&dl=0&rlkey=7kzg91tqfgjskxf5aji8khicx </ref> Danish <ref>https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&dl=0&rlkey=x63w8oqvarz284zg2btq2johv </ref> Spanish <ref> https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&dl=0&rlkey=popmr1fnodh1v951v9l7k9ezv </ref></small></span> * <span lang="en" dir="ltr"><small>'''Digital Integration:''' Research protocols actively investigate the efficacy of AI-powered digital lifestyle programs against traditional human coaching metrics, referencing landmark data showing massive risk reductions in prediabetic cohorts. <ref>Mathioudakis, Nestoras, Benjamin Lalani, Mohammed S. Abusamaan, et al. «An AI-Powered Lifestyle Intervention vs Human Coaching in the Diabetes Prevention Program: A Randomized Clinical Trial». JAMA 334, n.º 23 (2025): 2079-89. https://doi.org/10.1001/jama.2025.19563.</ref></small></span> * <span lang="en" dir="ltr"><small>'''Institutional Alignment:''' Local operations align with evidence-based resources from major entities like the American Diabetes Association (ADA) and regional Health Promoting Schools networks.</small></span> '''<small>Organisation</small>''' [[The John Snow Prediabetes Institute/The Maritime Health database design|1.1 The Maritime Health database design]] 1.2 Nursing Students Health Database 1.3. Medical student's Health Database 1.4. School childrens` Health database 1.5 Excel recoding (pre diabetes, hypertension etc) <small>- [[/The John Snow Institute bylaws /|The John Snow Institute bylaws]]</small> <small>- General research protocol draft <ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref></small> <small>- Health Promoting Schools <ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref> '''Publications''' <ref name=":0"> https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf </ref></small> [[Category:Prediabetes ]] [[Category:Prediabetes ]] <span lang="en" dir="ltr"><big>The global initiative is powered by an international network of clinical and public health experts:</big></span>''':''' <small>Prof. Dr. Olaf Jensen, MD, MPH, PhD; Prof. Ing. MSc. Nailet Delgado Mujica; Prof. Magda Medir Mb, Tarragona Spain, Prof. MSc.Ph.D. Bishal Gyawali SDU; MSc.PhD Vivi Just-Nørregaard; Prof. Dr. Johan Hviid Andersen MD, PhD. Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Bruno Nørdam, Randers; Maritime Doctor. Erik Haarløv MD, Assens, Denmark; Dr. Maite Duque MD Venezuela; Dr. Indira Santos Panama MD; Med.Stud. Ashley Lezcano, Panama; Dr. Joseph Abesamis MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dra.Marj Gab Huerte, MD, Filippines:</small> <small>Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona and Cadiz, Spain; Dr. Alejandro Martinez, MD, MPH, Costa Rica; Christian Acheampong, MBA, New Jersey US; Dr. Med. Sci Finn Gyntelberg NFA.and Bispebj. Hosp. Denmark: Dr. Solomon Ching MD Filippines: Dr. Marcel Alcaraz MD Filippines;</small> ==References== fm93j94rffluvqc7h7fjwspb6el6xls 2844462 2844461 2026-11-04T13:01:38Z JSINST 3110286 2844462 wikitext text/x-wiki <span lang="en" dir="ltr"><small>'''The John Snow Prediabetes Institute''' is an international research network that operates through a collaborative framework of global health professionals, academic institutions, and structured lifestyle interventions to identify and reverse early metabolic risks.</small></span> '''<small>Core Structure and Research Methodologies</small>''' <span lang="en" dir="ltr"><small>The organization addresses the rise of metabolic syndrome—including prediabetes, prehypertension, overweight and early arteriosclerosis—in individuals under the age of 30. It coordinates its efforts through several distinct vectors:</small></span> * <span lang="en" dir="ltr"><small>'''Prevalence Tracking:''' The network establishes comprehensive health databases by conducting repeated studies across high-risk adolescent and young adult populations. Key data streams include the Maritime Health Database alongside specialized tracking for nursing, medical, and school-aged cohorts.</small></span> * <span lang="en" dir="ltr"><small>'''Intervention Frameworks:''' The institute utilizes a standardized 16-week human-caused lifestyle course available globally in English, Danish, and Spanish. This curriculum combines video-guided education, dietary self-evaluation, and systematic blood glucose monitoring. English <ref>https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&dl=0&rlkey=7kzg91tqfgjskxf5aji8khicx </ref> Danish <ref>https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&dl=0&rlkey=x63w8oqvarz284zg2btq2johv </ref> Spanish <ref> https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&dl=0&rlkey=popmr1fnodh1v951v9l7k9ezv </ref></small> * <span lang="en" dir="ltr"><small>'''Digital Integration:''' Research protocols actively investigate the efficacy of AI-powered digital lifestyle programs against traditional human coaching metrics, referencing landmark data showing massive risk reductions in prediabetic cohorts.<ref>Mathioudakis, Nestoras, Benjamin Lalani, Mohammed S. Abusamaan, et al. «An AI-Powered Lifestyle Intervention vs Human Coaching in the Diabetes Prevention Program: A Randomized Clinical Trial». JAMA 334, n.º 23 (2025): 2079-89. https://doi.org/10.1001/jama.2025.19563.</ref></small> * <span lang="en" dir="ltr"><small>'''Institutional Alignment:''' Local operations align with evidence-based resources from major entities like the American Diabetes Association (ADA) and regional Health Promoting Schools networks.</small></span> '''<small>Organisation</small>''' [[The John Snow Prediabetes Institute/The Maritime Health database design|1.1 The Maritime Health database design]] 1.2 Nursing Students Health Database 1.3. Medical student's Health Database 1.4. School childrens` Health database 1.5 Excel recoding (pre diabetes, hypertension etc) <small>- [[/The John Snow Institute bylaws /|The John Snow Institute bylaws]]</small> <small>- General research protocol draft <ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref></small> <small>- Health Promoting Schools <ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref> '''Publications''' <ref name=":0"> https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf </ref></small> <span lang="en" dir="ltr"><big>The global initiative is powered by an international network of clinical and public health experts:</big></span>''':''' '''<small>Prof. Dr. Olaf Jensen, MD, MPH, PhD; Prof. Ing. MSc. Nailet Delgado Mujica; Prof. Magda Medir Mb, Tarragona Spain, Prof. MSc.Ph.D. Bishal Gyawali SDU; MSc.PhD Vivi Just-Nørregaard; Prof. Dr. Johan Hviid Andersen MD, PhD. Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Bruno Nørdam, Randers; Maritime Doctor. Erik Haarløv MD, Assens, Denmark; Dr. Maite Duque MD Venezuela; Dr. Indira Santos Panama MD; Med.Stud. Ashley Lezcano, Panama; Dr. Joseph Abesamis MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dra.Marj Gab Huerte, MD, Filippines:</small> <small>Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona and Cadiz, Spain; Dr. Alejandro Martinez, MD, MPH, Costa Rica; Christian Acheampong, MBA, New Jersey US; Dr. Med. Sci Finn Gyntelberg NFA.and Bispebj. Hosp. Denmark: Dr. Solomon Ching MD Filippines: Dr. Marcel Alcaraz MD Filippines;</small>''' [[Category:Prediabetes ]] [[Category:Prediabetes ]] ==References== a1tp6b6yaozbmqmt3fovsshoi2x9oef 2844463 2844462 2026-11-04T13:05:02Z JSINST 3110286 2844463 wikitext text/x-wiki <span lang="en" dir="ltr"><small>'''The John Snow Prediabetes Institute''' is an international research network that operates through a collaborative framework of global health professionals, academic institutions, and structured lifestyle interventions to identify and reverse early metabolic risks.</small></span> '''<small>Core Structure and Research Methodologies</small>''' <span lang="en" dir="ltr"><small>The organization addresses the rise of metabolic syndrome—including prediabetes, prehypertension, overweight and early arteriosclerosis, symptomless with the same etiologies: too small physical activity and overeating of —in individuals under the age of 30. It coordinates its efforts through several distinct vectors:</small></span> * <span lang="en" dir="ltr"><small>'''Prevalence Tracking:''' The network establishes comprehensive health databases by conducting repeated studies across high-risk adolescent and young adult populations. Key data streams include the Maritime Health Database alongside specialized tracking for nursing, medical, and school-aged cohorts.</small></span> * <span lang="en" dir="ltr"><small>'''Intervention Frameworks:''' The institute utilizes a standardized 16-week human-caused lifestyle course available globally in English, Danish, and Spanish. This curriculum combines video-guided education, dietary self-evaluation, and systematic blood glucose monitoring. English <ref>https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&dl=0&rlkey=7kzg91tqfgjskxf5aji8khicx </ref> Danish, <ref>https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&dl=0&rlkey=x63w8oqvarz284zg2btq2johv </ref> Spanish <ref> https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&dl=0&rlkey=popmr1fnodh1v951v9l7k9ezv </ref></small> * <span lang="en" dir="ltr"><small>'''Digital Integration:''' Research protocols actively investigate the efficacy of AI-powered digital lifestyle programs against traditional human coaching metrics, referencing landmark data showing massive risk reductions in prediabetic cohorts.<ref>Mathioudakis, Nestoras, Benjamin Lalani, Mohammed S. Abusamaan, et al. «An AI-Powered Lifestyle Intervention vs Human Coaching in the Diabetes Prevention Program: A Randomized Clinical Trial». JAMA 334, n.º 23 (2025): 2079-89. https://doi.org/10.1001/jama.2025.19563.</ref></small> * <span lang="en" dir="ltr"><small>'''Institutional Alignment:''' Local operations align with evidence-based resources from major entities like the American Diabetes Association (ADA) and regional Health Promoting Schools networks.</small></span> '''<small>Organisation</small>''' [[The John Snow Prediabetes Institute/The Maritime Health database design|1.1 The Maritime Health database design]] 1.2 Nursing Students Health Database 1.3. Medical student's Health Database 1.4. School childrens` Health database 1.5 Excel recoding (pre diabetes, hypertension etc) <small>- [[/The John Snow Institute bylaws /|The John Snow Institute bylaws]]</small> <small>- General research protocol draft <ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref></small> <small>- Health Promoting Schools <ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref> '''Publications''' <ref name=":0"> https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf </ref></small> <span lang="en" dir="ltr"><big>The global initiative is powered by an international network of clinical and public health experts:</big></span>''':''' '''<small>Prof. Dr. Olaf Jensen, MD, MPH, PhD; Prof. Ing. MSc. Nailet Delgado Mujica; Prof. Magda Medir Mb, Tarragona Spain, Prof. MSc.Ph.D. Bishal Gyawali SDU; MSc.PhD Vivi Just-Nørregaard; Prof. Dr. Johan Hviid Andersen MD, PhD. Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Bruno Nørdam, Randers; Maritime Doctor. Erik Haarløv MD, Assens, Denmark; Dr. Maite Duque MD Venezuela; Dr. Indira Santos Panama MD; Med.Stud. Ashley Lezcano, Panama; Dr. Joseph Abesamis MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dra.Marj Gab Huerte, MD, Filippines:</small> <small>Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona and Cadiz, Spain; Dr. Alejandro Martinez, MD, MPH, Costa Rica; Christian Acheampong, MBA, New Jersey US; Dr. Med. Sci Finn Gyntelberg NFA.and Bispebj. Hosp. Denmark: Dr. Solomon Ching MD Filippines: Dr. Marcel Alcaraz MD Filippines;</small>''' [[Category:Prediabetes ]] [[Category:Prediabetes ]] ==References== tbqspg24zlnhfdfbnuqgv6vfndia6rn 2844464 2844463 2026-11-04T13:06:16Z JSINST 3110286 2844464 wikitext text/x-wiki <span lang="en" dir="ltr"><small>'''The John Snow Prediabetes Institute''' is an international research network that operates through a collaborative framework of global health professionals, academic institutions, and structured lifestyle interventions to identify and reverse early metabolic risks.</small></span> '''<small>Core Structure and Research Methodologies</small>''' <span lang="en" dir="ltr"><small>The organization addresses the rise of metabolic syndrome—including prediabetes, prehypertension, overweight and early arteriosclerosis, symptomless with the same etiologies: too small physical activity and overeating of —in individuals under the age of 30. It coordinates its efforts through several distinct vectors:</small></span> * <span lang="en" dir="ltr"><small>'''Prevalence Tracking:''' The network establishes comprehensive health databases by conducting repeated studies across high-risk adolescent and young adult populations. Key data streams include the Maritime Health Database alongside specialized tracking for nursing, medical, and school-aged cohorts.</small></span> * <span lang="en" dir="ltr"><small>'''Intervention Frameworks:''' The institute utilizes a standardized 16-week human-caused lifestyle course available globally in English, Danish, and Spanish. This curriculum combines video-guided education, dietary self-evaluation, and systematic blood glucose monitoring. English <ref>https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&dl=0&rlkey=7kzg91tqfgjskxf5aji8khicx </ref> Danish, <ref>https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&dl=0&rlkey=x63w8oqvarz284zg2btq2johv </ref> Spanish <ref> https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&dl=0&rlkey=popmr1fnodh1v951v9l7k9ezv </ref></small> * <span lang="en" dir="ltr"><small>'''Digital Integration:''' Research protocols actively investigate the efficacy of AI-powered digital lifestyle programs against traditional human coaching metrics, referencing landmark data showing massive risk reductions in prediabetic cohorts.<ref>Mathioudakis, Nestoras, Benjamin Lalani, Mohammed S. Abusamaan, et al. «An AI-Powered Lifestyle Intervention vs Human Coaching in the Diabetes Prevention Program: A Randomized Clinical Trial». JAMA 334, n.º 23 (2025): 2079-89. https://doi.org/10.1001/jama.2025.19563.</ref></small> * <span lang="en" dir="ltr"><small>'''Institutional Alignment:''' Local operations align with evidence-based resources from major entities like the American Diabetes Association (ADA) and regional Health Promoting Schools networks.</small></span> '''<small>Organisation</small>''' [[The John Snow Prediabetes Institute/The Maritime Health database design|1.1 The Maritime Health database design]] 1.2 Nursing Students Health Database 1.3. Medical student's Health Database 1.4. School childrens` Health database 1.5 Excel recoding (pre diabetes, hypertension etc) <small>- [[/The John Snow Institute bylaws /|The John Snow Institute bylaws]]</small> <small>- General research protocol draft <ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref></small> <small>- Health Promoting Schools <ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref> '''Publications''' <ref name=":0"> https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf </ref></small> <span lang="en" dir="ltr"><big>'''The global initiative is powered by an international network of clinical and public health experts:'''</big></span>''':''' '''<small>Prof. Dr. Olaf Jensen, MD, MPH, PhD; Prof. Ing. MSc. Nailet Delgado Mujica; Prof. Magda Medir Mb, Tarragona Spain, Prof. MSc.Ph.D. Bishal Gyawali SDU; MSc.PhD Vivi Just-Nørregaard; Prof. Dr. Johan Hviid Andersen MD, PhD. Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Bruno Nørdam, Randers; Maritime Doctor. Erik Haarløv MD, Assens, Denmark; Dr. Maite Duque MD Venezuela; Dr. Indira Santos Panama MD; Med.Stud. Ashley Lezcano, Panama; Dr. Joseph Abesamis MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dra.Marj Gab Huerte, MD, Filippines:</small> <small>Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona and Cadiz, Spain; Dr. Alejandro Martinez, MD, MPH, Costa Rica; Christian Acheampong, MBA, New Jersey US; Dr. Med. Sci Finn Gyntelberg NFA.and Bispebj. Hosp. Denmark: Dr. Solomon Ching MD Filippines: Dr. Marcel Alcaraz MD Filippines;</small> [[Category:Prediabetes ]] [[Category:Prediabetes ]] ==References== pkhqjrmmsk9gahcgnijb4klzt9o3wb7 2844514 2844464 2026-11-04T21:59:51Z ~2026-46935-18 3110252 2844514 wikitext text/x-wiki <span lang="en" dir="ltr"><small>'''The John Snow Prediabetes Institute''' is an international research network that operates through a collaborative framework of global health professionals, academic institutions, and structured lifestyle interventions to identify and reverse early metabolic risks.</small></span> '''<small>Core Structure and Research Methodologies</small>''' <span lang="en" dir="ltr"><small>The organization addresses the rise of metabolic syndrome—including prediabetes, prehypertension, overweight and early arteriosclerosis, symptomless with the same etiologies: too small physical activity and overeating unhealthy food, under the age of 30. It coordinates its efforts through several distinct vectors:</small></span> * <span lang="en" dir="ltr"><small>'''Prevalence Tracking:''' The network establishes comprehensive health databases by conducting repeated studies across high-risk adolescent and young adult populations. Key data streams include the Maritime Health Database alongside specialized tracking for nursing, medical, and school-aged cohorts.</small></span> * <span lang="en" dir="ltr"><small>'''Intervention Frameworks:''' The institute utilizes a standardized 16-week human-caused lifestyle course available globally in English, Danish, and Spanish. This curriculum combines video-guided education, dietary self-evaluation, and systematic blood glucose monitoring. English <ref>https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&dl=0&rlkey=7kzg91tqfgjskxf5aji8khicx </ref> Danish, <ref>https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&dl=0&rlkey=x63w8oqvarz284zg2btq2johv </ref> Spanish <ref> https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&dl=0&rlkey=popmr1fnodh1v951v9l7k9ezv </ref></small> * <span lang="en" dir="ltr"><small>'''Digital Integration:''' Research protocols actively investigate the efficacy of AI-powered digital lifestyle programs against traditional human coaching metrics, referencing landmark data showing massive risk reductions in prediabetic cohorts.<ref>Mathioudakis, Nestoras, Benjamin Lalani, Mohammed S. Abusamaan, et al. «An AI-Powered Lifestyle Intervention vs Human Coaching in the Diabetes Prevention Program: A Randomized Clinical Trial». JAMA 334, n.º 23 (2025): 2079-89. https://doi.org/10.1001/jama.2025.19563.</ref></small> * <span lang="en" dir="ltr"><small>'''Institutional Alignment:''' Local operations align with evidence-based resources from major entities like the American Diabetes Association (ADA) and regional Health Promoting Schools networks.</small></span> '''<small>Organisation</small>''' [[The John Snow Prediabetes Institute/The Maritime Health database design|1.1 The Maritime Health database design]] 1.2 Nursing Students Health Database 1.3. Medical student's Health Database 1.4. School childrens` Health database 1.5 Excel recoding (pre diabetes, hypertension etc) <small>- [[/The John Snow Institute bylaws /|The John Snow Institute bylaws]]</small> <small>- General research protocol draft <ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref></small> <small>- Health Promoting Schools <ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref> '''Publications''' <ref name=":0"> https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf </ref></small> <span lang="en" dir="ltr"><big>'''The global initiative is powered by an international network of clinical and public health experts:'''</big></span>''':''' '''<small>Prof. Dr. Olaf Jensen, MD, MPH, PhD; Prof. Ing. MSc. Nailet Delgado Mujica; Prof. Magda Medir Mb, Tarragona Spain, Prof. MSc.Ph.D. Bishal Gyawali SDU; MSc.PhD Vivi Just-Nørregaard; Prof. Dr. Johan Hviid Andersen MD, PhD. Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Bruno Nørdam, Randers; Maritime Doctor. Erik Haarløv MD, Assens, Denmark; Dr. Maite Duque MD Venezuela; Dr. Indira Santos Panama MD; Med.Stud. Ashley Lezcano, Panama; Dr. Joseph Abesamis MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dra.Marj Gab Huerte, MD, Filippines:</small> <small>Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona and Cadiz, Spain; Dr. Alejandro Martinez, MD, MPH, Costa Rica; Christian Acheampong, MBA, New Jersey US; Dr. Med. Sci Finn Gyntelberg NFA.and Bispebj. Hosp. Denmark: Dr. Solomon Ching MD Filippines: Dr. Marcel Alcaraz MD Filippines;</small> [[Category:Prediabetes ]] [[Category:Prediabetes ]] ==References== 50k8315oh26p3aafql0ex1xiprhruw9 2844595 2844514 2026-11-05T10:42:58Z JSINST 3110286 2844595 wikitext text/x-wiki <big>The John Snow prediabetes Institute is an international research network focused on early risk identification of prediabetes, prehypertension and the other types of metabolic syndrome in the early ages for remission, by lifestyle health promotion. (https://w.wiki/Skm7)</big> '''<big>Background:</big>''' <big>Millions are at increased risk of developing metabolic syndrome starting with arteriosclerosis, pre-diabetes, type 2 Diabetes Mellitus, pre hypertension, hypertension and overweight before the age of 30 without any simptomes.<ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref> The diseases is the background for the cardiovascular diseases (CVDs) which is the leading cause of death and disability worldwide. Early diagnosis and prevention is possible and a cost-effective preventive strategy that can improve long-term health outcomes. The '''prevalence''' '''studies''' to identify the high risk groups is done by repeated studies from the adolescent's schools, as example for -seafarers, -nurses and medical students. Alternatively, data from the academic literature is used by asking the authors to reanalyse their data to identify the risk groups. The 16-weeks '''intervention study''' include learning by video sequences and self-monitoring of blood sugar with Glukometer and self-evaluation of diet and physical activity <ref>https://www.sciencedirect.com/science/article/abs/pii/S1751991825001068 </ref>. The new challenging intervention studies include the AI-Digital lifestyle programs.<ref> «Digital Lifestyle Program Cuts Diabetes Risk by 46% in Prediabetics, Study of 130k+ Adults Reveals». News-Medical, 14 de abril de 2025. https://www.news-medical.net/news/20250414/Digital-lifestyle-program-cuts-diabetes-risk-by-4625-in-prediabetics-study-of-130k2b-adults-reveals.aspx.</ref> Educational materials is available from the international diabetes organisations e.g the ADA:<ref>https://professional.diabetes.org/diabetes-support-resources</ref> and the Health Promoting Schools.<ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref></big>[[File:Lifestyle Medicine Pillars.png|250px|right| <big>Lifestyle Medicine Pillars.png The focus of Lifestyle Medicine is on these 6 pillars</big> ]]'''<big>Prevalence studies</big>''' [[/The Maritime Health database design // |<big>1.1 The Maritime Health database design</big>]] <big>1.2 Nursing Students Health Database <ref> https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&dl=0&rlkey=onbjh4o8ko1lzdvgyi8nlrotk </ref></big> <big>1.3. Medical student's Health Database <ref>https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&dl=0&rlkey=xyfqen5trdc5lniaovipl548n </ref></big> <big>1.4. School childrens` Health database <ref> https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&dl=0&rlkey=zlyz5wn673wf7owettq3nx3h5 </ref></big> <big>1.5 Excel recoding of the clinical data (pre diabetes, hypertension etc) <ref>https://www.dropbox.com/scl/fi/9qvqeccto2kbuqvx7mdkl?r=ACsntb2AuiFkJbEBLObVdo0HMD67UZBBUGCGl1RPCPy1X5Y3K3e9Lf4s7cG7i7E_iH0IK-WkynXw_FNCDWybHuCgaNl_OM8yche2w2aapJGk5BTHybbsLpLayBU8VKPPqxNa2zDaNslBdQg0MhmKTFSG</ref></big> <big>'''Personal prevention''' Personal reversal of prediabetes relies on targeted lifestyle changes like eating a healthy diet, exercising regularly, and losing a modest amount of excess weight. (local Diabetes Associations)</big> <span lang="en" dir="ltr"><big>'''Core Institutional Strategies''' Health authorities and organizations implement group-based or digital programs. The workplaces are prompted to make targeted interventions, including nutrition and exercise programs, to safeguard long-term cardiometabolic health in the workforces.</big></span> <big>16 weeks human caused Intervention studies: English <ref>https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&dl=0&rlkey=7kzg91tqfgjskxf5aji8khicx </ref> Danish <ref>https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&dl=0&rlkey=x63w8oqvarz284zg2btq2johv </ref> Spanish <ref> https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&dl=0&rlkey=popmr1fnodh1v951v9l7k9ezv </ref> and '''AI-Powered Lifestyle Intervention''' in the Diabetes Prevention <ref>Mathioudakis, Nestoras, Benjamin Lalani, Mohammed S. Abusamaan, et al. «An AI-Powered Lifestyle Intervention vs Human Coaching in the Diabetes Prevention Program: A Randomized Clinical Trial». JAMA 334, n.º 23 (2025): 2079-89. https://doi.org/10.1001/jama.2025.19563.</ref></big> '''<big>Organisation</big>''' <big>- [[/The John Snow Institute bylaws /|The John Snow Institute bylaws]]</big> <big>- General research protocol draft <ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref></big> <big>- Health Promoting Schools <ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref></big> [[Category:Prediabetes ]] [[Category:Prediabetes ]] <big>'''Publications''' <ref name=":0"> https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf </ref></big> '''The John Snow Prediabetes Institute Research Network:''' <small>Prof. Dr. Olaf Jensen, MD, MPH, PhD; Prof. Ing. MSc. Nailet Delgado Mujica; Prof. Magda Medir Mb, Tarragona Spain, Prof. MSc.Ph.D. Bishal Gyawali SDU; MSc.PhD Vivi Just-Nørregaard; Prof. Dr. Johan Hviid Andersen MD, PhD. Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Bruno Nørdam, Randers; Maritime Doctor. Erik Haarløv MD, Assens, Denmark; Dr. Maite Duque MD Venezuela; Dr. Indira Santos Panama MD; Med.Stud. Ashley Lezcano, Panama; Dr. Joseph Abesamis MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dra.Marj Gab Huerte, MD, Filippines:</small> <small>Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona and Cadiz, Spain; Dr. Alejandro Martinez, MD, MPH, Costa Rica; Christian Acheampong, MBA, New Jersey US; Dr. Med. Sci Finn Gyntelberg NFA.and Bispebj. Hosp. Denmark: Dr. Solomon Ching MD Filippines: Dr. Marcel Alcaraz MD Filippines;</small> ==References== m1yqkhhl3e6zigbne5o8ast7l5ezn1m 2844597 2844595 2026-11-05T10:47:13Z JSINST 3110286 2844597 wikitext text/x-wiki <big>The John Snow prediabetes Institute is an international research network focused on early identification of prediabetes, prehypertension and the other types of metabolic syndrome in the early ages for lifestyle health promotion. (https://w.wiki/Skm7)</big> '''<big>Background:</big>''' <big>Millions are at increased risk of developing metabolic syndrome starting with arteriosclerosis, pre-diabetes, type 2 Diabetes Mellitus, pre hypertension, hypertension and overweight before the age of 30 without any simptomes.<ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref> These diseases constitute the background for the cardiovascular diseases (CVDs) which is the leading cause of death and disability worldwide. Early diagnosis and prevention is possible and the cost-effective preventive strategy can improve long-term health outcomes. The '''prevalence''' '''studies''' are used to identify the high risk groups among some specific occcupational jobs with repeated studies from the adolescent's schools, as example -seafarers, -nurses and medical students. Alternatively, data from the academic literature is used by asking the authors to reanalyse their data to identify the risk groups.</big> <big>The 16-weeks '''intervention study''' include learning by video sequences and self-monitoring of blood sugar with Glukometer and self-evaluation of diet and physical activity <ref>https://www.sciencedirect.com/science/article/abs/pii/S1751991825001068 </ref>. The new challenging intervention studies include the AI-Digital lifestyle programs.<ref> «Digital Lifestyle Program Cuts Diabetes Risk by 46% in Prediabetics, Study of 130k+ Adults Reveals». News-Medical, 14 de abril de 2025. https://www.news-medical.net/news/20250414/Digital-lifestyle-program-cuts-diabetes-risk-by-4625-in-prediabetics-study-of-130k2b-adults-reveals.aspx.</ref> Educational materials is available from the international diabetes organisations e.g the ADA:<ref>https://professional.diabetes.org/diabetes-support-resources</ref> and the Health Promoting Schools.<ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref></big>[[File:Lifestyle Medicine Pillars.png|250px|right| <big>Lifestyle Medicine Pillars.png The focus of Lifestyle Medicine is on these 6 pillars</big> ]]'''<big>Prevalence studies</big>''' [[/The Maritime Health database design // |<big>1.1 The Maritime Health database design</big>]] <big>1.2 Nursing Students Health Database <ref> https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&dl=0&rlkey=onbjh4o8ko1lzdvgyi8nlrotk </ref></big> <big>1.3. Medical student's Health Database <ref>https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&dl=0&rlkey=xyfqen5trdc5lniaovipl548n </ref></big> <big>1.4. School childrens` Health database <ref> https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&dl=0&rlkey=zlyz5wn673wf7owettq3nx3h5 </ref></big> <big>1.5 Excel recoding of the clinical data (pre diabetes, hypertension etc) <ref>https://www.dropbox.com/scl/fi/9qvqeccto2kbuqvx7mdkl?r=ACsntb2AuiFkJbEBLObVdo0HMD67UZBBUGCGl1RPCPy1X5Y3K3e9Lf4s7cG7i7E_iH0IK-WkynXw_FNCDWybHuCgaNl_OM8yche2w2aapJGk5BTHybbsLpLayBU8VKPPqxNa2zDaNslBdQg0MhmKTFSG</ref></big> <big>'''Personal prevention''' Personal reversal of prediabetes relies on targeted lifestyle changes like eating a healthy diet, exercising regularly, and losing a modest amount of excess weight. (local Diabetes Associations)</big> <span lang="en" dir="ltr"><big>'''Core Institutional Strategies''' Health authorities and organizations implement group-based or digital programs. The workplaces are prompted to make targeted interventions, including nutrition and exercise programs, to safeguard long-term cardiometabolic health in the workforces.</big></span> <big>16 weeks human caused Intervention studies: English <ref>https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&dl=0&rlkey=7kzg91tqfgjskxf5aji8khicx </ref> Danish <ref>https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&dl=0&rlkey=x63w8oqvarz284zg2btq2johv </ref> Spanish <ref> https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&dl=0&rlkey=popmr1fnodh1v951v9l7k9ezv </ref> and '''AI-Powered Lifestyle Intervention''' in the Diabetes Prevention <ref>Mathioudakis, Nestoras, Benjamin Lalani, Mohammed S. Abusamaan, et al. «An AI-Powered Lifestyle Intervention vs Human Coaching in the Diabetes Prevention Program: A Randomized Clinical Trial». JAMA 334, n.º 23 (2025): 2079-89. https://doi.org/10.1001/jama.2025.19563.</ref></big> '''<big>Organisation</big>''' <big>- [[/The John Snow Institute bylaws /|The John Snow Institute bylaws]]</big> <big>- General research protocol draft <ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref></big> <big>- Health Promoting Schools <ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref></big> [[Category:Prediabetes ]] [[Category:Prediabetes ]] <big>'''Publications''' <ref name=":0"> https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf </ref></big> '''The John Snow Prediabetes Institute Research Network:''' <small>Prof. Dr. Olaf Jensen, MD, MPH, PhD; Prof. Ing. MSc. Nailet Delgado Mujica; Prof. Magda Medir Mb, Tarragona Spain, Prof. MSc.Ph.D. Bishal Gyawali SDU; MSc.PhD Vivi Just-Nørregaard; Prof. Dr. Johan Hviid Andersen MD, PhD. Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Bruno Nørdam, Randers; Maritime Doctor. Erik Haarløv MD, Assens, Denmark; Dr. Maite Duque MD Venezuela; Dr. Indira Santos Panama MD; Med.Stud. Ashley Lezcano, Panama; Dr. Joseph Abesamis MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dra.Marj Gab Huerte, MD, Filippines:</small> <small>Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona and Cadiz, Spain; Dr. Alejandro Martinez, MD, MPH, Costa Rica; Christian Acheampong, MBA, New Jersey US; Dr. Med. Sci Finn Gyntelberg NFA.and Bispebj. Hosp. Denmark: Dr. Solomon Ching MD Filippines: Dr. Marcel Alcaraz MD Filippines;</small> ==References== syq9ko1czwncfaa400bedddjqlmoor7 2844600 2844597 2026-11-05T10:59:05Z JSINST 3110286 2844600 wikitext text/x-wiki <big>The John Snow prediabetes Institute is an international research network focused on early identification of prediabetes, prehypertension and the other types of metabolic syndrome in the early ages for lifestyle health promotion. (https://w.wiki/Skm7)</big> <big>'''Background:''' Millions of people are at an increased risk of developing metabolic syndrome, starting with atherosclerosis, pre-diabetes, type 2 diabetes mellitus, pre-hypertension, hypertension, and being overweight before the age of 30, often without any symptoms.[1] These conditions constitute the background for cardiovascular diseases (CVDs), which are the leading cause of death and disability worldwide. Early diagnosis and prevention is possible, and a cost-effective preventive strategy can improve long-term health outcomes. '''Prevalence studies''' are used to identify high-risk groups among specific occupational groups, with repeated studies conducted in schools and universities—targeting, for example, seafarers, nurses, and medical students. Alternatively, data from academic literature can be utilized to identify these high risk group by requesting authors of relevant publications to re-analyze their data.</big> <big>The 16-weeks '''intervention study''' include learning by video sequences and self-monitoring of blood sugar with Glukometer and self-evaluation of diet and physical activity <ref>https://www.sciencedirect.com/science/article/abs/pii/S1751991825001068 </ref>. The new challenging intervention studies include the AI-Digital lifestyle programs.<ref> «Digital Lifestyle Program Cuts Diabetes Risk by 46% in Prediabetics, Study of 130k+ Adults Reveals». News-Medical, 14 de abril de 2025. https://www.news-medical.net/news/20250414/Digital-lifestyle-program-cuts-diabetes-risk-by-4625-in-prediabetics-study-of-130k2b-adults-reveals.aspx.</ref> Educational materials is available from the international diabetes organisations e.g the ADA:<ref>https://professional.diabetes.org/diabetes-support-resources</ref> and the Health Promoting Schools.<ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref></big>[[File:Lifestyle Medicine Pillars.png|250px|right| <big>Lifestyle Medicine Pillars.png The focus of Lifestyle Medicine is on these 6 pillars</big> ]]'''<big>Prevalence studies</big>''' [[/The Maritime Health database design // |<big>1.1 The Maritime Health database design</big>]] <big>1.2 Nursing Students Health Database <ref> https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&dl=0&rlkey=onbjh4o8ko1lzdvgyi8nlrotk </ref></big> <big>1.3. Medical student's Health Database <ref>https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&dl=0&rlkey=xyfqen5trdc5lniaovipl548n </ref></big> <big>1.4. School childrens` Health database <ref> https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&dl=0&rlkey=zlyz5wn673wf7owettq3nx3h5 </ref></big> <big>1.5 Excel recoding of the clinical data (pre diabetes, hypertension etc) <ref>https://www.dropbox.com/scl/fi/9qvqeccto2kbuqvx7mdkl?r=ACsntb2AuiFkJbEBLObVdo0HMD67UZBBUGCGl1RPCPy1X5Y3K3e9Lf4s7cG7i7E_iH0IK-WkynXw_FNCDWybHuCgaNl_OM8yche2w2aapJGk5BTHybbsLpLayBU8VKPPqxNa2zDaNslBdQg0MhmKTFSG</ref></big> <big>'''Personal prevention''' Personal reversal of prediabetes relies on targeted lifestyle changes like eating a healthy diet, exercising regularly, and losing a modest amount of excess weight. (local Diabetes Associations)</big> <span lang="en" dir="ltr"><big>'''Core Institutional Strategies''' Health authorities and organizations implement group-based or digital programs. The workplaces are prompted to make targeted interventions, including nutrition and exercise programs, to safeguard long-term cardiometabolic health in the workforces.</big></span> <big>16 weeks human caused Intervention studies: English <ref>https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&dl=0&rlkey=7kzg91tqfgjskxf5aji8khicx </ref> Danish <ref>https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&dl=0&rlkey=x63w8oqvarz284zg2btq2johv </ref> Spanish <ref> https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&dl=0&rlkey=popmr1fnodh1v951v9l7k9ezv </ref> and '''AI-Powered Lifestyle Intervention''' in the Diabetes Prevention <ref>Mathioudakis, Nestoras, Benjamin Lalani, Mohammed S. Abusamaan, et al. «An AI-Powered Lifestyle Intervention vs Human Coaching in the Diabetes Prevention Program: A Randomized Clinical Trial». JAMA 334, n.º 23 (2025): 2079-89. https://doi.org/10.1001/jama.2025.19563.</ref></big> '''<big>Organisation</big>''' <big>- [[/The John Snow Institute bylaws /|The John Snow Institute bylaws]]</big> <big>- General research protocol draft <ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref></big> <big>- Health Promoting Schools <ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref></big> [[Category:Prediabetes ]] [[Category:Prediabetes ]] <big>'''Publications''' <ref name=":0"> https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf </ref></big> '''The John Snow Prediabetes Institute Research Network:''' <small>Prof. Dr. Olaf Jensen, MD, MPH, PhD; Prof. Ing. MSc. Nailet Delgado Mujica; Prof. Magda Medir Mb, Tarragona Spain, Prof. MSc.Ph.D. Bishal Gyawali SDU; MSc.PhD Vivi Just-Nørregaard; Prof. Dr. Johan Hviid Andersen MD, PhD. Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Bruno Nørdam, Randers; Maritime Doctor. Erik Haarløv MD, Assens, Denmark; Dr. Maite Duque MD Venezuela; Dr. Indira Santos Panama MD; Med.Stud. Ashley Lezcano, Panama; Dr. Joseph Abesamis MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dra.Marj Gab Huerte, MD, Filippines:</small> <small>Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona and Cadiz, Spain; Dr. Alejandro Martinez, MD, MPH, Costa Rica; Christian Acheampong, MBA, New Jersey US; Dr. Med. Sci Finn Gyntelberg NFA.and Bispebj. Hosp. Denmark: Dr. Solomon Ching MD Filippines: Dr. Marcel Alcaraz MD Filippines;</small> ==References== moesm3um727zv39mxjwdd8b88c5y9wv 2844601 2844600 2026-11-05T10:59:32Z JSINST 3110286 2844601 wikitext text/x-wiki <big>The John Snow prediabetes Institute is an international research network focused on early identification of prediabetes, prehypertension and the other types of metabolic syndrome in the early ages for lifestyle health promotion. (https://w.wiki/Skm7)</big> <big>'''Background:''' Millions of people are at an increased risk of developing metabolic syndrome, starting with atherosclerosis, pre-diabetes, type 2 diabetes mellitus, pre-hypertension, hypertension, and being overweight before the age of 30, often without any symptoms.[1] These conditions constitute the background for cardiovascular diseases (CVDs), which are the leading cause of death and disability worldwide. Early diagnosis and prevention is possible, and a cost-effective preventive strategy can improve long-term health outcomes. '''Prevalence studies''' are used to identify high-risk groups among specific occupational groups, with repeated studies conducted in schools and universities—targeting, for example, seafarers, nurses, and medical students. Alternatively, data from academic literature can be utilized to identify these high risk group by requesting authors of relevant publications to re-analyze their data.</big> <big>The 16-weeks '''intervention studies''' include learning by video sequences and self-monitoring of blood sugar with Glukometer and self-evaluation of diet and physical activity <ref>https://www.sciencedirect.com/science/article/abs/pii/S1751991825001068 </ref>. The new challenging intervention studies include the AI-Digital lifestyle programs.<ref> «Digital Lifestyle Program Cuts Diabetes Risk by 46% in Prediabetics, Study of 130k+ Adults Reveals». News-Medical, 14 de abril de 2025. https://www.news-medical.net/news/20250414/Digital-lifestyle-program-cuts-diabetes-risk-by-4625-in-prediabetics-study-of-130k2b-adults-reveals.aspx.</ref> Educational materials is available from the international diabetes organisations e.g the ADA:<ref>https://professional.diabetes.org/diabetes-support-resources</ref> and the Health Promoting Schools.<ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref></big>[[File:Lifestyle Medicine Pillars.png|250px|right| <big>Lifestyle Medicine Pillars.png The focus of Lifestyle Medicine is on these 6 pillars</big> ]]'''<big>Prevalence studies</big>''' [[/The Maritime Health database design // |<big>1.1 The Maritime Health database design</big>]] <big>1.2 Nursing Students Health Database <ref> https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&dl=0&rlkey=onbjh4o8ko1lzdvgyi8nlrotk </ref></big> <big>1.3. Medical student's Health Database <ref>https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&dl=0&rlkey=xyfqen5trdc5lniaovipl548n </ref></big> <big>1.4. School childrens` Health database <ref> https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&dl=0&rlkey=zlyz5wn673wf7owettq3nx3h5 </ref></big> <big>1.5 Excel recoding of the clinical data (pre diabetes, hypertension etc) <ref>https://www.dropbox.com/scl/fi/9qvqeccto2kbuqvx7mdkl?r=ACsntb2AuiFkJbEBLObVdo0HMD67UZBBUGCGl1RPCPy1X5Y3K3e9Lf4s7cG7i7E_iH0IK-WkynXw_FNCDWybHuCgaNl_OM8yche2w2aapJGk5BTHybbsLpLayBU8VKPPqxNa2zDaNslBdQg0MhmKTFSG</ref></big> <big>'''Personal prevention''' Personal reversal of prediabetes relies on targeted lifestyle changes like eating a healthy diet, exercising regularly, and losing a modest amount of excess weight. (local Diabetes Associations)</big> <span lang="en" dir="ltr"><big>'''Core Institutional Strategies''' Health authorities and organizations implement group-based or digital programs. The workplaces are prompted to make targeted interventions, including nutrition and exercise programs, to safeguard long-term cardiometabolic health in the workforces.</big></span> <big>16 weeks human caused Intervention studies: English <ref>https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&dl=0&rlkey=7kzg91tqfgjskxf5aji8khicx </ref> Danish <ref>https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&dl=0&rlkey=x63w8oqvarz284zg2btq2johv </ref> Spanish <ref> https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&dl=0&rlkey=popmr1fnodh1v951v9l7k9ezv </ref> and '''AI-Powered Lifestyle Intervention''' in the Diabetes Prevention <ref>Mathioudakis, Nestoras, Benjamin Lalani, Mohammed S. Abusamaan, et al. «An AI-Powered Lifestyle Intervention vs Human Coaching in the Diabetes Prevention Program: A Randomized Clinical Trial». JAMA 334, n.º 23 (2025): 2079-89. https://doi.org/10.1001/jama.2025.19563.</ref></big> '''<big>Organisation</big>''' <big>- [[/The John Snow Institute bylaws /|The John Snow Institute bylaws]]</big> <big>- General research protocol draft <ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref></big> <big>- Health Promoting Schools <ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref></big> [[Category:Prediabetes ]] [[Category:Prediabetes ]] <big>'''Publications''' <ref name=":0"> https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf </ref></big> '''The John Snow Prediabetes Institute Research Network:''' <small>Prof. Dr. Olaf Jensen, MD, MPH, PhD; Prof. Ing. MSc. Nailet Delgado Mujica; Prof. Magda Medir Mb, Tarragona Spain, Prof. MSc.Ph.D. Bishal Gyawali SDU; MSc.PhD Vivi Just-Nørregaard; Prof. Dr. Johan Hviid Andersen MD, PhD. Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Bruno Nørdam, Randers; Maritime Doctor. Erik Haarløv MD, Assens, Denmark; Dr. Maite Duque MD Venezuela; Dr. Indira Santos Panama MD; Med.Stud. Ashley Lezcano, Panama; Dr. Joseph Abesamis MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dra.Marj Gab Huerte, MD, Filippines:</small> <small>Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona and Cadiz, Spain; Dr. Alejandro Martinez, MD, MPH, Costa Rica; Christian Acheampong, MBA, New Jersey US; Dr. Med. Sci Finn Gyntelberg NFA.and Bispebj. Hosp. Denmark: Dr. Solomon Ching MD Filippines: Dr. Marcel Alcaraz MD Filippines;</small> ==References== lt4naj2v812jd0zugtzcyqwndhg5e1a 0 331104 2844473 2841832 2026-11-04T13:37:50Z U3292769 3106878 /* Physiological and psychological processes of excitement */ Fixing spelling mistake 2844473 wikitext text/x-wiki {{title|Excitement as an emotion:<br>What is the emotion of excitement and how does it influence behaviour and well-being?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}} [[File:2022-07-03 Basketball, Männer, European Qualifiers, Deutschland - Polen 1DX 1386 by Stepro.jpg|right|thumb|200px|'''Figure 1'''. Steve is ready and excited to play basketball.]] ; Consider this scenario It's 11:30 p.m. the night before Steve's championship basketball game for the state title. As the undefeated team's captain and leading scorer, he has been preparing his whole life for this moment, and tomorrow, he will get to showcase his skills towards a lifelong dream in front of teammates, coaches, fans, family, and professional basketball team recruiters. Instead of feeling nervous, he is suddenly excited. His heart beats faster. He feels energetic, alert, and unusually motivated. He starts to imagine himself having a great game. He practises his shooting form in the mirror one more time, sends basketball highlights and tape to his teammates for review and feedback, and even begins planning what he'll do and say if his team wins the game. The next morning, he wakes up before his alarm. Steve is still buzzing with anticipation. As he walks onto the court, his attention narrows onto the basket. He feels ready to act (see Figure 1). But there is a problem. Steve's excitement becomes so intense that he starts rushing his teams plays too quickly. He manoeuvres through bodies, overlooks an open shot, and finds himself making decisions and passes without thinking them through. So, was his excitement helpful or harmful? And why can it motivate us to pursue opportunities while also influencing how we think, behave, and experience well-being? {{RoundBoxBottom}} In Steve's case, excitement primes his body and propels him to prepare for his big game but this also comes with some unanticipated consequences as he begins to rush and miss information. The following chapter explores excitement as an emotion and how it can influence our behaviours and wellbeing. By examining how and why we get excited, and its physiological and psychological affects, we can better understand how we perceive and respond to it. This understanding may consciously allows us to reframe and engage with excitement as an active tool to shape our behaviours, experiences, and well-being. {{RoundBoxTop|theme=3}} '''Focus questions''' * What is excitement as an emotion, and how does it differ from other emotions? * What are the physiological and psychological processes that underpin excitement? * How does excitement influence motivation, cognition, and behaviour? * How does excitement contribute to, or undermine psychological well-being across individual and cultural contexts? {{RoundBoxBottom}} ==Understanding excitement as an emotion== Excitement is more than the feeling of happiness, it is generally characterised to be a positive, high-arousal emotional state associated with preparing an individual to engage with something anticipated or desired and is linked to things like interest, motivation and even risk-taking (Fredrickson, 2001; Ketonen et al., 2023). It is simultaneously unique from other emotions as it often involves relatively high levels of physiological and psychological activation, but similar, as many of its behaviours, internal feelings and physiological affects can be attributed to other emotional states such as anxiety, or even anger. Excitement is also affected by both the subjective experiences of the emotion and the physiological processes that accompany it as experienced by the individual (Schachter & Singer, 1962, as cited in Brooks, 2014; Russel, 1980). Early works by Stratton (1928) identified excitement as a distinct state and a unique emotion characterised by the the intense impulse to be directed towards a specific goal. Stratton noticed that excited individuals often were more organised for a specific task at hand and had greater motor movement and specifically targeted behaviour, proposing that excitement was a purposeful psychophysical state that dictates cognitive and motor responses. === The circumplex model of affect === One framework to further understand excitement is Russell's (1980) [[Motivation and emotion/Book/2019/Circumplex model of affect|circumplex model of affect]] which places excitement on a set of continuous dimensions: * [[w:Valence_(psychology)|Valence]], the appraisal of a situation or stimulus as either pleasant (positive) or unpleasant (negaitve). * [[wikipedia:Arousal|Arousal]], the degree by which a stimulus places the bodys nervous system into physical and mental readiness ranging from a deactived state (e.g. sleepiness) to an activated state (e.g. high alert). [[File:Valence-Arousal Circumplex.jpg|thumb|'''Figure 2.''' The circumplex model of affect (Russell, 1980). |266x266px]] Within this model, excitement consistently places in the upper-right quandrant, seen as high-arousal and with a positive valence (see Figure 2). Across Russell's experiment to investigate emotions, participants interacted with 28 tasks relating to the categorisation of emotions and their word meanings. Russell's model has helped propose the idea that emotions can act as affective states that are systematically interrelated, and not individual. Through the circumplex model of affect, excitement is best able to be understood and characterised compared to other emotions (see Table 1). ;Table 1 ''Comparison of Selected Emotions Along Valence and Arousal Dimensions'' {| class="wikitable" |+ !Emotion !Valence !Arousal/Activation Level !Primary Motivational Oreintation |- |Excitement |Positive (Pleasant) |High |High Apporach (Goal Pursuit) (Gable & Dreisbach, 2021) |- |Happiness |Positve (Pleasant) |Variable (Moderate to High) |Moderate Approach (Tugade et al., 2014) |- |Contentment |Positive (Pleasant) |Low |Low Approach (Post-Goal Savouring) (Tugade et al., 2014) |- |Anxiety |Negative (Unpleasant) |High |High Avoidance/Threat Vigilance (Brooks, 2014) |- |Sadness |Negative (Unpleasant) |Low |Low Avoidance/Withdrawal (Gable & Dreisbach, 2021) |} === The motivational dimensional model of affect (MDMA) === While the two-dimensional cirumplex model captures valence and arousal, Gable and Harmon-Jones' (2010) [[Motivation and emotion/Book/2025/Motivational dimensional model of affect|Motivational dimensional model of affect (MDMA)]] expands upon this, incorporating motivational intensity, the urge to approach or avoid a stimulus. MDMA demontrates that excitement is a high-intensity, approach-motivated positive state driven by the [[wikipedia:Gray's_biopsychological_theory_of_personality|Behavioural activation system (BAS)]]. Like excitement, high-intensity pre-goal states prepare organisms for direct action, wehereas low-intensity post-goal states like contentment signal reward and accomplishment (Gable & Dreisbach, 2021). === Excitement as an anticipatory emotion === The Japanese expression ''waku waku'' (ワクワク) describes a feeling of eager anticipation and excitement and is associated with the expection of something enjoyable (Hasada, 1994). Work by Macahizawa et al. (2020) proposes that excitement is fundamentally an anticipatory emotion driven by the predicting of positive future outcomes under uncertainty. To understand this temporal, future anticipatory dimension of excitement, electroencephalography (EEG) data was recorded from 28 adult participants when tested with different anticipatory tones while predicting an upcoming emotional image. From this data, the study developed a three-axial model integrating valence, arousal and a new cognitive dimension of expectation/anticipation. Through this study, Macahizawa et al., (2020) found that this added temporal dimension was significant to the emotional experience of excitement. {{RoundBoxTop|theme=3}} '''Scenario application''' Steve loves basketball, is skilled at the sport and thrives on competition. His subjective perception of his upcoming final game has a positive valence, high arousal and he is motivated, goal driven and anticpatory, signalling excitement. Steve's girlfriend, who is not so into basketball and hates crowds might have a completely different emotional state if she were put into the same position. One that most likely reflects an unpleasant valence and high arousal making her avoid working towards this reality entirely. {{RoundBoxBottom}} ==Physiological and psychological processes of excitement== {{anchor|Physiological responses}} === Physiological responses === With high arousal, excitement triggers robust [[wikipedia:Autonomic_nervous_system|autonomic nervous sytem]] (ANS) activation, preparing the body for action (Oxendine, 1970). * A study tracking the biometric markers of 136 high school students compared to their self-report on emotion (i.e. excitement, calm, boredom etc.) found that after controlling for physical activity, self-reported excitement was associated with increased physiological arousal. This included increased heart rate, alertness and energy with reduced heart rate variability, signalling a heightened sympathetic nervous system (Ketonen et al., 2023). The corresponding associations for anxiety and calmness were not found to be signficant thus possibly signalling excitement as a unique emotional experience despite ununique physiological responses i.e., anxiety sharing similar physiological responses to excitement. * Emotional excitement may also increase blood pressure and/or cause this to spike acutely (Piira et al., 2011). Physiological arousal isn't inherently positive or negative; its interpretation depends partly on context which may also alter if the individual is perceiving the physiological and psychological changes as excitement or something else (Schachter & Singer, 1962). [[File:Excitement from Two-Factor Theory.png|thumb|'''Figure 3.''' Physiological and psychological changes of excitement altered by the individual's perceived context (Schacter & Singer, 1962)|417x417px]] {{anchor|Psychological processes}} === Psychological processes === Psychologically, excitement alters how individuals perceive an upcoming task or event (Brooks, 2014). *Anticipation and excitement shift cognitive resources toward potential rewards or desirable outcomes, and has been shown to boost self-efficacy and perceived readiness to act in the face of a pre-goal reward expectation (Brooks, 2014). * Excitement can increase an individual's perceived readiness to act and even alter momentary thought-actions encouraging problem-solving, and affecting an individual's exploration (Fredrickson, 2001). {{anchor|Anticipatory emotion}} {{Robelbox|theme=5|width=40%|title=Quiz Question|iconwidth=48px|icon=Nuvola_apps_korganizer.svg}}<div style="{{Robelbox/pad}}"> <quiz display=simple> '''What is an example of excitement based on Steve's physiological and psychological changes?''' |type="()"} - Steve feels calm, his heart rate slows, and he begins to feel sleepy + Steve feels ready to take on the world, he feels his heart start to race as he hops out of bed - Following a bad practice, Steve feels his heart rate and alertness increase, he wants to leave the arena immediately </quiz> </div> {{RoundBoxBottom}} ==The influence of excitement on motivation, cognition, and behaviour== {{anchor|Approach motivation}} === Approach motivation and goal pursuit === High affective states have been linked to more approach-oriented behaviour, that is to say, motivation to move towards a specific goal. As excitement has a high positive affective state, this emotion has been linked with approach-oriented behaviour with high approach motivation increasing an individual's persistence and goal-directed activation. Research suggests that the emotional state of excitement can influence the willingness to pursue desirable goals. In these instances, excitement may act as a catalyst for approach motivations further reinforcing behavioural activation towards long term goals and increasing persistence. Gable & Dreisbach (2021) suggest that this may be because anticipating a positive outcome can motivate action before the reward is actually obtained and further supports the model of excitement as an anticipatory emotion. From this research, it is important to note that other emotional affective states can also have goal directed behaviour, however, as excitement is a high positive affect (valence) state and anticipatory, it can be posited that excitement compared to some other emotions may be stronger in approach-oriented behaviour. {{RoundBoxTop|theme=3}}'''Scenario application''' Steve's excitement leads him to practice and refine his basketball moves and watch tape before the game. An active approach to the challenge rather than not engaging with it or avoiding it completely. {{RoundBoxBottom}} {{anchor|Cognitive and behavioural effects}} === Excitements affect on cognition === Excitement can have affects on cognition, as well as effecting ones ability to take in information and environmental cues, making this broader or narrow, referred to as cognitive scope. Work on positive affect emotions have found that these positively valenced states can alter attentional focus. Fredrickson (2001) describes low-intensity, positive emotions (e.g. contentment, joy, pride) as being able to broaden momentary thought-action repertoires, fostering creative exploration, known as the Broaden-and-build theory. In contrast, high-intensity approach positive affect such as excitement narrows cognitive scope. Excitement may instead concentrate attention on a central goal stimuli while filtering out peripheral details leading to greater goal performance at the cost of additional detail and exploration (Gable & Dreisbach, 2021). The consequences of positive emotion on cognition may depend partly on motivational intensity and not always translate into greater performance. Excitement may help in focusing in on a task, but an overly narrow focus may make it more difficult to take in necessary information and stimuli; see Table 2. '''Table 2''' ''Comparison of low-intensity positive affect on cognitive scope versus high (Fredrickson, 2001; Gable & Dreisbach, 2021)'' {| class="wikitable" style="margin: auto;" ! !Low-intensity positive affect !High-intensity positive affect |- |'''Cognitive scope''' |Broadens attention to access wider range of stimuli and information. |Narrows attention towards central goal potentially filtering out relevant information. |- |'''Potential benefits''' |Encourages creative exploration, flexible thinking, alternative ideas. |Focused processing of goal-related information, persistence and performance on tasks with focused attention. |- |'''Potential limitations''' |Broad focus may cause less action towards specific goal required, or lack intense concentration needed towards target. |Narrow focus may cause important stimuli to be lost or peripheral information to be overlooked. |} Furthermore, according to Joseph Oxendine's (1968) emotional arousal mode, excitement's effect on motor sports performance follows task-specific requirements. In this research, high excitement improved tasks relying on raw strength, speed, and endurance (e.g., sprinting and football tacking) whereas, fine motor tasks and decision-making were impaired. Activities like basketball free throws and archery that required fine motor precision, steady hand control and complex tactical judgement were negatively affected. === Excitement on behaviour === ==== Impulsivity and risk taking ==== Unregulated excitement can lead to a sort of behavioural overshooting. Galentino et al. (2017) demonstrates that positive emotional arousal significantly increases preferences for monetary risks by over-valuing rewards over their potential losses. This could extend into other behaviours outside of fiscally motivated rewards. Supporting this, Andrede, Odean & Lin (2016) tested 55 experimental asset markets over a number of participants and found that excitement generated significantly larger market bubbles in magnitiude and peak amplitude driving immediate overevaluation across a first round. While excitement can increase motivion and engagement, this heightened arousal may also influence judgement and behavioural control. Consequently, excitement may encourage productive action in some situations while contributing to impulsive or risky decisions in others (Sohn et al., 2015; Galentino, Bonini, & Savadori, 2017; Andrade, Odean, & Lin, 2016). Storebeck & Clore (2008) also saw high affective arousal impact individuals judgement leading to poorer choices, however, some functions of learning and memory were greater. When helpful, excitement may aid an individual towards a goal by motivating them to prepare, practice and engage. However, when unhelpful, a person may become excessively excited about an invested reward, with focus and fixation leading to an underestimation of risks and causing impulsive decisions. == Excitement and its influence on wellbeing == === Cognitively restructuring emotions as excitement === Excitement and anxiety share similar high-arousal physiological profiles. A study by Brooks (2014) identified that these similar profiles were difficult to suppress with a majority of of participants believing variations of 'calming down' was the best cognitive strategy to assist when under high-pressure tasks. Instead, this research demonstrated that reframing anxiety via simple self-statements such as "I am excited" or prompts such as "Get excited" shifted participants mindsets from threat to opportunity without requiring an unphysiological drop in responses such as increased heart rate. When simply employing this cognitive restructure, users found increases in activities such as karaoke singing accuracy, public speaking persuasiveness, and math accuracy under pressure. {{RoundBoxTop|theme=3}}'''Scenario application''' During the finals game, Steve continues to make mistakes and he starts to get anxious. Under this high pressure environment, Steve starts to reframe his mistakes as opprtunities to learn and chances to adjust to his competition. He uses some simple self-statements which he feels helps him connect to his game. His teammates notice his performance improve.{{RoundBoxBottom}} === Excitement across the lifespan === Excitement and individuals with the proclivity towards this emotional state may impact them across their lifespan. Liu et al. (2026) evaluated the differential impact on pleasant positive affect (low-arousal/harmony) versus energised positive affect (high-arousal/excitement) on life satisfaction. Using random intercept cross-lagged panel models across a 9-wave longitudinal study, pleasant positive affect exhibited reicprocal associations with life satisfaction that were twice as strong as energised positive affect. Liu et al. (2026) proposed that pleasant positive affect may stabilise life satisfaction whereas more energised positive affect may be present in individuals with gaps between a current and desired states. These may drive personal growth, agency, and goal adaptation as opposed to those with a baseline satisfaction. More research may be required to expand upon this idea, but most individuals may find themselves across both states dictated by life experiences, personal interest and their own perception of ability and performance across tasks. Overall, positive affect in general has been shown to increase life satisfaction and relationships over time (Liu et al., 2016). As excitement is a positive emotional experience, it too may contribute to ones subjective well-being. By making activities feel more engaging, this anticipation can lead to more engagement and possibly reward. Excitement may not necessarily improve well-being simply because it is a positive emotional state, rather, the emotion may encourage preparation, engagement and goal pursuit which can further contribute to a sense of accomplishment (Liu et al. 2026). === Excitement as viewed cross-culturally === [[File:Excitement and anticipation at the Reykjavík Children’s Culture Festival 2016 (26145642903).jpg|thumb|'''Figure 4.''' Excitement may be viewed differently across cultures]] Tsai (2024) and Lim (2016) highlight that emotional valuation may vary systematically across cultures, with some individualistic Westerm cultures seemingly valuing high arousal positive states, such as excitement and enthusiasm as compared to some Eastern cultures. This may be due to Western cultures individualism, that is to say, the desire to assert personal autononmy and influence on others. Eastern cultures on the other hand might value low arousal positive states such as calmness, serenity and peacefulness as their ideal affect to promote social harmony so as to better support collectivist values. As excitement is a perception of an event and an individual's own physiological and psychological states, one's own culture and upbringing may begin to give rise to how to react and value this emotional state across a lifespan but also population. {{anchor|Does excitement help or hinder behaviour?}} === Does excitement help or hinder behaviour? === With helpful excitement, an individual might move towards a goal by preparing, practising, engaging and performing behaviours that lead to a desired outcome. However, with unhelpful excitement a person may become excessively excited about an invested reward, this focus and fixation may lead them to underestimate risks and form and impulsive decision (Galentino et al., 2017). Extremely high emotional arousal may become overwhelming, with the consequences of excitement dependent on the context and the individual's ability to regulate their behaviour. Excitement should not necessarily be conceptualised as beneficial simply because it is a positive emotion. Excitement can support well-being when it facilitates meaningful engagement, motivation and goal pursuit, but its effects depends on intensity, context and behavioural regulation. {{RoundBoxTop|theme=9}}'''Intervention application''' To reduce the effects of overwhelm during high emotional arousal, interventions should work in teaching individuals to identify when excitement is too great and use cognitive restructures in the form of strategies like 'simple statements'.{{RoundBoxBottom}} ==Conclusion== Excitement is a positively valenced, high-arousal emotional state associated with anticipation, approach motivation and readiness to act. It can encourage preparation, persistence and engagement with meaningful goals. Although, following in similar physiological traits to other emotions such as anxiety, excitement may differ from other emotions due to its combination of positive valence, high activation/arousal, approach to goals and anticipatory nature (Russell, 1980; Gable & Harmon-Jones, 2010; Macahizawa et al., 2020). Excitement as a state triggers activation of the autonomic nervous system, preparing itself for some kind of action (Oxendine, 1970). Common physiological responses include elevated heart rate, alertness, energy and rise in blood pressure (Ketonen et al., 2023). Psychologically, excitement can boost an individual's readiness to act, perceived self-efficacy, and cognitive scope (Brooks, 2014; Fredrickson, 2001), while also causing them to underestimate or overevaluate certain aspects of the situation (Galentino, Bonini, & Savadori, 2017; Andrade, Odean, & Lin, 2016). As a catalyst to engaging with actions towards a certain goal, excitement can effect ones cognition, narrowing or widening its scope (Fredrickson, 2001), and possibly leading to greater impulsive or risk taking behaviours (e.g., Galentino et al., 2017). Excitement can also be a powerful tool in restructuring other emotions, and as a positive emotion, may increase satisfaction across the lifespan (Liu et al., 2016). With its value differing across culture, excitement can support well-being by promoting engagement and pursuit of rewarding experiences, although excessive arousal or poorly regulated excitement may produce less beneficial outcomes. Understanding excitement therefore requires recognising it not simply as a pleasurable feeling, but as an emotion that prepares people to act and can shape how they think, behave and experience their well-being. {{RoundBoxTop|theme=3}} '''Take-home message''' Excitement is a unique emotional state that can be as much of a powerful tool in goal pursuit, action and well-being as it is simply a positive state of being. An event that strikes fear into one, may be exciting for another, therefore, excitement and emotions as whole come from the unique interplay of our active and passive shaping of the significance we assign to our experiences. {{RoundBoxBottom}} {{RoundBoxTop|theme=11|title=Quiz}}<quiz display=simple> {'''Chapter quiz:''' Steve and his team begin to pull away in the final minutes of the game. A smile emerges from his face as he anticipates the messages and coverage he will receive from the crowd and media. On the otherhand, his teammate Bob, hates attention and he begins to feel stressed. According to the chapter, what best explains this situation? |type="()"} +Their individual perceptions and interpretations of the sitatuion influence whether they experience heightened arousal as excitement or stress. -Everyone experiences excitement the same way. -Their emotional responses are determined entirely by physiological arousal, rather than how they interpret the situation. </quiz> {{RoundBoxBottom}} ==See also== * [[w:Affect_(psychology)|Affect (psychology)]] (Wikipedia) * [[Motivation and emotion/Book/2019/Circumplex model of affect|Circumplex model of affect]] (Book chapter, 2019) * [[Motivation and emotion/Book/2025/Motivational dimensional model of affect|Motivational dimensional model of affect]] (Book chapter, 2025) ==References== {{Hanging indent|1= Andrade, E. B., Odean, T., & Lin, S. (2016). Bubbling with Excitement: An Experiment. 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A., & Peterson, B. S. (2005). The circumplex model of affect: An integrative approach to affective neuroscience, cognitive development, and psychopathology. ''Development and Psychopathology, 17''(3), 715–734. https://doi.org/10.1017/S0954579405050340 Robinson, M. D., & Irvin, R. L. (2022). Out of the shell, into the world: preferences for openness as a substrate of approach motivation and positive affect. ''Motivation and Emotion, 46''(3), 307–318. https://doi.org/10.1007/s11031-022-09934-5 Russell, J. A. (1980). A circumplex model of affect. Journal of Personality and Social Psychology, 39(6), 1161–1178. https://doi.org/10.1037/h0077714 Schachter, S., & Singer, J. (1962). Cognitive, social, and physiological determinants of emotional state. ''Psychological Review, 69''(5), 379–399. https://doi.org/10.1037/h0046234 Smith, K. E., Woodard, K., & Pollak, S. D. (2025). Arousal May Not Be Anything to Get Excited About. ''Emotion Review, 17''(1), 3–15. https://doi.org/10.1177/17540739241303499 Sohn, J.-H., Kim, H.-E., Sohn, S., Seok, J.-W., Choi, D., & Watanuki, S. (2015). Effect of emotional arousal on inter-temporal decision-making: an fMRI study. ''Journal of Physiological Anthropology, 34''(1), Article 8. https://doi.org/10.1186/s40101-015-0047-5 Storbeck, J., & Clore, G. L. (2008). Affective Arousal as Information: How Affective Arousal Influences Judgments, Learning, and Memory. ''Social and Personality Psychology Compass, 2''(5), 1824–1843. https://doi.org/10.1111/j.1751-9004.2008.00138.x Stratton, G. M. (1928). The function of emotion as shown particularly in excitement. ''Psychological Review, 35''(5), 351–366. https://doi.org/10.1037/h0071406 Tsai, J. L. (2024). Investigating culture and emotion: From responses to ideals. In M. J. Gelfand, C.-Y. Chiu, & Y.-Y. Hong (Eds.), ''Handbook of advances in culture and psychology'' (pp. 54–117). Oxford University Press. Tugade, M. M., Shiota, M. N., & Kirby, L. D. (Eds.). (2014). ''Handbook of positive emotions'' (1st ed.). Guilford Press. }} ==External links== * [https://www.youtube.com/watch?v=S9tBnXz2B1Q How to turn anxiety into excitement] (SciShow Psych) * [https://fi.edu/en/blog/science-emotions-excitement?ct=1787858646114 Science of emotions: excitement] (The Franklin Institue) * [https://www.simplypsychology.com/articles/anxiety-reappraisal-strategies Cognitive reappraisal for anxiety: evidence-based strategies that work] (Simply Psychology) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] dag2k0p15t89fm6t06bdcwlc7wik9zb 0 331106 2844558 2844342 2026-11-05T02:10:35Z Jtneill 10242 /* Social exchange theory (Thibaut & Kelley, 1959) */ Move citation 2844558 wikitext text/x-wiki {{title|Self-disclosure and emotional intimacy:<br>How does self-disclosure foster emotional closeness in relationships?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}} [[File:Stockcouple.jpg|thumb|'''Figure 1'''. Image of couple hugging {{ic|Explain what this has to do with self-disclosure}/}]] ;Picture this... Jim and Pam (see Figure 1) have been dating for a few months; things have been going well but at times there is noticeable emotional distance between them. One night Jim tells Pam that “I haven’t talked to anyone about this, but I’m terrified one day you will get bored and leave”, {{g}} this confession opens the flood gates for Jim and Pam to share their deepest fears and anxieties. In this moment, an invisible barrier dissolves, they have discovered a new profound sense of trust and understanding. A new and deeper layer of their relationship has been unlocked; there is an emotional closeness that wasn’t there before. Why does this self-disclosure from Jim create this emotional closeness between him and Pam? {{RoundBoxBottom}} [[wikipedia:Self-disclosure|Self-disclosure]] is the voluntary process of sharing personal thoughts, feeling and information with another person. Self-disclosure is one of the most significant and effective ways to initiate and enhance relationship intimacy, and to maintain positive personal relationships (Forgas, 2011). Intimacy or intimate come from the Latin terms ''intimus'' meaning innermost, and ''intimare'' meaning to make the innermost known (Reis, 2018). [[wikipedia:Emotional_intimacy|Emotional intimacy]] refers to a shared sense of understanding, trust and closeness within an interpersonal relationship. This chapter discusses what both self-disclosure and emotional intimacy mean, and how self-disclosures can be encouraged and responded to, with reference to psychological theories and research. {{RoundBoxTop|theme=11}} ;Focus questions [[File:Crystal Clear app ktip.svg|left|20px|]] *What is the self-disclosure? *What is emotional intimacy and why is it important? *How can self-disclosure be fostered to improve emotional intimacy in relationships? {{RoundBoxBottom}} == Self-disclosure == Self-disclosure is defined as the process of divulging personal experiences, thoughts and feelings to others (Yu & Li, 2026). Self-disclosure has long been discussed and researched for its relationship to emotional intimacy {{f}}. According to the research, we tend to like people who disclose more, we disclose more to those we like, and when we disclose, we like those we disclosed to more afterwards (Utz, 2015) {{expand}}. === Social exchange theory === [[File:Costs vs rewards.png|thumb|247x247px|'''Figure 2.''' Social Exchange Theory, costs versus rewards {{g}}|right]] [[wikipedia:Social_exchange_theory|Social Exchange Theory]] is a psychological framework that states human behaviours and relationships work through a cost-benefit analysis (see Figure 2) (Thibaut & Kelley, 1959). This core concept can be applied to that of self-disclosure, in which individuals assess the potential rewards and risks of sharing personal information. In the case of self-disclosure, possible rewards or benefits of divulging personal information may be validation, emotional relief and enhanced closeness {{f}}. If we assess that these are likely outcomes of self-disclosing we are more likely to do so. In contrast, potential costs or risks of self-disclosure may be a loss of privacy, judgement or fear of rejection and in this circumstance we are much less likely to self disclose. Interpersonal relationships thrive when self-disclosure remains profitable or rewarding. {{f}} ==== Norm of reciprocity ==== Within Social Exchange Theory is the concept of the [[wikipedia:Norm_of_reciprocity|norm of reciprocity]] {{f}}. This entails the universal social expectation that people should return or repay favours, gifts and acts of kindness with similar treatment {{f}}. For example, if someone gifts you an expensive item for your birthday, you will likely feel an internal obligation to get them a similarly expensive item for their birthday. This norm enables smooth, equal and fair social exchanges (Burger et al., 2009). This concept is also applicable to self-disclosure, {{g}} research has shown that self-disclosure is reciprocal in existing relationships and in the context of strangers (Sprecher et al., 2013). Hence, reciprocal self-disclosure is the process where one person shares personal thoughts, feelings or information and the other person responds with similar personal details in return. === Social penetration theory (Altman & Taylor, 1973) === [[File:Bdspt.model.webp|'''Figure 3.''' Social Penetration Theory Model (Pecune, 2013).|alt=Figure 2. Social Penetration Theory Model (Pecune, 2013).|thumb|300x300px]] [[wikipedia:Social_penetration_theory|Social Penetration Theory]] is a social psychological model that explains how interpersonal relationships evolve from superficial or surface level interactions to deep emotional intimacy through self-disclosure {{f}}. Social Penetration Theory uses an onion model (see Figure 3) as a metaphor to describe how the social penetration process works, in which individuals "peel back" others layers of personal information through social interaction (self-disclosures) {{f}}. One aspect of Social Penetration Theory is the types of information shared, which is referred to as levels {{f}}. The first {{g}} being the Superficial level or in other words, surface level information which includes visible attributes. The second level is the Intimate or peripheral level information, which is information that people usually feel free to share in any general social situation, such as their name and where they are from. Personal or Intermediate level information is the third level, in which includes information that is not always readily shared but that is also not very private, such as values and goals. Lastly, the Core or central level information is regarded as highly private and is shared when someone feels close to the other person or believes they are very trustworthy (Manning, 2019). An example of Core level information may be one's insecurities or fears. This theory states that for interpersonal relationships to develop there must be information exchanged between those within the relationship. A significant aspect of the theory is that of [[wikipedia:Social_penetration_theory#Breadth_and_depth|breadth and depth]] {{f}}. Breadth refers to the number of topics discussed, whereas depth refers to the level of intimacy of these discussions. For example, breadth may include discussing hobbies, family, interests and depth may be discussing feelings surrounding family issues or future goals (Carpenter & Greene, 2016), depth discussions may increase intimacy levels {{why}} {{g}}. According to this {{what}} theory {{g}} self-disclosure passes through phases as an interpersonal relationship progresses, {{g}} these stages include orientation, exploratory affective exchange, affective exchange, stable exchange and in some cases depenetration (see Table 1). The first stage is orientation, in which individuals only share superficial level information. Exploratory affective exchange is the second stage, where individuals may share beyond superficial information and may increase the breadth of topics discussed. The third stage is affective exchange, which includes more casual self-disclosures, communication and information from deeper levels, this stage is often reached with close friends or romantic partners. Stable exchange is the fourth stage, in which communication is open and honest, self-disclosures are continuous, and there is breadth and depth throughout conversations. This stage is maintained with few relationships such as romantic partners, close family and friends. Finally, depenetration or dissolution is a possible stage, whereby self-disclosure is reduced as a result of conflict or stress (Carpenter & Greene, 2016). {| class="wikitable" |+Table 1. The five stages of social penetration {{fact}} and examples. !Five stages of social penetration !Examples: |- |Orientation |Small talk, adhering to social norms. |- |Exploratory affective exchange |Personal attitudes & preferences. |- |Affective exchange |Casual relaxed communication, honest opinions, deeper emotional disclosures. |- |Stable exchange |Continuous open connection, deep core values, emotional reactions are predictable, breadth and depth explored. |- |Depenetration |Optional phase, communication breadth & depth withdrawal. |} == Emotional intimacy & its importance == Emotional intimacy is a part of interpersonal relationships, defined as a shared sense of closeness, trust and mutual understanding. Emotional intimacy happens between individuals when they consistently show each other who they really are and respond to each other in ways that promote feelings of safety, security and understanding. Thus, emotional intimacy can be experienced in a variety of different kind of relationships including, romantic partners, friends and family members. {{f}} === The intimacy process model (Reis & Shaver, 1988) === Reis and Shaver (1988) argue that intimacy is a dynamic, moment to moment transactional and cyclical process. The intimacy process model shows how emotional intimacy builds through a cyclical interactive progression between two individuals. This model uses self-disclosure, responsiveness, and perceived responsiveness as core components of emotional intimacy development. [[File:Intimacy process model.png|thumb|420x420px|'''Figure 4.''' The intimacy process model (Reis & Shaver, 1988).|left]] In this model {{g}} an interaction occurs between two people A and B, or for this example we will use the names Annabel and Bridgette. The model shows a single interaction in which Annabel is self-disclosing and Bridgette is responding (see Figure 4), however in most situations both individuals would freely move between self-disclosing and responding. In this example, the intimacy process begins with Annabel. Annabel has some mixture of motives, goals or fears that results in a self disclosure, whereby Bridgette must respond. As the role of the responder Bridgette is influenced by a mixture of concerns, which can be positive or negative {{g}}. Bridgette may want to be a good friend or partner or on the other hand, Bridgette may fear commitment, feel that Annabel is dependent on them and so forth. Annabel may intend {{sp}} on one thing, where Bridgette may perceive another. As Bridgette interprets the disclosure and expressions from Annabel, a complex mixture of motives and goals is at work, {{g}} this is referred to as the interpretive filter. In this model {{g}} Reis and Shaver (1988) argue that Bridgette's response to Annabel's self-disclosure is just as important as Annabel's initial disclosure, and will determine whether the interaction will become emotionally intimate. Suitable responses will enhance feelings of connectedness between the individuals, and inappropriate responses or non-responsiveness will keep the individuals at a distance. However, what matters most is Annabel's interpretation and reaction to Bridgette's response, for example if Annabel interprets Bridgette's response as unsupportive or too intrusive, greater intimacy is unlikely to be achieved (Reis & Shaver, 1988). === The importance of emotional intimacy === Emotional intimacy is intrinsically gratifying, and involves feelings of safety, trust, understanding, validation, acceptance and overall feeling cared for {{f}}. Studies have found associations between emotional intimacy, communication and relationship satisfaction, such that partners' perception of their emotional intimacy predicts both their own relationship satisfaction as well as their partners (Yoo et al., 2014). Similarly, those with strong intimate relationships tend to have higher life satisfaction (Amati et al., 2018) and wellbeing (Greene et al., 2006). Overall, intimacy can result in reciprocal intimacy, mutual liking and relationship progress (Forgas, 2011). Emotional intimacy has been found to play a role in reducing stress responses (Phillips et al., 2009). It has also been found that loneliness and depression partly stem from a lack of emotional intimacy (Rokach & Ohiliber-Lignieres, 2015). This research highlights just how important emotional intimacy is in everyday life, therefore it is important to learn how to improve emotional intimacy. {{RoundBoxTop|theme=3}} ;Key points: [[File:Crystal Clear app ktip.svg|left|20px|]] *Intimacy is related to relationship and life satisfaction. *Emotional intimacy is related to higher wellbeing and reduced stress responses. *Emotional intimacy can facilitate reciprocal intimacy, mutual liking and relationship progression. {{RoundBoxBottom}} ==Fostering self-disclosure== Rewarding relationships are practically inconceivable without self-disclosure. Self-disclosure is important in initiating, maintaining and enhancing emotional intimacy within interpersonal relationships, which has been linked to relationship success and satisfaction, as well as improved mental health (Forgas, 2011). Therefore, learning to foster and encourage self-disclosure is essential for everyday life. === How to encourage & respond to self-disclosures === In order to encourage self-disclosures from others we must remember Social Exchange Theory (Thibaut & Kelley, 1959) and the cost-benefit analysis. If someone wanted to disclose something to you, think to yourself, would it be rewarding for them? Making a self-disclosure conversation rewarding can be done in a multitude of ways. Social cues such as enquiries of concern (asking if someone is okay for example) or initiating with a self-disclosure displays inviting cues to the other person to disclose (Greene et al., 2006). It has also been found that moods have an effect of self-disclosure, such that positive moods increase self-disclosure intimacy, abstractness and breadth, whereas negative moods reduce intimacy but improve reciprocity of self-disclosure (Forgas, 2011), in which is an important aspect to take into account. Additionally, adherence to the norm of reciprocity, in which you can respond with a similar self-disclosure. Similarly, the intimacy process model (Reis & Shaver, 1988) highlights the underlying cause of a self-disclosure as well as the importance of responsiveness to self-disclosures. It is helpful to remember that there is an underlying motive, goal, fear or reason for someone to self-disclose. In the case of responsiveness, High or good responsiveness is comprised of the content of the response (elaborating on or matching disclosure), the style of response (showing concern and care) as well as the timing of the response (immediate or delayed). However, it is extremely important to take into account boundaries and assess the appropriateness of the situation, such as the physical environment where the interaction is occurring (Greene et al., 2006). {{RoundBoxTop|theme=3}} ;Key points: [[File:Crystal Clear app ktip.svg|left|20px|]] How to encourage and respond to self-disclosure: *The cost-benefit analysis: make the interaction rewarding through listening and responsiveness. *Social cues: enquiries of concern and initiating self-disclosure. *The norm of reciprocity: use reciprocal self-disclosure. *Boundaries: assess the appropriateness of the situation. {{RoundBoxBottom}} ==Conclusion== Self-disclosure creates a sense of understanding, safety, and trust that is essential for emotionally intimate interpersonal relationships. This self-disclosure process is intrinsic and crucial to progress interpersonal relationships. Self-disclosure drives emotional intimacy by creating a loop of vulnerability, validation and mutual understanding. Emotional intimacy has effects on aspects of everyday life including on communication and relationship satisfaction, such that perception of emotional intimacy can predict both one's own relationship satisfaction as well as their partners. Strong emotionally intimate relationships tend to have higher life satisfaction and wellbeing. Additionally, intimacy can result in reciprocal intimacy, mutual liking and relationship progress. Emotional intimacy has also been found to play a role in reducing stress responses and a lack of emotional intimacy can play a role in loneliness and depression. Therefore, it is important to improve emotional intimacy through learning to foster and encourage self-disclosure. Self-disclosure can be fostered through social cues, making the interaction rewarding and validating, using the norm of reciprocity and taking responsiveness into account. {{RoundBoxTop|theme=11}} ;Take home messages [[File:Crystal Clear app ktip.svg|left|20px|]] *Self-disclosure is the process of divulging personal experiences, thoughts and feelings to others. *Emotional intimacy is a shared sense of closeness, trust and mutual understanding, that can have effects on life and relationship satisfaction, wellbeing, stress and loneliness. *Self-disclosure can be fostered to improve emotional intimacy in relationships through reciprocal self-disclosure, responsiveness, and social cues. {{RoundBoxBottom}} ==See also== * [[Motivation and emotion/Book/2026/Responsiveness and interpersonal trust|Responsiveness and interpersonal trust]] (Book chapter, 2026) * [[Motivation and emotion/Book/2025/Self-disclosure and well-being|Self-disclosure and well-being]] (Book chapter, 2025) * [[wikipedia:Social_exchange_theory|Social exchange theory]] (Wikipedia) * [[wikipedia:Social_penetration_theory|Social penetration theory]] (Wikipedia) ==References== {{Hanging indent|1= Altman, I., & Taylor, D. (1973). ''Social penetration: The development of interpersonal relationships''. Halsted Press. https://psycnet.apa.org/record/1973-28661-000 Amati, V., Meggiolaro, S., Rivellini, G., & Zaccarin. (2018). Social relations and life satisfaction: the role of friends. ''Genus, 74''(7). 1-18. https://doi.org/10.1186/s41118-018-0032-z Burger, J. M., Sanches, J., Imberi, J. E., & Grande L. R. (2009). The norm of reciprocity as an internalized social norm: Returning favors even when no one finds out. ''Social Influence, 4''(1). 11-17. https://doi.org/10.1080/15534510802131004 Carpenter, A., & Greene, K. (2016). Social penetration theory. In C. R. Berger & M. E. Roloff (Eds.), ''The International Encyclopedia of Interpersonal Communication'' (1st ed., pp. 1-4). John Wiley & Sons. https://doi.org/10.1002/9781118540190.wbeic0160 Derlega, V. J., Winstead, B. A., & Greene, K. (2008). Self-disclosure and starting a close relationship. In S. Sprecher, A. Wenzel, & J. Harvey (Eds.), ''Handbook of relationship initiation''. Psychology Press. 53–174. https://psycnet.apa.org/record/2008-09972-008 Forgas, J. P. (2011). Affective influences on self-disclosure: Mood effects on the intimacy and reciprocity of disclosing personal information. ''Journal of Personality and Social Psychology, 100''(3). 449-461. https://doi.org/10.1037/a0021129 Greene, K., Derlega, V. J., & Mathews, A. (2006). Self-disclosure in personal relationships. In A. L. Vangelisti & D. Perlman (Eds.), ''The Cambridge handbook of personal relationships.'' (pp. 409-427). Cambridge University Press. https://doi.org/10.1017/CBO9780511606632.023 Manning, J. (2019). Thinking about interpersonal relationships and social penetration theory: Is it the same for lesbian, gay or bisexual people? In C. J. Liberman, A. S. Rancer & T. A. Avtgis (Eds.), ''Casing Communication Theory'' (pp. 293-306). Kendall Hunt Publishing Company. Pecune, F. (2013). Toward a computational model of social relations for artificial companions. ''Proceedings - 2013 Humaine Association Conference on Affective Computing and Intelligent Interaction''. 677-682. [https://www.researchgate.net/publication/261271664%20Toward%20a%20Computational%20Model%20of%20Social%20Relations%20for%20Artificial%20Companions DOI:10.1109/ACII.2013.118] Phillips, A., Gallagher, S., & Carroll, D. (2009) Social support, social intimacy, and cardiovascular reactions to acute psychological stress. ''Annals of Behavioural Medicine, 37''. 38-45. http://dx.doi.org/10.1007/s12160-008-9077-0 Reis, H. T. (2018). Intimacy as an interpersonal process. In H. T. Reis (Ed.), ''Relationships, Well-Being and Behaviour: Selected works of Harry T. Reis'' (pp. 209-244). Routledge. https://doi.org/10.4324/9780203732496 Reis, H. T., & Shaver, P. (1988). Intimacy as an interpersonal process. In S. Duck, D. H. Fay, J. K. Simpson, & J. C. Murray (Eds.), ''Handbook of personal relationships: Theory, research and applications'' (p. 367–389). John Wiley & Sons. Rokach, A., & Philibert-Lignieres, G. (2015). Intimacy, loneliness & infidelity. ''The Open Psychology Journal, 8''. 71-77. DOI: 10.2174/1874350101508010071 Sprecher, S., Treger, S., Wondra, J. D., Hilaire, N., & Wallpe, K. (2013). Taking turns: Reciprocal self-disclosure promotes liking in initial interactions. ''Journal of Experimental Social Psychology, 49''(5). 860-866. https://doi.org/10.1016/j.jesp.2013.03.017. Tolstedt, B. E., & Stokes, J. P. (1984). Self-disclosure, intimacy, and the depenetration process. ''Journal of Personality and Social Psychology, 46''(1). 84-90. https://doi.org/10.1037/0022-3514.46.1.84 Utz, S. (2015). The function of self-disclosure on social network sites: Not only intimate, but also positive and entertaining self-disclosures increase the feeling of connection. ''Computers in Human Behaviour, 45''. 1-10. https://doi.org/10.1016/j.chb.2014.11.076. Yoo, H., Bartle-Haring, S., Day. R., & Gangamma, R. (2014). Couple communication, emotional and sexual intimacy, and relationship satisfaction. ''Journal of Sex & Marital Therapy, 40''(4), 275-293. https://doi.org/10.1080/0092623X.2012.751072 Yu, Y., & Li, W. (2026). A fNIRS hyperscanning study on enhancing intimacy through emotional self-disclosure from the disclosure-responder perspective. ''BMC Psychology, 14''. https://doi.org/10.1186/s40359-026-04257-3 }} ==External links== * [https://www.youtube.com/watch?v=Z4bRXozREKU&t=90s Self-disclosure in personal relationships: How to stop oversharing] (youtube.com) * [https://www.psychologytoday.com/au/blog/between-the-sheets/202011/why-romantic-intimacy-requires-self-disclosure Why romantic intimacy requires self-disclosure] (psychologytoday.com) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Relationships]] i9gvs9ce5f0oveqn54f52pqsvbjwur2 2844559 2844558 2026-11-05T02:11:35Z Jtneill 10242 /* Social penetration theory (Altman & Taylor, 1973) */ Move citation 2844559 wikitext text/x-wiki {{title|Self-disclosure and emotional intimacy:<br>How does self-disclosure foster emotional closeness in relationships?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}} [[File:Stockcouple.jpg|thumb|'''Figure 1'''. Image of couple hugging {{ic|Explain what this has to do with self-disclosure}/}]] ;Picture this... Jim and Pam (see Figure 1) have been dating for a few months; things have been going well but at times there is noticeable emotional distance between them. One night Jim tells Pam that “I haven’t talked to anyone about this, but I’m terrified one day you will get bored and leave”, {{g}} this confession opens the flood gates for Jim and Pam to share their deepest fears and anxieties. In this moment, an invisible barrier dissolves, they have discovered a new profound sense of trust and understanding. A new and deeper layer of their relationship has been unlocked; there is an emotional closeness that wasn’t there before. Why does this self-disclosure from Jim create this emotional closeness between him and Pam? {{RoundBoxBottom}} [[wikipedia:Self-disclosure|Self-disclosure]] is the voluntary process of sharing personal thoughts, feeling and information with another person. Self-disclosure is one of the most significant and effective ways to initiate and enhance relationship intimacy, and to maintain positive personal relationships (Forgas, 2011). Intimacy or intimate come from the Latin terms ''intimus'' meaning innermost, and ''intimare'' meaning to make the innermost known (Reis, 2018). [[wikipedia:Emotional_intimacy|Emotional intimacy]] refers to a shared sense of understanding, trust and closeness within an interpersonal relationship. This chapter discusses what both self-disclosure and emotional intimacy mean, and how self-disclosures can be encouraged and responded to, with reference to psychological theories and research. {{RoundBoxTop|theme=11}} ;Focus questions [[File:Crystal Clear app ktip.svg|left|20px|]] *What is the self-disclosure? *What is emotional intimacy and why is it important? *How can self-disclosure be fostered to improve emotional intimacy in relationships? {{RoundBoxBottom}} == Self-disclosure == Self-disclosure is defined as the process of divulging personal experiences, thoughts and feelings to others (Yu & Li, 2026). Self-disclosure has long been discussed and researched for its relationship to emotional intimacy {{f}}. According to the research, we tend to like people who disclose more, we disclose more to those we like, and when we disclose, we like those we disclosed to more afterwards (Utz, 2015) {{expand}}. === Social exchange theory === [[File:Costs vs rewards.png|thumb|247x247px|'''Figure 2.''' Social Exchange Theory, costs versus rewards {{g}}|right]] [[wikipedia:Social_exchange_theory|Social Exchange Theory]] is a psychological framework that states human behaviours and relationships work through a cost-benefit analysis (see Figure 2) (Thibaut & Kelley, 1959). This core concept can be applied to that of self-disclosure, in which individuals assess the potential rewards and risks of sharing personal information. In the case of self-disclosure, possible rewards or benefits of divulging personal information may be validation, emotional relief and enhanced closeness {{f}}. If we assess that these are likely outcomes of self-disclosing we are more likely to do so. In contrast, potential costs or risks of self-disclosure may be a loss of privacy, judgement or fear of rejection and in this circumstance we are much less likely to self disclose. Interpersonal relationships thrive when self-disclosure remains profitable or rewarding. {{f}} ==== Norm of reciprocity ==== Within Social Exchange Theory is the concept of the [[wikipedia:Norm_of_reciprocity|norm of reciprocity]] {{f}}. This entails the universal social expectation that people should return or repay favours, gifts and acts of kindness with similar treatment {{f}}. For example, if someone gifts you an expensive item for your birthday, you will likely feel an internal obligation to get them a similarly expensive item for their birthday. This norm enables smooth, equal and fair social exchanges (Burger et al., 2009). This concept is also applicable to self-disclosure, {{g}} research has shown that self-disclosure is reciprocal in existing relationships and in the context of strangers (Sprecher et al., 2013). Hence, reciprocal self-disclosure is the process where one person shares personal thoughts, feelings or information and the other person responds with similar personal details in return. === Social penetration theory === [[File:Bdspt.model.webp|'''Figure 3.''' Social Penetration Theory Model (Pecune, 2013).|alt=Figure 2. Social Penetration Theory Model (Pecune, 2013).|thumb|300x300px]] [[wikipedia:Social_penetration_theory|Social Penetration Theory]] is a social psychological model that explains how interpersonal relationships evolve from superficial or surface level interactions to deep emotional intimacy through self-disclosure (Altman & Taylor, 1973). Social Penetration Theory uses an onion model (see Figure 3) as a metaphor to describe how the social penetration process works, in which individuals "peel back" others layers of personal information through social interaction (self-disclosures) {{f}}. One aspect of Social Penetration Theory is the types of information shared, which is referred to as levels {{f}}. The first {{g}} being the Superficial level or in other words, surface level information which includes visible attributes. The second level is the Intimate or peripheral level information, which is information that people usually feel free to share in any general social situation, such as their name and where they are from. Personal or Intermediate level information is the third level, in which includes information that is not always readily shared but that is also not very private, such as values and goals. Lastly, the Core or central level information is regarded as highly private and is shared when someone feels close to the other person or believes they are very trustworthy (Manning, 2019). An example of Core level information may be one's insecurities or fears. This theory states that for interpersonal relationships to develop there must be information exchanged between those within the relationship. A significant aspect of the theory is that of [[wikipedia:Social_penetration_theory#Breadth_and_depth|breadth and depth]] {{f}}. Breadth refers to the number of topics discussed, whereas depth refers to the level of intimacy of these discussions. For example, breadth may include discussing hobbies, family, interests and depth may be discussing feelings surrounding family issues or future goals (Carpenter & Greene, 2016), depth discussions may increase intimacy levels {{why}} {{g}}. According to this {{what}} theory {{g}} self-disclosure passes through phases as an interpersonal relationship progresses, {{g}} these stages include orientation, exploratory affective exchange, affective exchange, stable exchange and in some cases depenetration (see Table 1). The first stage is orientation, in which individuals only share superficial level information. Exploratory affective exchange is the second stage, where individuals may share beyond superficial information and may increase the breadth of topics discussed. The third stage is affective exchange, which includes more casual self-disclosures, communication and information from deeper levels, this stage is often reached with close friends or romantic partners. Stable exchange is the fourth stage, in which communication is open and honest, self-disclosures are continuous, and there is breadth and depth throughout conversations. This stage is maintained with few relationships such as romantic partners, close family and friends. Finally, depenetration or dissolution is a possible stage, whereby self-disclosure is reduced as a result of conflict or stress (Carpenter & Greene, 2016). {| class="wikitable" |+Table 1. The five stages of social penetration {{fact}} and examples. !Five stages of social penetration !Examples: |- |Orientation |Small talk, adhering to social norms. |- |Exploratory affective exchange |Personal attitudes & preferences. |- |Affective exchange |Casual relaxed communication, honest opinions, deeper emotional disclosures. |- |Stable exchange |Continuous open connection, deep core values, emotional reactions are predictable, breadth and depth explored. |- |Depenetration |Optional phase, communication breadth & depth withdrawal. |} == Emotional intimacy & its importance == Emotional intimacy is a part of interpersonal relationships, defined as a shared sense of closeness, trust and mutual understanding. Emotional intimacy happens between individuals when they consistently show each other who they really are and respond to each other in ways that promote feelings of safety, security and understanding. Thus, emotional intimacy can be experienced in a variety of different kind of relationships including, romantic partners, friends and family members. {{f}} === The intimacy process model (Reis & Shaver, 1988) === Reis and Shaver (1988) argue that intimacy is a dynamic, moment to moment transactional and cyclical process. The intimacy process model shows how emotional intimacy builds through a cyclical interactive progression between two individuals. This model uses self-disclosure, responsiveness, and perceived responsiveness as core components of emotional intimacy development. [[File:Intimacy process model.png|thumb|420x420px|'''Figure 4.''' The intimacy process model (Reis & Shaver, 1988).|left]] In this model {{g}} an interaction occurs between two people A and B, or for this example we will use the names Annabel and Bridgette. The model shows a single interaction in which Annabel is self-disclosing and Bridgette is responding (see Figure 4), however in most situations both individuals would freely move between self-disclosing and responding. In this example, the intimacy process begins with Annabel. Annabel has some mixture of motives, goals or fears that results in a self disclosure, whereby Bridgette must respond. As the role of the responder Bridgette is influenced by a mixture of concerns, which can be positive or negative {{g}}. Bridgette may want to be a good friend or partner or on the other hand, Bridgette may fear commitment, feel that Annabel is dependent on them and so forth. Annabel may intend {{sp}} on one thing, where Bridgette may perceive another. As Bridgette interprets the disclosure and expressions from Annabel, a complex mixture of motives and goals is at work, {{g}} this is referred to as the interpretive filter. In this model {{g}} Reis and Shaver (1988) argue that Bridgette's response to Annabel's self-disclosure is just as important as Annabel's initial disclosure, and will determine whether the interaction will become emotionally intimate. Suitable responses will enhance feelings of connectedness between the individuals, and inappropriate responses or non-responsiveness will keep the individuals at a distance. However, what matters most is Annabel's interpretation and reaction to Bridgette's response, for example if Annabel interprets Bridgette's response as unsupportive or too intrusive, greater intimacy is unlikely to be achieved (Reis & Shaver, 1988). === The importance of emotional intimacy === Emotional intimacy is intrinsically gratifying, and involves feelings of safety, trust, understanding, validation, acceptance and overall feeling cared for {{f}}. Studies have found associations between emotional intimacy, communication and relationship satisfaction, such that partners' perception of their emotional intimacy predicts both their own relationship satisfaction as well as their partners (Yoo et al., 2014). Similarly, those with strong intimate relationships tend to have higher life satisfaction (Amati et al., 2018) and wellbeing (Greene et al., 2006). Overall, intimacy can result in reciprocal intimacy, mutual liking and relationship progress (Forgas, 2011). Emotional intimacy has been found to play a role in reducing stress responses (Phillips et al., 2009). It has also been found that loneliness and depression partly stem from a lack of emotional intimacy (Rokach & Ohiliber-Lignieres, 2015). This research highlights just how important emotional intimacy is in everyday life, therefore it is important to learn how to improve emotional intimacy. {{RoundBoxTop|theme=3}} ;Key points: [[File:Crystal Clear app ktip.svg|left|20px|]] *Intimacy is related to relationship and life satisfaction. *Emotional intimacy is related to higher wellbeing and reduced stress responses. *Emotional intimacy can facilitate reciprocal intimacy, mutual liking and relationship progression. {{RoundBoxBottom}} ==Fostering self-disclosure== Rewarding relationships are practically inconceivable without self-disclosure. Self-disclosure is important in initiating, maintaining and enhancing emotional intimacy within interpersonal relationships, which has been linked to relationship success and satisfaction, as well as improved mental health (Forgas, 2011). Therefore, learning to foster and encourage self-disclosure is essential for everyday life. === How to encourage & respond to self-disclosures === In order to encourage self-disclosures from others we must remember Social Exchange Theory (Thibaut & Kelley, 1959) and the cost-benefit analysis. If someone wanted to disclose something to you, think to yourself, would it be rewarding for them? Making a self-disclosure conversation rewarding can be done in a multitude of ways. Social cues such as enquiries of concern (asking if someone is okay for example) or initiating with a self-disclosure displays inviting cues to the other person to disclose (Greene et al., 2006). It has also been found that moods have an effect of self-disclosure, such that positive moods increase self-disclosure intimacy, abstractness and breadth, whereas negative moods reduce intimacy but improve reciprocity of self-disclosure (Forgas, 2011), in which is an important aspect to take into account. Additionally, adherence to the norm of reciprocity, in which you can respond with a similar self-disclosure. Similarly, the intimacy process model (Reis & Shaver, 1988) highlights the underlying cause of a self-disclosure as well as the importance of responsiveness to self-disclosures. It is helpful to remember that there is an underlying motive, goal, fear or reason for someone to self-disclose. In the case of responsiveness, High or good responsiveness is comprised of the content of the response (elaborating on or matching disclosure), the style of response (showing concern and care) as well as the timing of the response (immediate or delayed). However, it is extremely important to take into account boundaries and assess the appropriateness of the situation, such as the physical environment where the interaction is occurring (Greene et al., 2006). {{RoundBoxTop|theme=3}} ;Key points: [[File:Crystal Clear app ktip.svg|left|20px|]] How to encourage and respond to self-disclosure: *The cost-benefit analysis: make the interaction rewarding through listening and responsiveness. *Social cues: enquiries of concern and initiating self-disclosure. *The norm of reciprocity: use reciprocal self-disclosure. *Boundaries: assess the appropriateness of the situation. {{RoundBoxBottom}} ==Conclusion== Self-disclosure creates a sense of understanding, safety, and trust that is essential for emotionally intimate interpersonal relationships. This self-disclosure process is intrinsic and crucial to progress interpersonal relationships. Self-disclosure drives emotional intimacy by creating a loop of vulnerability, validation and mutual understanding. Emotional intimacy has effects on aspects of everyday life including on communication and relationship satisfaction, such that perception of emotional intimacy can predict both one's own relationship satisfaction as well as their partners. Strong emotionally intimate relationships tend to have higher life satisfaction and wellbeing. Additionally, intimacy can result in reciprocal intimacy, mutual liking and relationship progress. Emotional intimacy has also been found to play a role in reducing stress responses and a lack of emotional intimacy can play a role in loneliness and depression. Therefore, it is important to improve emotional intimacy through learning to foster and encourage self-disclosure. Self-disclosure can be fostered through social cues, making the interaction rewarding and validating, using the norm of reciprocity and taking responsiveness into account. {{RoundBoxTop|theme=11}} ;Take home messages [[File:Crystal Clear app ktip.svg|left|20px|]] *Self-disclosure is the process of divulging personal experiences, thoughts and feelings to others. *Emotional intimacy is a shared sense of closeness, trust and mutual understanding, that can have effects on life and relationship satisfaction, wellbeing, stress and loneliness. *Self-disclosure can be fostered to improve emotional intimacy in relationships through reciprocal self-disclosure, responsiveness, and social cues. {{RoundBoxBottom}} ==See also== * [[Motivation and emotion/Book/2026/Responsiveness and interpersonal trust|Responsiveness and interpersonal trust]] (Book chapter, 2026) * [[Motivation and emotion/Book/2025/Self-disclosure and well-being|Self-disclosure and well-being]] (Book chapter, 2025) * [[wikipedia:Social_exchange_theory|Social exchange theory]] (Wikipedia) * [[wikipedia:Social_penetration_theory|Social penetration theory]] (Wikipedia) ==References== {{Hanging indent|1= Altman, I., & Taylor, D. (1973). ''Social penetration: The development of interpersonal relationships''. Halsted Press. https://psycnet.apa.org/record/1973-28661-000 Amati, V., Meggiolaro, S., Rivellini, G., & Zaccarin. (2018). Social relations and life satisfaction: the role of friends. ''Genus, 74''(7). 1-18. https://doi.org/10.1186/s41118-018-0032-z Burger, J. M., Sanches, J., Imberi, J. E., & Grande L. R. (2009). The norm of reciprocity as an internalized social norm: Returning favors even when no one finds out. ''Social Influence, 4''(1). 11-17. https://doi.org/10.1080/15534510802131004 Carpenter, A., & Greene, K. (2016). Social penetration theory. In C. R. Berger & M. E. Roloff (Eds.), ''The International Encyclopedia of Interpersonal Communication'' (1st ed., pp. 1-4). John Wiley & Sons. https://doi.org/10.1002/9781118540190.wbeic0160 Derlega, V. J., Winstead, B. A., & Greene, K. (2008). Self-disclosure and starting a close relationship. In S. Sprecher, A. Wenzel, & J. Harvey (Eds.), ''Handbook of relationship initiation''. Psychology Press. 53–174. https://psycnet.apa.org/record/2008-09972-008 Forgas, J. P. (2011). Affective influences on self-disclosure: Mood effects on the intimacy and reciprocity of disclosing personal information. ''Journal of Personality and Social Psychology, 100''(3). 449-461. https://doi.org/10.1037/a0021129 Greene, K., Derlega, V. J., & Mathews, A. (2006). Self-disclosure in personal relationships. In A. L. Vangelisti & D. Perlman (Eds.), ''The Cambridge handbook of personal relationships.'' (pp. 409-427). Cambridge University Press. https://doi.org/10.1017/CBO9780511606632.023 Manning, J. (2019). Thinking about interpersonal relationships and social penetration theory: Is it the same for lesbian, gay or bisexual people? In C. J. Liberman, A. S. Rancer & T. A. Avtgis (Eds.), ''Casing Communication Theory'' (pp. 293-306). Kendall Hunt Publishing Company. Pecune, F. (2013). Toward a computational model of social relations for artificial companions. ''Proceedings - 2013 Humaine Association Conference on Affective Computing and Intelligent Interaction''. 677-682. [https://www.researchgate.net/publication/261271664%20Toward%20a%20Computational%20Model%20of%20Social%20Relations%20for%20Artificial%20Companions DOI:10.1109/ACII.2013.118] Phillips, A., Gallagher, S., & Carroll, D. (2009) Social support, social intimacy, and cardiovascular reactions to acute psychological stress. ''Annals of Behavioural Medicine, 37''. 38-45. http://dx.doi.org/10.1007/s12160-008-9077-0 Reis, H. T. (2018). Intimacy as an interpersonal process. In H. T. Reis (Ed.), ''Relationships, Well-Being and Behaviour: Selected works of Harry T. Reis'' (pp. 209-244). Routledge. https://doi.org/10.4324/9780203732496 Reis, H. T., & Shaver, P. (1988). Intimacy as an interpersonal process. In S. Duck, D. H. Fay, J. K. Simpson, & J. C. Murray (Eds.), ''Handbook of personal relationships: Theory, research and applications'' (p. 367–389). John Wiley & Sons. Rokach, A., & Philibert-Lignieres, G. (2015). Intimacy, loneliness & infidelity. ''The Open Psychology Journal, 8''. 71-77. DOI: 10.2174/1874350101508010071 Sprecher, S., Treger, S., Wondra, J. D., Hilaire, N., & Wallpe, K. (2013). Taking turns: Reciprocal self-disclosure promotes liking in initial interactions. ''Journal of Experimental Social Psychology, 49''(5). 860-866. https://doi.org/10.1016/j.jesp.2013.03.017. Tolstedt, B. E., & Stokes, J. P. (1984). Self-disclosure, intimacy, and the depenetration process. ''Journal of Personality and Social Psychology, 46''(1). 84-90. https://doi.org/10.1037/0022-3514.46.1.84 Utz, S. (2015). The function of self-disclosure on social network sites: Not only intimate, but also positive and entertaining self-disclosures increase the feeling of connection. ''Computers in Human Behaviour, 45''. 1-10. https://doi.org/10.1016/j.chb.2014.11.076. Yoo, H., Bartle-Haring, S., Day. R., & Gangamma, R. (2014). Couple communication, emotional and sexual intimacy, and relationship satisfaction. ''Journal of Sex & Marital Therapy, 40''(4), 275-293. https://doi.org/10.1080/0092623X.2012.751072 Yu, Y., & Li, W. (2026). A fNIRS hyperscanning study on enhancing intimacy through emotional self-disclosure from the disclosure-responder perspective. ''BMC Psychology, 14''. https://doi.org/10.1186/s40359-026-04257-3 }} ==External links== * [https://www.youtube.com/watch?v=Z4bRXozREKU&t=90s Self-disclosure in personal relationships: How to stop oversharing] (youtube.com) * [https://www.psychologytoday.com/au/blog/between-the-sheets/202011/why-romantic-intimacy-requires-self-disclosure Why romantic intimacy requires self-disclosure] (psychologytoday.com) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Relationships]] ngkydn6dr9avx8zku5772ywkyskhzoo 2844562 2844559 2026-11-05T02:15:11Z Jtneill 10242 /* Overview */ Grammar fix 2844562 wikitext text/x-wiki {{title|Self-disclosure and emotional intimacy:<br>How does self-disclosure foster emotional closeness in relationships?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}} [[File:Stockcouple.jpg|thumb|150px|'''Figure 1'''. Image of couple hugging {{ic|Explain what this has to do with self-disclosure}/}]] ;Picture this... Jim and Pam (see Figure 1) have been dating for a few months; things have been going well but at times there is noticeable emotional distance between them. One night Jim tells Pam that “I haven’t talked to anyone about this, but I’m terrified one day you will get bored and leave”. This confession opens the flood gates for Jim and Pam to share their deepest fears and anxieties. In this moment, an invisible barrier dissolves, they have discovered a new profound sense of trust and understanding. A new and deeper layer of their relationship has been unlocked; there is an emotional closeness that wasn’t there before. Why does this self-disclosure from Jim create this emotional closeness between him and Pam? {{RoundBoxBottom}} [[wikipedia:Self-disclosure|Self-disclosure]] is the voluntary process of sharing personal thoughts, feeling and information with another person. Self-disclosure is one of the most significant and effective ways to initiate and enhance relationship intimacy, and to maintain positive personal relationships (Forgas, 2011). Intimacy or intimate come from the Latin terms ''intimus'' meaning innermost, and ''intimare'' meaning to make the innermost known (Reis, 2018). [[wikipedia:Emotional_intimacy|Emotional intimacy]] refers to a shared sense of understanding, trust and closeness within an interpersonal relationship. This chapter discusses what both self-disclosure and emotional intimacy mean, and how self-disclosures can be encouraged and responded to, with reference to psychological theories and research. {{RoundBoxTop|theme=11}} ;Focus questions [[File:Crystal Clear app ktip.svg|left|20px|]] *What is the self-disclosure? *What is emotional intimacy and why is it important? *How can self-disclosure be fostered to improve emotional intimacy in relationships? {{RoundBoxBottom}} == Self-disclosure == Self-disclosure is defined as the process of divulging personal experiences, thoughts and feelings to others (Yu & Li, 2026). Self-disclosure has long been discussed and researched for its relationship to emotional intimacy {{f}}. According to the research, we tend to like people who disclose more, we disclose more to those we like, and when we disclose, we like those we disclosed to more afterwards (Utz, 2015) {{expand}}. === Social exchange theory === [[File:Costs vs rewards.png|thumb|247x247px|'''Figure 2.''' Social Exchange Theory, costs versus rewards {{g}}|right]] [[wikipedia:Social_exchange_theory|Social Exchange Theory]] is a psychological framework that states human behaviours and relationships work through a cost-benefit analysis (see Figure 2) (Thibaut & Kelley, 1959). This core concept can be applied to that of self-disclosure, in which individuals assess the potential rewards and risks of sharing personal information. In the case of self-disclosure, possible rewards or benefits of divulging personal information may be validation, emotional relief and enhanced closeness {{f}}. If we assess that these are likely outcomes of self-disclosing we are more likely to do so. In contrast, potential costs or risks of self-disclosure may be a loss of privacy, judgement or fear of rejection and in this circumstance we are much less likely to self disclose. Interpersonal relationships thrive when self-disclosure remains profitable or rewarding. {{f}} ==== Norm of reciprocity ==== Within Social Exchange Theory is the concept of the [[wikipedia:Norm_of_reciprocity|norm of reciprocity]] {{f}}. This entails the universal social expectation that people should return or repay favours, gifts and acts of kindness with similar treatment {{f}}. For example, if someone gifts you an expensive item for your birthday, you will likely feel an internal obligation to get them a similarly expensive item for their birthday. This norm enables smooth, equal and fair social exchanges (Burger et al., 2009). This concept is also applicable to self-disclosure, {{g}} research has shown that self-disclosure is reciprocal in existing relationships and in the context of strangers (Sprecher et al., 2013). Hence, reciprocal self-disclosure is the process where one person shares personal thoughts, feelings or information and the other person responds with similar personal details in return. === Social penetration theory === [[File:Bdspt.model.webp|'''Figure 3.''' Social Penetration Theory Model (Pecune, 2013).|alt=Figure 2. Social Penetration Theory Model (Pecune, 2013).|thumb|300x300px]] [[wikipedia:Social_penetration_theory|Social Penetration Theory]] is a social psychological model that explains how interpersonal relationships evolve from superficial or surface level interactions to deep emotional intimacy through self-disclosure (Altman & Taylor, 1973). Social Penetration Theory uses an onion model (see Figure 3) as a metaphor to describe how the social penetration process works, in which individuals "peel back" others layers of personal information through social interaction (self-disclosures) {{f}}. One aspect of Social Penetration Theory is the types of information shared, which is referred to as levels {{f}}. The first {{g}} being the Superficial level or in other words, surface level information which includes visible attributes. The second level is the Intimate or peripheral level information, which is information that people usually feel free to share in any general social situation, such as their name and where they are from. Personal or Intermediate level information is the third level, in which includes information that is not always readily shared but that is also not very private, such as values and goals. Lastly, the Core or central level information is regarded as highly private and is shared when someone feels close to the other person or believes they are very trustworthy (Manning, 2019). An example of Core level information may be one's insecurities or fears. This theory states that for interpersonal relationships to develop there must be information exchanged between those within the relationship. A significant aspect of the theory is that of [[wikipedia:Social_penetration_theory#Breadth_and_depth|breadth and depth]] {{f}}. Breadth refers to the number of topics discussed, whereas depth refers to the level of intimacy of these discussions. For example, breadth may include discussing hobbies, family, interests and depth may be discussing feelings surrounding family issues or future goals (Carpenter & Greene, 2016), depth discussions may increase intimacy levels {{why}} {{g}}. According to this {{what}} theory {{g}} self-disclosure passes through phases as an interpersonal relationship progresses, {{g}} these stages include orientation, exploratory affective exchange, affective exchange, stable exchange and in some cases depenetration (see Table 1). The first stage is orientation, in which individuals only share superficial level information. Exploratory affective exchange is the second stage, where individuals may share beyond superficial information and may increase the breadth of topics discussed. The third stage is affective exchange, which includes more casual self-disclosures, communication and information from deeper levels, this stage is often reached with close friends or romantic partners. Stable exchange is the fourth stage, in which communication is open and honest, self-disclosures are continuous, and there is breadth and depth throughout conversations. This stage is maintained with few relationships such as romantic partners, close family and friends. Finally, depenetration or dissolution is a possible stage, whereby self-disclosure is reduced as a result of conflict or stress (Carpenter & Greene, 2016). {| class="wikitable" |+Table 1. The five stages of social penetration {{fact}} and examples. !Five stages of social penetration !Examples: |- |Orientation |Small talk, adhering to social norms. |- |Exploratory affective exchange |Personal attitudes & preferences. |- |Affective exchange |Casual relaxed communication, honest opinions, deeper emotional disclosures. |- |Stable exchange |Continuous open connection, deep core values, emotional reactions are predictable, breadth and depth explored. |- |Depenetration |Optional phase, communication breadth & depth withdrawal. |} == Emotional intimacy & its importance == Emotional intimacy is a part of interpersonal relationships, defined as a shared sense of closeness, trust and mutual understanding. Emotional intimacy happens between individuals when they consistently show each other who they really are and respond to each other in ways that promote feelings of safety, security and understanding. Thus, emotional intimacy can be experienced in a variety of different kind of relationships including, romantic partners, friends and family members. {{f}} === The intimacy process model (Reis & Shaver, 1988) === Reis and Shaver (1988) argue that intimacy is a dynamic, moment to moment transactional and cyclical process. The intimacy process model shows how emotional intimacy builds through a cyclical interactive progression between two individuals. This model uses self-disclosure, responsiveness, and perceived responsiveness as core components of emotional intimacy development. [[File:Intimacy process model.png|thumb|420x420px|'''Figure 4.''' The intimacy process model (Reis & Shaver, 1988).|left]] In this model {{g}} an interaction occurs between two people A and B, or for this example we will use the names Annabel and Bridgette. The model shows a single interaction in which Annabel is self-disclosing and Bridgette is responding (see Figure 4), however in most situations both individuals would freely move between self-disclosing and responding. In this example, the intimacy process begins with Annabel. Annabel has some mixture of motives, goals or fears that results in a self disclosure, whereby Bridgette must respond. As the role of the responder Bridgette is influenced by a mixture of concerns, which can be positive or negative {{g}}. Bridgette may want to be a good friend or partner or on the other hand, Bridgette may fear commitment, feel that Annabel is dependent on them and so forth. Annabel may intend {{sp}} on one thing, where Bridgette may perceive another. As Bridgette interprets the disclosure and expressions from Annabel, a complex mixture of motives and goals is at work, {{g}} this is referred to as the interpretive filter. In this model {{g}} Reis and Shaver (1988) argue that Bridgette's response to Annabel's self-disclosure is just as important as Annabel's initial disclosure, and will determine whether the interaction will become emotionally intimate. Suitable responses will enhance feelings of connectedness between the individuals, and inappropriate responses or non-responsiveness will keep the individuals at a distance. However, what matters most is Annabel's interpretation and reaction to Bridgette's response, for example if Annabel interprets Bridgette's response as unsupportive or too intrusive, greater intimacy is unlikely to be achieved (Reis & Shaver, 1988). === The importance of emotional intimacy === Emotional intimacy is intrinsically gratifying, and involves feelings of safety, trust, understanding, validation, acceptance and overall feeling cared for {{f}}. Studies have found associations between emotional intimacy, communication and relationship satisfaction, such that partners' perception of their emotional intimacy predicts both their own relationship satisfaction as well as their partners (Yoo et al., 2014). Similarly, those with strong intimate relationships tend to have higher life satisfaction (Amati et al., 2018) and wellbeing (Greene et al., 2006). Overall, intimacy can result in reciprocal intimacy, mutual liking and relationship progress (Forgas, 2011). Emotional intimacy has been found to play a role in reducing stress responses (Phillips et al., 2009). It has also been found that loneliness and depression partly stem from a lack of emotional intimacy (Rokach & Ohiliber-Lignieres, 2015). This research highlights just how important emotional intimacy is in everyday life, therefore it is important to learn how to improve emotional intimacy. {{RoundBoxTop|theme=3}} ;Key points: [[File:Crystal Clear app ktip.svg|left|20px|]] *Intimacy is related to relationship and life satisfaction. *Emotional intimacy is related to higher wellbeing and reduced stress responses. *Emotional intimacy can facilitate reciprocal intimacy, mutual liking and relationship progression. {{RoundBoxBottom}} ==Fostering self-disclosure== Rewarding relationships are practically inconceivable without self-disclosure. Self-disclosure is important in initiating, maintaining and enhancing emotional intimacy within interpersonal relationships, which has been linked to relationship success and satisfaction, as well as improved mental health (Forgas, 2011). Therefore, learning to foster and encourage self-disclosure is essential for everyday life. === How to encourage & respond to self-disclosures === In order to encourage self-disclosures from others we must remember Social Exchange Theory (Thibaut & Kelley, 1959) and the cost-benefit analysis. If someone wanted to disclose something to you, think to yourself, would it be rewarding for them? Making a self-disclosure conversation rewarding can be done in a multitude of ways. Social cues such as enquiries of concern (asking if someone is okay for example) or initiating with a self-disclosure displays inviting cues to the other person to disclose (Greene et al., 2006). It has also been found that moods have an effect of self-disclosure, such that positive moods increase self-disclosure intimacy, abstractness and breadth, whereas negative moods reduce intimacy but improve reciprocity of self-disclosure (Forgas, 2011), in which is an important aspect to take into account. Additionally, adherence to the norm of reciprocity, in which you can respond with a similar self-disclosure. Similarly, the intimacy process model (Reis & Shaver, 1988) highlights the underlying cause of a self-disclosure as well as the importance of responsiveness to self-disclosures. It is helpful to remember that there is an underlying motive, goal, fear or reason for someone to self-disclose. In the case of responsiveness, High or good responsiveness is comprised of the content of the response (elaborating on or matching disclosure), the style of response (showing concern and care) as well as the timing of the response (immediate or delayed). However, it is extremely important to take into account boundaries and assess the appropriateness of the situation, such as the physical environment where the interaction is occurring (Greene et al., 2006). {{RoundBoxTop|theme=3}} ;Key points: [[File:Crystal Clear app ktip.svg|left|20px|]] How to encourage and respond to self-disclosure: *The cost-benefit analysis: make the interaction rewarding through listening and responsiveness. *Social cues: enquiries of concern and initiating self-disclosure. *The norm of reciprocity: use reciprocal self-disclosure. *Boundaries: assess the appropriateness of the situation. {{RoundBoxBottom}} ==Conclusion== Self-disclosure creates a sense of understanding, safety, and trust that is essential for emotionally intimate interpersonal relationships. This self-disclosure process is intrinsic and crucial to progress interpersonal relationships. Self-disclosure drives emotional intimacy by creating a loop of vulnerability, validation and mutual understanding. Emotional intimacy has effects on aspects of everyday life including on communication and relationship satisfaction, such that perception of emotional intimacy can predict both one's own relationship satisfaction as well as their partners. Strong emotionally intimate relationships tend to have higher life satisfaction and wellbeing. Additionally, intimacy can result in reciprocal intimacy, mutual liking and relationship progress. Emotional intimacy has also been found to play a role in reducing stress responses and a lack of emotional intimacy can play a role in loneliness and depression. Therefore, it is important to improve emotional intimacy through learning to foster and encourage self-disclosure. Self-disclosure can be fostered through social cues, making the interaction rewarding and validating, using the norm of reciprocity and taking responsiveness into account. {{RoundBoxTop|theme=11}} ;Take home messages [[File:Crystal Clear app ktip.svg|left|20px|]] *Self-disclosure is the process of divulging personal experiences, thoughts and feelings to others. *Emotional intimacy is a shared sense of closeness, trust and mutual understanding, that can have effects on life and relationship satisfaction, wellbeing, stress and loneliness. *Self-disclosure can be fostered to improve emotional intimacy in relationships through reciprocal self-disclosure, responsiveness, and social cues. {{RoundBoxBottom}} ==See also== * [[Motivation and emotion/Book/2026/Responsiveness and interpersonal trust|Responsiveness and interpersonal trust]] (Book chapter, 2026) * [[Motivation and emotion/Book/2025/Self-disclosure and well-being|Self-disclosure and well-being]] (Book chapter, 2025) * [[wikipedia:Social_exchange_theory|Social exchange theory]] (Wikipedia) * [[wikipedia:Social_penetration_theory|Social penetration theory]] (Wikipedia) ==References== {{Hanging indent|1= Altman, I., & Taylor, D. (1973). ''Social penetration: The development of interpersonal relationships''. Halsted Press. https://psycnet.apa.org/record/1973-28661-000 Amati, V., Meggiolaro, S., Rivellini, G., & Zaccarin. (2018). Social relations and life satisfaction: the role of friends. ''Genus, 74''(7). 1-18. https://doi.org/10.1186/s41118-018-0032-z Burger, J. M., Sanches, J., Imberi, J. E., & Grande L. R. (2009). The norm of reciprocity as an internalized social norm: Returning favors even when no one finds out. ''Social Influence, 4''(1). 11-17. https://doi.org/10.1080/15534510802131004 Carpenter, A., & Greene, K. (2016). Social penetration theory. In C. R. Berger & M. E. Roloff (Eds.), ''The International Encyclopedia of Interpersonal Communication'' (1st ed., pp. 1-4). John Wiley & Sons. https://doi.org/10.1002/9781118540190.wbeic0160 Derlega, V. J., Winstead, B. A., & Greene, K. (2008). Self-disclosure and starting a close relationship. In S. Sprecher, A. Wenzel, & J. Harvey (Eds.), ''Handbook of relationship initiation''. Psychology Press. 53–174. https://psycnet.apa.org/record/2008-09972-008 Forgas, J. P. (2011). Affective influences on self-disclosure: Mood effects on the intimacy and reciprocity of disclosing personal information. ''Journal of Personality and Social Psychology, 100''(3). 449-461. https://doi.org/10.1037/a0021129 Greene, K., Derlega, V. J., & Mathews, A. (2006). Self-disclosure in personal relationships. In A. L. Vangelisti & D. Perlman (Eds.), ''The Cambridge handbook of personal relationships.'' (pp. 409-427). Cambridge University Press. https://doi.org/10.1017/CBO9780511606632.023 Manning, J. (2019). Thinking about interpersonal relationships and social penetration theory: Is it the same for lesbian, gay or bisexual people? In C. J. Liberman, A. S. Rancer & T. A. Avtgis (Eds.), ''Casing Communication Theory'' (pp. 293-306). Kendall Hunt Publishing Company. Pecune, F. (2013). Toward a computational model of social relations for artificial companions. ''Proceedings - 2013 Humaine Association Conference on Affective Computing and Intelligent Interaction''. 677-682. [https://www.researchgate.net/publication/261271664%20Toward%20a%20Computational%20Model%20of%20Social%20Relations%20for%20Artificial%20Companions DOI:10.1109/ACII.2013.118] Phillips, A., Gallagher, S., & Carroll, D. (2009) Social support, social intimacy, and cardiovascular reactions to acute psychological stress. ''Annals of Behavioural Medicine, 37''. 38-45. http://dx.doi.org/10.1007/s12160-008-9077-0 Reis, H. T. (2018). Intimacy as an interpersonal process. In H. T. Reis (Ed.), ''Relationships, Well-Being and Behaviour: Selected works of Harry T. Reis'' (pp. 209-244). Routledge. https://doi.org/10.4324/9780203732496 Reis, H. T., & Shaver, P. (1988). Intimacy as an interpersonal process. In S. Duck, D. H. Fay, J. K. Simpson, & J. C. Murray (Eds.), ''Handbook of personal relationships: Theory, research and applications'' (p. 367–389). John Wiley & Sons. Rokach, A., & Philibert-Lignieres, G. (2015). Intimacy, loneliness & infidelity. ''The Open Psychology Journal, 8''. 71-77. DOI: 10.2174/1874350101508010071 Sprecher, S., Treger, S., Wondra, J. D., Hilaire, N., & Wallpe, K. (2013). Taking turns: Reciprocal self-disclosure promotes liking in initial interactions. ''Journal of Experimental Social Psychology, 49''(5). 860-866. https://doi.org/10.1016/j.jesp.2013.03.017. Tolstedt, B. E., & Stokes, J. P. (1984). Self-disclosure, intimacy, and the depenetration process. ''Journal of Personality and Social Psychology, 46''(1). 84-90. https://doi.org/10.1037/0022-3514.46.1.84 Utz, S. (2015). The function of self-disclosure on social network sites: Not only intimate, but also positive and entertaining self-disclosures increase the feeling of connection. ''Computers in Human Behaviour, 45''. 1-10. https://doi.org/10.1016/j.chb.2014.11.076. Yoo, H., Bartle-Haring, S., Day. R., & Gangamma, R. (2014). Couple communication, emotional and sexual intimacy, and relationship satisfaction. ''Journal of Sex & Marital Therapy, 40''(4), 275-293. https://doi.org/10.1080/0092623X.2012.751072 Yu, Y., & Li, W. (2026). A fNIRS hyperscanning study on enhancing intimacy through emotional self-disclosure from the disclosure-responder perspective. ''BMC Psychology, 14''. https://doi.org/10.1186/s40359-026-04257-3 }} ==External links== * [https://www.youtube.com/watch?v=Z4bRXozREKU&t=90s Self-disclosure in personal relationships: How to stop oversharing] (youtube.com) * [https://www.psychologytoday.com/au/blog/between-the-sheets/202011/why-romantic-intimacy-requires-self-disclosure Why romantic intimacy requires self-disclosure] (psychologytoday.com) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Relationships]] qa6kqmqchldqwyd09l2fp8qgg3mtrju 2844563 2844562 2026-11-05T02:15:36Z Jtneill 10242 /* Overview */ 2844563 wikitext text/x-wiki {{title|Self-disclosure and emotional intimacy:<br>How does self-disclosure foster emotional closeness in relationships?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}} [[File:Stockcouple.jpg|thumb|right|150px|'''Figure 1'''. Image of couple hugging {{ic|Explain what this has to do with self-disclosure}/}]] ;Picture this... Jim and Pam (see Figure 1) have been dating for a few months; things have been going well but at times there is noticeable emotional distance between them. One night Jim tells Pam that “I haven’t talked to anyone about this, but I’m terrified one day you will get bored and leave”. This confession opens the flood gates for Jim and Pam to share their deepest fears and anxieties. In this moment, an invisible barrier dissolves, they have discovered a new profound sense of trust and understanding. A new and deeper layer of their relationship has been unlocked; there is an emotional closeness that wasn’t there before. Why does this self-disclosure from Jim create this emotional closeness between him and Pam? {{RoundBoxBottom}} [[wikipedia:Self-disclosure|Self-disclosure]] is the voluntary process of sharing personal thoughts, feeling and information with another person. Self-disclosure is one of the most significant and effective ways to initiate and enhance relationship intimacy, and to maintain positive personal relationships (Forgas, 2011). Intimacy or intimate come from the Latin terms ''intimus'' meaning innermost, and ''intimare'' meaning to make the innermost known (Reis, 2018). [[wikipedia:Emotional_intimacy|Emotional intimacy]] refers to a shared sense of understanding, trust and closeness within an interpersonal relationship. This chapter discusses what both self-disclosure and emotional intimacy mean, and how self-disclosures can be encouraged and responded to, with reference to psychological theories and research. {{RoundBoxTop|theme=11}} ;Focus questions [[File:Crystal Clear app ktip.svg|left|20px|]] *What is the self-disclosure? *What is emotional intimacy and why is it important? *How can self-disclosure be fostered to improve emotional intimacy in relationships? {{RoundBoxBottom}} == Self-disclosure == Self-disclosure is defined as the process of divulging personal experiences, thoughts and feelings to others (Yu & Li, 2026). Self-disclosure has long been discussed and researched for its relationship to emotional intimacy {{f}}. According to the research, we tend to like people who disclose more, we disclose more to those we like, and when we disclose, we like those we disclosed to more afterwards (Utz, 2015) {{expand}}. === Social exchange theory === [[File:Costs vs rewards.png|thumb|247x247px|'''Figure 2.''' Social Exchange Theory, costs versus rewards {{g}}|right]] [[wikipedia:Social_exchange_theory|Social Exchange Theory]] is a psychological framework that states human behaviours and relationships work through a cost-benefit analysis (see Figure 2) (Thibaut & Kelley, 1959). This core concept can be applied to that of self-disclosure, in which individuals assess the potential rewards and risks of sharing personal information. In the case of self-disclosure, possible rewards or benefits of divulging personal information may be validation, emotional relief and enhanced closeness {{f}}. If we assess that these are likely outcomes of self-disclosing we are more likely to do so. In contrast, potential costs or risks of self-disclosure may be a loss of privacy, judgement or fear of rejection and in this circumstance we are much less likely to self disclose. Interpersonal relationships thrive when self-disclosure remains profitable or rewarding. {{f}} ==== Norm of reciprocity ==== Within Social Exchange Theory is the concept of the [[wikipedia:Norm_of_reciprocity|norm of reciprocity]] {{f}}. This entails the universal social expectation that people should return or repay favours, gifts and acts of kindness with similar treatment {{f}}. For example, if someone gifts you an expensive item for your birthday, you will likely feel an internal obligation to get them a similarly expensive item for their birthday. This norm enables smooth, equal and fair social exchanges (Burger et al., 2009). This concept is also applicable to self-disclosure, {{g}} research has shown that self-disclosure is reciprocal in existing relationships and in the context of strangers (Sprecher et al., 2013). Hence, reciprocal self-disclosure is the process where one person shares personal thoughts, feelings or information and the other person responds with similar personal details in return. === Social penetration theory === [[File:Bdspt.model.webp|'''Figure 3.''' Social Penetration Theory Model (Pecune, 2013).|alt=Figure 2. Social Penetration Theory Model (Pecune, 2013).|thumb|300x300px]] [[wikipedia:Social_penetration_theory|Social Penetration Theory]] is a social psychological model that explains how interpersonal relationships evolve from superficial or surface level interactions to deep emotional intimacy through self-disclosure (Altman & Taylor, 1973). Social Penetration Theory uses an onion model (see Figure 3) as a metaphor to describe how the social penetration process works, in which individuals "peel back" others layers of personal information through social interaction (self-disclosures) {{f}}. One aspect of Social Penetration Theory is the types of information shared, which is referred to as levels {{f}}. The first {{g}} being the Superficial level or in other words, surface level information which includes visible attributes. The second level is the Intimate or peripheral level information, which is information that people usually feel free to share in any general social situation, such as their name and where they are from. Personal or Intermediate level information is the third level, in which includes information that is not always readily shared but that is also not very private, such as values and goals. Lastly, the Core or central level information is regarded as highly private and is shared when someone feels close to the other person or believes they are very trustworthy (Manning, 2019). An example of Core level information may be one's insecurities or fears. This theory states that for interpersonal relationships to develop there must be information exchanged between those within the relationship. A significant aspect of the theory is that of [[wikipedia:Social_penetration_theory#Breadth_and_depth|breadth and depth]] {{f}}. Breadth refers to the number of topics discussed, whereas depth refers to the level of intimacy of these discussions. For example, breadth may include discussing hobbies, family, interests and depth may be discussing feelings surrounding family issues or future goals (Carpenter & Greene, 2016), depth discussions may increase intimacy levels {{why}} {{g}}. According to this {{what}} theory {{g}} self-disclosure passes through phases as an interpersonal relationship progresses, {{g}} these stages include orientation, exploratory affective exchange, affective exchange, stable exchange and in some cases depenetration (see Table 1). The first stage is orientation, in which individuals only share superficial level information. Exploratory affective exchange is the second stage, where individuals may share beyond superficial information and may increase the breadth of topics discussed. The third stage is affective exchange, which includes more casual self-disclosures, communication and information from deeper levels, this stage is often reached with close friends or romantic partners. Stable exchange is the fourth stage, in which communication is open and honest, self-disclosures are continuous, and there is breadth and depth throughout conversations. This stage is maintained with few relationships such as romantic partners, close family and friends. Finally, depenetration or dissolution is a possible stage, whereby self-disclosure is reduced as a result of conflict or stress (Carpenter & Greene, 2016). {| class="wikitable" |+Table 1. The five stages of social penetration {{fact}} and examples. !Five stages of social penetration !Examples: |- |Orientation |Small talk, adhering to social norms. |- |Exploratory affective exchange |Personal attitudes & preferences. |- |Affective exchange |Casual relaxed communication, honest opinions, deeper emotional disclosures. |- |Stable exchange |Continuous open connection, deep core values, emotional reactions are predictable, breadth and depth explored. |- |Depenetration |Optional phase, communication breadth & depth withdrawal. |} == Emotional intimacy & its importance == Emotional intimacy is a part of interpersonal relationships, defined as a shared sense of closeness, trust and mutual understanding. Emotional intimacy happens between individuals when they consistently show each other who they really are and respond to each other in ways that promote feelings of safety, security and understanding. Thus, emotional intimacy can be experienced in a variety of different kind of relationships including, romantic partners, friends and family members. {{f}} === The intimacy process model (Reis & Shaver, 1988) === Reis and Shaver (1988) argue that intimacy is a dynamic, moment to moment transactional and cyclical process. The intimacy process model shows how emotional intimacy builds through a cyclical interactive progression between two individuals. This model uses self-disclosure, responsiveness, and perceived responsiveness as core components of emotional intimacy development. [[File:Intimacy process model.png|thumb|420x420px|'''Figure 4.''' The intimacy process model (Reis & Shaver, 1988).|left]] In this model {{g}} an interaction occurs between two people A and B, or for this example we will use the names Annabel and Bridgette. The model shows a single interaction in which Annabel is self-disclosing and Bridgette is responding (see Figure 4), however in most situations both individuals would freely move between self-disclosing and responding. In this example, the intimacy process begins with Annabel. Annabel has some mixture of motives, goals or fears that results in a self disclosure, whereby Bridgette must respond. As the role of the responder Bridgette is influenced by a mixture of concerns, which can be positive or negative {{g}}. Bridgette may want to be a good friend or partner or on the other hand, Bridgette may fear commitment, feel that Annabel is dependent on them and so forth. Annabel may intend {{sp}} on one thing, where Bridgette may perceive another. As Bridgette interprets the disclosure and expressions from Annabel, a complex mixture of motives and goals is at work, {{g}} this is referred to as the interpretive filter. In this model {{g}} Reis and Shaver (1988) argue that Bridgette's response to Annabel's self-disclosure is just as important as Annabel's initial disclosure, and will determine whether the interaction will become emotionally intimate. Suitable responses will enhance feelings of connectedness between the individuals, and inappropriate responses or non-responsiveness will keep the individuals at a distance. However, what matters most is Annabel's interpretation and reaction to Bridgette's response, for example if Annabel interprets Bridgette's response as unsupportive or too intrusive, greater intimacy is unlikely to be achieved (Reis & Shaver, 1988). === The importance of emotional intimacy === Emotional intimacy is intrinsically gratifying, and involves feelings of safety, trust, understanding, validation, acceptance and overall feeling cared for {{f}}. Studies have found associations between emotional intimacy, communication and relationship satisfaction, such that partners' perception of their emotional intimacy predicts both their own relationship satisfaction as well as their partners (Yoo et al., 2014). Similarly, those with strong intimate relationships tend to have higher life satisfaction (Amati et al., 2018) and wellbeing (Greene et al., 2006). Overall, intimacy can result in reciprocal intimacy, mutual liking and relationship progress (Forgas, 2011). Emotional intimacy has been found to play a role in reducing stress responses (Phillips et al., 2009). It has also been found that loneliness and depression partly stem from a lack of emotional intimacy (Rokach & Ohiliber-Lignieres, 2015). This research highlights just how important emotional intimacy is in everyday life, therefore it is important to learn how to improve emotional intimacy. {{RoundBoxTop|theme=3}} ;Key points: [[File:Crystal Clear app ktip.svg|left|20px|]] *Intimacy is related to relationship and life satisfaction. *Emotional intimacy is related to higher wellbeing and reduced stress responses. *Emotional intimacy can facilitate reciprocal intimacy, mutual liking and relationship progression. {{RoundBoxBottom}} ==Fostering self-disclosure== Rewarding relationships are practically inconceivable without self-disclosure. Self-disclosure is important in initiating, maintaining and enhancing emotional intimacy within interpersonal relationships, which has been linked to relationship success and satisfaction, as well as improved mental health (Forgas, 2011). Therefore, learning to foster and encourage self-disclosure is essential for everyday life. === How to encourage & respond to self-disclosures === In order to encourage self-disclosures from others we must remember Social Exchange Theory (Thibaut & Kelley, 1959) and the cost-benefit analysis. If someone wanted to disclose something to you, think to yourself, would it be rewarding for them? Making a self-disclosure conversation rewarding can be done in a multitude of ways. Social cues such as enquiries of concern (asking if someone is okay for example) or initiating with a self-disclosure displays inviting cues to the other person to disclose (Greene et al., 2006). It has also been found that moods have an effect of self-disclosure, such that positive moods increase self-disclosure intimacy, abstractness and breadth, whereas negative moods reduce intimacy but improve reciprocity of self-disclosure (Forgas, 2011), in which is an important aspect to take into account. Additionally, adherence to the norm of reciprocity, in which you can respond with a similar self-disclosure. Similarly, the intimacy process model (Reis & Shaver, 1988) highlights the underlying cause of a self-disclosure as well as the importance of responsiveness to self-disclosures. It is helpful to remember that there is an underlying motive, goal, fear or reason for someone to self-disclose. In the case of responsiveness, High or good responsiveness is comprised of the content of the response (elaborating on or matching disclosure), the style of response (showing concern and care) as well as the timing of the response (immediate or delayed). However, it is extremely important to take into account boundaries and assess the appropriateness of the situation, such as the physical environment where the interaction is occurring (Greene et al., 2006). {{RoundBoxTop|theme=3}} ;Key points: [[File:Crystal Clear app ktip.svg|left|20px|]] How to encourage and respond to self-disclosure: *The cost-benefit analysis: make the interaction rewarding through listening and responsiveness. *Social cues: enquiries of concern and initiating self-disclosure. *The norm of reciprocity: use reciprocal self-disclosure. *Boundaries: assess the appropriateness of the situation. {{RoundBoxBottom}} ==Conclusion== Self-disclosure creates a sense of understanding, safety, and trust that is essential for emotionally intimate interpersonal relationships. This self-disclosure process is intrinsic and crucial to progress interpersonal relationships. Self-disclosure drives emotional intimacy by creating a loop of vulnerability, validation and mutual understanding. Emotional intimacy has effects on aspects of everyday life including on communication and relationship satisfaction, such that perception of emotional intimacy can predict both one's own relationship satisfaction as well as their partners. Strong emotionally intimate relationships tend to have higher life satisfaction and wellbeing. Additionally, intimacy can result in reciprocal intimacy, mutual liking and relationship progress. Emotional intimacy has also been found to play a role in reducing stress responses and a lack of emotional intimacy can play a role in loneliness and depression. Therefore, it is important to improve emotional intimacy through learning to foster and encourage self-disclosure. Self-disclosure can be fostered through social cues, making the interaction rewarding and validating, using the norm of reciprocity and taking responsiveness into account. {{RoundBoxTop|theme=11}} ;Take home messages [[File:Crystal Clear app ktip.svg|left|20px|]] *Self-disclosure is the process of divulging personal experiences, thoughts and feelings to others. *Emotional intimacy is a shared sense of closeness, trust and mutual understanding, that can have effects on life and relationship satisfaction, wellbeing, stress and loneliness. *Self-disclosure can be fostered to improve emotional intimacy in relationships through reciprocal self-disclosure, responsiveness, and social cues. {{RoundBoxBottom}} ==See also== * [[Motivation and emotion/Book/2026/Responsiveness and interpersonal trust|Responsiveness and interpersonal trust]] (Book chapter, 2026) * [[Motivation and emotion/Book/2025/Self-disclosure and well-being|Self-disclosure and well-being]] (Book chapter, 2025) * [[wikipedia:Social_exchange_theory|Social exchange theory]] (Wikipedia) * [[wikipedia:Social_penetration_theory|Social penetration theory]] (Wikipedia) ==References== {{Hanging indent|1= Altman, I., & Taylor, D. (1973). ''Social penetration: The development of interpersonal relationships''. Halsted Press. https://psycnet.apa.org/record/1973-28661-000 Amati, V., Meggiolaro, S., Rivellini, G., & Zaccarin. (2018). Social relations and life satisfaction: the role of friends. ''Genus, 74''(7). 1-18. https://doi.org/10.1186/s41118-018-0032-z Burger, J. M., Sanches, J., Imberi, J. E., & Grande L. R. (2009). The norm of reciprocity as an internalized social norm: Returning favors even when no one finds out. ''Social Influence, 4''(1). 11-17. https://doi.org/10.1080/15534510802131004 Carpenter, A., & Greene, K. (2016). Social penetration theory. In C. R. Berger & M. E. Roloff (Eds.), ''The International Encyclopedia of Interpersonal Communication'' (1st ed., pp. 1-4). John Wiley & Sons. https://doi.org/10.1002/9781118540190.wbeic0160 Derlega, V. J., Winstead, B. A., & Greene, K. (2008). Self-disclosure and starting a close relationship. In S. Sprecher, A. Wenzel, & J. Harvey (Eds.), ''Handbook of relationship initiation''. Psychology Press. 53–174. https://psycnet.apa.org/record/2008-09972-008 Forgas, J. P. (2011). Affective influences on self-disclosure: Mood effects on the intimacy and reciprocity of disclosing personal information. ''Journal of Personality and Social Psychology, 100''(3). 449-461. https://doi.org/10.1037/a0021129 Greene, K., Derlega, V. J., & Mathews, A. (2006). Self-disclosure in personal relationships. In A. L. Vangelisti & D. Perlman (Eds.), ''The Cambridge handbook of personal relationships.'' (pp. 409-427). Cambridge University Press. https://doi.org/10.1017/CBO9780511606632.023 Manning, J. (2019). Thinking about interpersonal relationships and social penetration theory: Is it the same for lesbian, gay or bisexual people? In C. J. Liberman, A. S. Rancer & T. A. Avtgis (Eds.), ''Casing Communication Theory'' (pp. 293-306). Kendall Hunt Publishing Company. Pecune, F. (2013). Toward a computational model of social relations for artificial companions. ''Proceedings - 2013 Humaine Association Conference on Affective Computing and Intelligent Interaction''. 677-682. [https://www.researchgate.net/publication/261271664%20Toward%20a%20Computational%20Model%20of%20Social%20Relations%20for%20Artificial%20Companions DOI:10.1109/ACII.2013.118] Phillips, A., Gallagher, S., & Carroll, D. (2009) Social support, social intimacy, and cardiovascular reactions to acute psychological stress. ''Annals of Behavioural Medicine, 37''. 38-45. http://dx.doi.org/10.1007/s12160-008-9077-0 Reis, H. T. (2018). Intimacy as an interpersonal process. In H. T. Reis (Ed.), ''Relationships, Well-Being and Behaviour: Selected works of Harry T. Reis'' (pp. 209-244). Routledge. https://doi.org/10.4324/9780203732496 Reis, H. T., & Shaver, P. (1988). Intimacy as an interpersonal process. In S. Duck, D. H. Fay, J. K. Simpson, & J. C. Murray (Eds.), ''Handbook of personal relationships: Theory, research and applications'' (p. 367–389). John Wiley & Sons. Rokach, A., & Philibert-Lignieres, G. (2015). Intimacy, loneliness & infidelity. ''The Open Psychology Journal, 8''. 71-77. DOI: 10.2174/1874350101508010071 Sprecher, S., Treger, S., Wondra, J. D., Hilaire, N., & Wallpe, K. (2013). Taking turns: Reciprocal self-disclosure promotes liking in initial interactions. ''Journal of Experimental Social Psychology, 49''(5). 860-866. https://doi.org/10.1016/j.jesp.2013.03.017. Tolstedt, B. E., & Stokes, J. P. (1984). Self-disclosure, intimacy, and the depenetration process. ''Journal of Personality and Social Psychology, 46''(1). 84-90. https://doi.org/10.1037/0022-3514.46.1.84 Utz, S. (2015). The function of self-disclosure on social network sites: Not only intimate, but also positive and entertaining self-disclosures increase the feeling of connection. ''Computers in Human Behaviour, 45''. 1-10. https://doi.org/10.1016/j.chb.2014.11.076. Yoo, H., Bartle-Haring, S., Day. R., & Gangamma, R. (2014). Couple communication, emotional and sexual intimacy, and relationship satisfaction. ''Journal of Sex & Marital Therapy, 40''(4), 275-293. https://doi.org/10.1080/0092623X.2012.751072 Yu, Y., & Li, W. (2026). A fNIRS hyperscanning study on enhancing intimacy through emotional self-disclosure from the disclosure-responder perspective. ''BMC Psychology, 14''. https://doi.org/10.1186/s40359-026-04257-3 }} ==External links== * [https://www.youtube.com/watch?v=Z4bRXozREKU&t=90s Self-disclosure in personal relationships: How to stop oversharing] (youtube.com) * [https://www.psychologytoday.com/au/blog/between-the-sheets/202011/why-romantic-intimacy-requires-self-disclosure Why romantic intimacy requires self-disclosure] (psychologytoday.com) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Relationships]] 1io4vtasjqd511gsdqjhyhr8lofg2g9 2844567 2844563 2026-11-05T02:19:23Z Jtneill 10242 /* The intimacy process model (Reis & Shaver, 1988) */ 2844567 wikitext text/x-wiki {{title|Self-disclosure and emotional intimacy:<br>How does self-disclosure foster emotional closeness in relationships?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}} [[File:Stockcouple.jpg|thumb|right|150px|'''Figure 1'''. Image of couple hugging {{ic|Explain what this has to do with self-disclosure}/}]] ;Picture this... Jim and Pam (see Figure 1) have been dating for a few months; things have been going well but at times there is noticeable emotional distance between them. One night Jim tells Pam that “I haven’t talked to anyone about this, but I’m terrified one day you will get bored and leave”. This confession opens the flood gates for Jim and Pam to share their deepest fears and anxieties. In this moment, an invisible barrier dissolves, they have discovered a new profound sense of trust and understanding. A new and deeper layer of their relationship has been unlocked; there is an emotional closeness that wasn’t there before. Why does this self-disclosure from Jim create this emotional closeness between him and Pam? {{RoundBoxBottom}} [[wikipedia:Self-disclosure|Self-disclosure]] is the voluntary process of sharing personal thoughts, feeling and information with another person. Self-disclosure is one of the most significant and effective ways to initiate and enhance relationship intimacy, and to maintain positive personal relationships (Forgas, 2011). Intimacy or intimate come from the Latin terms ''intimus'' meaning innermost, and ''intimare'' meaning to make the innermost known (Reis, 2018). [[wikipedia:Emotional_intimacy|Emotional intimacy]] refers to a shared sense of understanding, trust and closeness within an interpersonal relationship. This chapter discusses what both self-disclosure and emotional intimacy mean, and how self-disclosures can be encouraged and responded to, with reference to psychological theories and research. {{RoundBoxTop|theme=11}} ;Focus questions [[File:Crystal Clear app ktip.svg|left|20px|]] *What is the self-disclosure? *What is emotional intimacy and why is it important? *How can self-disclosure be fostered to improve emotional intimacy in relationships? {{RoundBoxBottom}} == Self-disclosure == Self-disclosure is defined as the process of divulging personal experiences, thoughts and feelings to others (Yu & Li, 2026). Self-disclosure has long been discussed and researched for its relationship to emotional intimacy {{f}}. According to the research, we tend to like people who disclose more, we disclose more to those we like, and when we disclose, we like those we disclosed to more afterwards (Utz, 2015) {{expand}}. === Social exchange theory === [[File:Costs vs rewards.png|thumb|247x247px|'''Figure 2.''' Social Exchange Theory, costs versus rewards {{g}}|right]] [[wikipedia:Social_exchange_theory|Social Exchange Theory]] is a psychological framework that states human behaviours and relationships work through a cost-benefit analysis (see Figure 2) (Thibaut & Kelley, 1959). This core concept can be applied to that of self-disclosure, in which individuals assess the potential rewards and risks of sharing personal information. In the case of self-disclosure, possible rewards or benefits of divulging personal information may be validation, emotional relief and enhanced closeness {{f}}. If we assess that these are likely outcomes of self-disclosing we are more likely to do so. In contrast, potential costs or risks of self-disclosure may be a loss of privacy, judgement or fear of rejection and in this circumstance we are much less likely to self disclose. Interpersonal relationships thrive when self-disclosure remains profitable or rewarding. {{f}} ==== Norm of reciprocity ==== Within Social Exchange Theory is the concept of the [[wikipedia:Norm_of_reciprocity|norm of reciprocity]] {{f}}. This entails the universal social expectation that people should return or repay favours, gifts and acts of kindness with similar treatment {{f}}. For example, if someone gifts you an expensive item for your birthday, you will likely feel an internal obligation to get them a similarly expensive item for their birthday. This norm enables smooth, equal and fair social exchanges (Burger et al., 2009). This concept is also applicable to self-disclosure, {{g}} research has shown that self-disclosure is reciprocal in existing relationships and in the context of strangers (Sprecher et al., 2013). Hence, reciprocal self-disclosure is the process where one person shares personal thoughts, feelings or information and the other person responds with similar personal details in return. === Social penetration theory === [[File:Bdspt.model.webp|'''Figure 3.''' Social Penetration Theory Model (Pecune, 2013).|alt=Figure 2. Social Penetration Theory Model (Pecune, 2013).|thumb|300x300px]] [[wikipedia:Social_penetration_theory|Social Penetration Theory]] is a social psychological model that explains how interpersonal relationships evolve from superficial or surface level interactions to deep emotional intimacy through self-disclosure (Altman & Taylor, 1973). Social Penetration Theory uses an onion model (see Figure 3) as a metaphor to describe how the social penetration process works, in which individuals "peel back" others layers of personal information through social interaction (self-disclosures) {{f}}. One aspect of Social Penetration Theory is the types of information shared, which is referred to as levels {{f}}. The first {{g}} being the Superficial level or in other words, surface level information which includes visible attributes. The second level is the Intimate or peripheral level information, which is information that people usually feel free to share in any general social situation, such as their name and where they are from. Personal or Intermediate level information is the third level, in which includes information that is not always readily shared but that is also not very private, such as values and goals. Lastly, the Core or central level information is regarded as highly private and is shared when someone feels close to the other person or believes they are very trustworthy (Manning, 2019). An example of Core level information may be one's insecurities or fears. This theory states that for interpersonal relationships to develop there must be information exchanged between those within the relationship. A significant aspect of the theory is that of [[wikipedia:Social_penetration_theory#Breadth_and_depth|breadth and depth]] {{f}}. Breadth refers to the number of topics discussed, whereas depth refers to the level of intimacy of these discussions. For example, breadth may include discussing hobbies, family, interests and depth may be discussing feelings surrounding family issues or future goals (Carpenter & Greene, 2016), depth discussions may increase intimacy levels {{why}} {{g}}. According to this {{what}} theory {{g}} self-disclosure passes through phases as an interpersonal relationship progresses, {{g}} these stages include orientation, exploratory affective exchange, affective exchange, stable exchange and in some cases depenetration (see Table 1). The first stage is orientation, in which individuals only share superficial level information. Exploratory affective exchange is the second stage, where individuals may share beyond superficial information and may increase the breadth of topics discussed. The third stage is affective exchange, which includes more casual self-disclosures, communication and information from deeper levels, this stage is often reached with close friends or romantic partners. Stable exchange is the fourth stage, in which communication is open and honest, self-disclosures are continuous, and there is breadth and depth throughout conversations. This stage is maintained with few relationships such as romantic partners, close family and friends. Finally, depenetration or dissolution is a possible stage, whereby self-disclosure is reduced as a result of conflict or stress (Carpenter & Greene, 2016). {| class="wikitable" |+Table 1. The five stages of social penetration {{fact}} and examples. !Five stages of social penetration !Examples: |- |Orientation |Small talk, adhering to social norms. |- |Exploratory affective exchange |Personal attitudes & preferences. |- |Affective exchange |Casual relaxed communication, honest opinions, deeper emotional disclosures. |- |Stable exchange |Continuous open connection, deep core values, emotional reactions are predictable, breadth and depth explored. |- |Depenetration |Optional phase, communication breadth & depth withdrawal. |} == Emotional intimacy & its importance == Emotional intimacy is a part of interpersonal relationships, defined as a shared sense of closeness, trust and mutual understanding. Emotional intimacy happens between individuals when they consistently show each other who they really are and respond to each other in ways that promote feelings of safety, security and understanding. Thus, emotional intimacy can be experienced in a variety of different kind of relationships including, romantic partners, friends and family members. {{f}} === The intimacy process model === Reis and Shaver (1988) argue that intimacy is a dynamic, moment to moment transactional and cyclical process. The intimacy process model shows how emotional intimacy builds through a cyclical interactive progression between two individuals. This model uses self-disclosure, responsiveness, and perceived responsiveness as core components of emotional intimacy development. [[File:Intimacy process model.png|thumb|420x420px|'''Figure 4.''' The intimacy process model (Reis & Shaver, 1988).|left]] In this model {{g}} an interaction occurs between two people A and B, or for this example we will use the names Annabel and Bridgette. The model shows a single interaction in which Annabel is self-disclosing and Bridgette is responding (see Figure 4), however in most situations both individuals would freely move between self-disclosing and responding. In this example, the intimacy process begins with Annabel. Annabel has some mixture of motives, goals or fears that results in a self disclosure, whereby Bridgette must respond. As the role of the responder Bridgette is influenced by a mixture of concerns, which can be positive or negative {{g}}. Bridgette may want to be a good friend or partner or on the other hand, Bridgette may fear commitment, feel that Annabel is dependent on them and so forth. Annabel may intend on one thing, where Bridgette may perceive another. As Bridgette interprets the disclosure and expressions from Annabel, a complex mixture of motives and goals is at work, {{g}} this is referred to as the interpretive filter. In this model {{g}} Reis and Shaver (1988) argue that Bridgette's response to Annabel's self-disclosure is just as important as Annabel's initial disclosure, and will determine whether the interaction will become emotionally intimate. Suitable responses will enhance feelings of connectedness between the individuals, and inappropriate responses or non-responsiveness will keep the individuals at a distance. However, what matters most is Annabel's interpretation and reaction to Bridgette's response, for example if Annabel interprets Bridgette's response as unsupportive or too intrusive, greater intimacy is unlikely to be achieved (Reis & Shaver, 1988). === The importance of emotional intimacy === Emotional intimacy is intrinsically gratifying, and involves feelings of safety, trust, understanding, validation, acceptance and overall feeling cared for {{f}}. Studies have found associations between emotional intimacy, communication and relationship satisfaction, such that partners' perception of their emotional intimacy predicts both their own relationship satisfaction as well as their partners (Yoo et al., 2014). Similarly, those with strong intimate relationships tend to have higher life satisfaction (Amati et al., 2018) and wellbeing (Greene et al., 2006). Overall, intimacy can result in reciprocal intimacy, mutual liking and relationship progress (Forgas, 2011). Emotional intimacy has been found to play a role in reducing stress responses (Phillips et al., 2009). It has also been found that loneliness and depression partly stem from a lack of emotional intimacy (Rokach & Ohiliber-Lignieres, 2015). This research highlights just how important emotional intimacy is in everyday life, therefore it is important to learn how to improve emotional intimacy. {{RoundBoxTop|theme=3}} ;Key points: [[File:Crystal Clear app ktip.svg|left|20px|]] *Intimacy is related to relationship and life satisfaction. *Emotional intimacy is related to higher wellbeing and reduced stress responses. *Emotional intimacy can facilitate reciprocal intimacy, mutual liking and relationship progression. {{RoundBoxBottom}} ==Fostering self-disclosure== Rewarding relationships are practically inconceivable without self-disclosure. Self-disclosure is important in initiating, maintaining and enhancing emotional intimacy within interpersonal relationships, which has been linked to relationship success and satisfaction, as well as improved mental health (Forgas, 2011). Therefore, learning to foster and encourage self-disclosure is essential for everyday life. === How to encourage & respond to self-disclosures === In order to encourage self-disclosures from others we must remember Social Exchange Theory (Thibaut & Kelley, 1959) and the cost-benefit analysis. If someone wanted to disclose something to you, think to yourself, would it be rewarding for them? Making a self-disclosure conversation rewarding can be done in a multitude of ways. Social cues such as enquiries of concern (asking if someone is okay for example) or initiating with a self-disclosure displays inviting cues to the other person to disclose (Greene et al., 2006). It has also been found that moods have an effect of self-disclosure, such that positive moods increase self-disclosure intimacy, abstractness and breadth, whereas negative moods reduce intimacy but improve reciprocity of self-disclosure (Forgas, 2011), in which is an important aspect to take into account. Additionally, adherence to the norm of reciprocity, in which you can respond with a similar self-disclosure. Similarly, the intimacy process model (Reis & Shaver, 1988) highlights the underlying cause of a self-disclosure as well as the importance of responsiveness to self-disclosures. It is helpful to remember that there is an underlying motive, goal, fear or reason for someone to self-disclose. In the case of responsiveness, High or good responsiveness is comprised of the content of the response (elaborating on or matching disclosure), the style of response (showing concern and care) as well as the timing of the response (immediate or delayed). However, it is extremely important to take into account boundaries and assess the appropriateness of the situation, such as the physical environment where the interaction is occurring (Greene et al., 2006). {{RoundBoxTop|theme=3}} ;Key points: [[File:Crystal Clear app ktip.svg|left|20px|]] How to encourage and respond to self-disclosure: *The cost-benefit analysis: make the interaction rewarding through listening and responsiveness. *Social cues: enquiries of concern and initiating self-disclosure. *The norm of reciprocity: use reciprocal self-disclosure. *Boundaries: assess the appropriateness of the situation. {{RoundBoxBottom}} ==Conclusion== Self-disclosure creates a sense of understanding, safety, and trust that is essential for emotionally intimate interpersonal relationships. This self-disclosure process is intrinsic and crucial to progress interpersonal relationships. Self-disclosure drives emotional intimacy by creating a loop of vulnerability, validation and mutual understanding. Emotional intimacy has effects on aspects of everyday life including on communication and relationship satisfaction, such that perception of emotional intimacy can predict both one's own relationship satisfaction as well as their partners. Strong emotionally intimate relationships tend to have higher life satisfaction and wellbeing. Additionally, intimacy can result in reciprocal intimacy, mutual liking and relationship progress. Emotional intimacy has also been found to play a role in reducing stress responses and a lack of emotional intimacy can play a role in loneliness and depression. Therefore, it is important to improve emotional intimacy through learning to foster and encourage self-disclosure. Self-disclosure can be fostered through social cues, making the interaction rewarding and validating, using the norm of reciprocity and taking responsiveness into account. {{RoundBoxTop|theme=11}} ;Take home messages [[File:Crystal Clear app ktip.svg|left|20px|]] *Self-disclosure is the process of divulging personal experiences, thoughts and feelings to others. *Emotional intimacy is a shared sense of closeness, trust and mutual understanding, that can have effects on life and relationship satisfaction, wellbeing, stress and loneliness. *Self-disclosure can be fostered to improve emotional intimacy in relationships through reciprocal self-disclosure, responsiveness, and social cues. {{RoundBoxBottom}} ==See also== * [[Motivation and emotion/Book/2026/Responsiveness and interpersonal trust|Responsiveness and interpersonal trust]] (Book chapter, 2026) * [[Motivation and emotion/Book/2025/Self-disclosure and well-being|Self-disclosure and well-being]] (Book chapter, 2025) * [[wikipedia:Social_exchange_theory|Social exchange theory]] (Wikipedia) * [[wikipedia:Social_penetration_theory|Social penetration theory]] (Wikipedia) ==References== {{Hanging indent|1= Altman, I., & Taylor, D. (1973). ''Social penetration: The development of interpersonal relationships''. Halsted Press. https://psycnet.apa.org/record/1973-28661-000 Amati, V., Meggiolaro, S., Rivellini, G., & Zaccarin. (2018). Social relations and life satisfaction: the role of friends. ''Genus, 74''(7). 1-18. https://doi.org/10.1186/s41118-018-0032-z Burger, J. M., Sanches, J., Imberi, J. E., & Grande L. R. (2009). The norm of reciprocity as an internalized social norm: Returning favors even when no one finds out. ''Social Influence, 4''(1). 11-17. https://doi.org/10.1080/15534510802131004 Carpenter, A., & Greene, K. (2016). Social penetration theory. In C. R. Berger & M. E. Roloff (Eds.), ''The International Encyclopedia of Interpersonal Communication'' (1st ed., pp. 1-4). John Wiley & Sons. https://doi.org/10.1002/9781118540190.wbeic0160 Derlega, V. J., Winstead, B. A., & Greene, K. (2008). Self-disclosure and starting a close relationship. In S. Sprecher, A. Wenzel, & J. Harvey (Eds.), ''Handbook of relationship initiation''. Psychology Press. 53–174. https://psycnet.apa.org/record/2008-09972-008 Forgas, J. P. (2011). Affective influences on self-disclosure: Mood effects on the intimacy and reciprocity of disclosing personal information. ''Journal of Personality and Social Psychology, 100''(3). 449-461. https://doi.org/10.1037/a0021129 Greene, K., Derlega, V. J., & Mathews, A. (2006). Self-disclosure in personal relationships. In A. L. Vangelisti & D. Perlman (Eds.), ''The Cambridge handbook of personal relationships.'' (pp. 409-427). Cambridge University Press. https://doi.org/10.1017/CBO9780511606632.023 Manning, J. (2019). Thinking about interpersonal relationships and social penetration theory: Is it the same for lesbian, gay or bisexual people? In C. J. Liberman, A. S. Rancer & T. A. Avtgis (Eds.), ''Casing Communication Theory'' (pp. 293-306). Kendall Hunt Publishing Company. Pecune, F. (2013). Toward a computational model of social relations for artificial companions. ''Proceedings - 2013 Humaine Association Conference on Affective Computing and Intelligent Interaction''. 677-682. [https://www.researchgate.net/publication/261271664%20Toward%20a%20Computational%20Model%20of%20Social%20Relations%20for%20Artificial%20Companions DOI:10.1109/ACII.2013.118] Phillips, A., Gallagher, S., & Carroll, D. (2009) Social support, social intimacy, and cardiovascular reactions to acute psychological stress. ''Annals of Behavioural Medicine, 37''. 38-45. http://dx.doi.org/10.1007/s12160-008-9077-0 Reis, H. T. (2018). Intimacy as an interpersonal process. In H. T. Reis (Ed.), ''Relationships, Well-Being and Behaviour: Selected works of Harry T. Reis'' (pp. 209-244). Routledge. https://doi.org/10.4324/9780203732496 Reis, H. T., & Shaver, P. (1988). Intimacy as an interpersonal process. In S. Duck, D. H. Fay, J. K. Simpson, & J. C. Murray (Eds.), ''Handbook of personal relationships: Theory, research and applications'' (p. 367–389). John Wiley & Sons. Rokach, A., & Philibert-Lignieres, G. (2015). Intimacy, loneliness & infidelity. ''The Open Psychology Journal, 8''. 71-77. DOI: 10.2174/1874350101508010071 Sprecher, S., Treger, S., Wondra, J. D., Hilaire, N., & Wallpe, K. (2013). Taking turns: Reciprocal self-disclosure promotes liking in initial interactions. ''Journal of Experimental Social Psychology, 49''(5). 860-866. https://doi.org/10.1016/j.jesp.2013.03.017. Tolstedt, B. E., & Stokes, J. P. (1984). Self-disclosure, intimacy, and the depenetration process. ''Journal of Personality and Social Psychology, 46''(1). 84-90. https://doi.org/10.1037/0022-3514.46.1.84 Utz, S. (2015). The function of self-disclosure on social network sites: Not only intimate, but also positive and entertaining self-disclosures increase the feeling of connection. ''Computers in Human Behaviour, 45''. 1-10. https://doi.org/10.1016/j.chb.2014.11.076. Yoo, H., Bartle-Haring, S., Day. R., & Gangamma, R. (2014). Couple communication, emotional and sexual intimacy, and relationship satisfaction. ''Journal of Sex & Marital Therapy, 40''(4), 275-293. https://doi.org/10.1080/0092623X.2012.751072 Yu, Y., & Li, W. (2026). A fNIRS hyperscanning study on enhancing intimacy through emotional self-disclosure from the disclosure-responder perspective. ''BMC Psychology, 14''. https://doi.org/10.1186/s40359-026-04257-3 }} ==External links== * [https://www.youtube.com/watch?v=Z4bRXozREKU&t=90s Self-disclosure in personal relationships: How to stop oversharing] (youtube.com) * [https://www.psychologytoday.com/au/blog/between-the-sheets/202011/why-romantic-intimacy-requires-self-disclosure Why romantic intimacy requires self-disclosure] (psychologytoday.com) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Relationships]] iq3x5x7ydbyg4xoj4r3wezrsdbhyjli 2844568 2844567 2026-11-05T02:21:00Z Jtneill 10242 /* How to encourage & respond to self-disclosures */ 2844568 wikitext text/x-wiki {{title|Self-disclosure and emotional intimacy:<br>How does self-disclosure foster emotional closeness in relationships?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}} [[File:Stockcouple.jpg|thumb|right|150px|'''Figure 1'''. Image of couple hugging {{ic|Explain what this has to do with self-disclosure}/}]] ;Picture this... Jim and Pam (see Figure 1) have been dating for a few months; things have been going well but at times there is noticeable emotional distance between them. One night Jim tells Pam that “I haven’t talked to anyone about this, but I’m terrified one day you will get bored and leave”. This confession opens the flood gates for Jim and Pam to share their deepest fears and anxieties. In this moment, an invisible barrier dissolves, they have discovered a new profound sense of trust and understanding. A new and deeper layer of their relationship has been unlocked; there is an emotional closeness that wasn’t there before. Why does this self-disclosure from Jim create this emotional closeness between him and Pam? {{RoundBoxBottom}} [[wikipedia:Self-disclosure|Self-disclosure]] is the voluntary process of sharing personal thoughts, feeling and information with another person. Self-disclosure is one of the most significant and effective ways to initiate and enhance relationship intimacy, and to maintain positive personal relationships (Forgas, 2011). Intimacy or intimate come from the Latin terms ''intimus'' meaning innermost, and ''intimare'' meaning to make the innermost known (Reis, 2018). [[wikipedia:Emotional_intimacy|Emotional intimacy]] refers to a shared sense of understanding, trust and closeness within an interpersonal relationship. This chapter discusses what both self-disclosure and emotional intimacy mean, and how self-disclosures can be encouraged and responded to, with reference to psychological theories and research. {{RoundBoxTop|theme=11}} ;Focus questions [[File:Crystal Clear app ktip.svg|left|20px|]] *What is the self-disclosure? *What is emotional intimacy and why is it important? *How can self-disclosure be fostered to improve emotional intimacy in relationships? {{RoundBoxBottom}} == Self-disclosure == Self-disclosure is defined as the process of divulging personal experiences, thoughts and feelings to others (Yu & Li, 2026). Self-disclosure has long been discussed and researched for its relationship to emotional intimacy {{f}}. According to the research, we tend to like people who disclose more, we disclose more to those we like, and when we disclose, we like those we disclosed to more afterwards (Utz, 2015) {{expand}}. === Social exchange theory === [[File:Costs vs rewards.png|thumb|247x247px|'''Figure 2.''' Social Exchange Theory, costs versus rewards {{g}}|right]] [[wikipedia:Social_exchange_theory|Social Exchange Theory]] is a psychological framework that states human behaviours and relationships work through a cost-benefit analysis (see Figure 2) (Thibaut & Kelley, 1959). This core concept can be applied to that of self-disclosure, in which individuals assess the potential rewards and risks of sharing personal information. In the case of self-disclosure, possible rewards or benefits of divulging personal information may be validation, emotional relief and enhanced closeness {{f}}. If we assess that these are likely outcomes of self-disclosing we are more likely to do so. In contrast, potential costs or risks of self-disclosure may be a loss of privacy, judgement or fear of rejection and in this circumstance we are much less likely to self disclose. Interpersonal relationships thrive when self-disclosure remains profitable or rewarding. {{f}} ==== Norm of reciprocity ==== Within Social Exchange Theory is the concept of the [[wikipedia:Norm_of_reciprocity|norm of reciprocity]] {{f}}. This entails the universal social expectation that people should return or repay favours, gifts and acts of kindness with similar treatment {{f}}. For example, if someone gifts you an expensive item for your birthday, you will likely feel an internal obligation to get them a similarly expensive item for their birthday. This norm enables smooth, equal and fair social exchanges (Burger et al., 2009). This concept is also applicable to self-disclosure, {{g}} research has shown that self-disclosure is reciprocal in existing relationships and in the context of strangers (Sprecher et al., 2013). Hence, reciprocal self-disclosure is the process where one person shares personal thoughts, feelings or information and the other person responds with similar personal details in return. === Social penetration theory === [[File:Bdspt.model.webp|'''Figure 3.''' Social Penetration Theory Model (Pecune, 2013).|alt=Figure 2. Social Penetration Theory Model (Pecune, 2013).|thumb|300x300px]] [[wikipedia:Social_penetration_theory|Social Penetration Theory]] is a social psychological model that explains how interpersonal relationships evolve from superficial or surface level interactions to deep emotional intimacy through self-disclosure (Altman & Taylor, 1973). Social Penetration Theory uses an onion model (see Figure 3) as a metaphor to describe how the social penetration process works, in which individuals "peel back" others layers of personal information through social interaction (self-disclosures) {{f}}. One aspect of Social Penetration Theory is the types of information shared, which is referred to as levels {{f}}. The first {{g}} being the Superficial level or in other words, surface level information which includes visible attributes. The second level is the Intimate or peripheral level information, which is information that people usually feel free to share in any general social situation, such as their name and where they are from. Personal or Intermediate level information is the third level, in which includes information that is not always readily shared but that is also not very private, such as values and goals. Lastly, the Core or central level information is regarded as highly private and is shared when someone feels close to the other person or believes they are very trustworthy (Manning, 2019). An example of Core level information may be one's insecurities or fears. This theory states that for interpersonal relationships to develop there must be information exchanged between those within the relationship. A significant aspect of the theory is that of [[wikipedia:Social_penetration_theory#Breadth_and_depth|breadth and depth]] {{f}}. Breadth refers to the number of topics discussed, whereas depth refers to the level of intimacy of these discussions. For example, breadth may include discussing hobbies, family, interests and depth may be discussing feelings surrounding family issues or future goals (Carpenter & Greene, 2016), depth discussions may increase intimacy levels {{why}} {{g}}. According to this {{what}} theory {{g}} self-disclosure passes through phases as an interpersonal relationship progresses, {{g}} these stages include orientation, exploratory affective exchange, affective exchange, stable exchange and in some cases depenetration (see Table 1). The first stage is orientation, in which individuals only share superficial level information. Exploratory affective exchange is the second stage, where individuals may share beyond superficial information and may increase the breadth of topics discussed. The third stage is affective exchange, which includes more casual self-disclosures, communication and information from deeper levels, this stage is often reached with close friends or romantic partners. Stable exchange is the fourth stage, in which communication is open and honest, self-disclosures are continuous, and there is breadth and depth throughout conversations. This stage is maintained with few relationships such as romantic partners, close family and friends. Finally, depenetration or dissolution is a possible stage, whereby self-disclosure is reduced as a result of conflict or stress (Carpenter & Greene, 2016). {| class="wikitable" |+Table 1. The five stages of social penetration {{fact}} and examples. !Five stages of social penetration !Examples: |- |Orientation |Small talk, adhering to social norms. |- |Exploratory affective exchange |Personal attitudes & preferences. |- |Affective exchange |Casual relaxed communication, honest opinions, deeper emotional disclosures. |- |Stable exchange |Continuous open connection, deep core values, emotional reactions are predictable, breadth and depth explored. |- |Depenetration |Optional phase, communication breadth & depth withdrawal. |} == Emotional intimacy & its importance == Emotional intimacy is a part of interpersonal relationships, defined as a shared sense of closeness, trust and mutual understanding. Emotional intimacy happens between individuals when they consistently show each other who they really are and respond to each other in ways that promote feelings of safety, security and understanding. Thus, emotional intimacy can be experienced in a variety of different kind of relationships including, romantic partners, friends and family members. {{f}} === The intimacy process model === Reis and Shaver (1988) argue that intimacy is a dynamic, moment to moment transactional and cyclical process. The intimacy process model shows how emotional intimacy builds through a cyclical interactive progression between two individuals. This model uses self-disclosure, responsiveness, and perceived responsiveness as core components of emotional intimacy development. [[File:Intimacy process model.png|thumb|420x420px|'''Figure 4.''' The intimacy process model (Reis & Shaver, 1988).|left]] In this model {{g}} an interaction occurs between two people A and B, or for this example we will use the names Annabel and Bridgette. The model shows a single interaction in which Annabel is self-disclosing and Bridgette is responding (see Figure 4), however in most situations both individuals would freely move between self-disclosing and responding. In this example, the intimacy process begins with Annabel. Annabel has some mixture of motives, goals or fears that results in a self disclosure, whereby Bridgette must respond. As the role of the responder Bridgette is influenced by a mixture of concerns, which can be positive or negative {{g}}. Bridgette may want to be a good friend or partner or on the other hand, Bridgette may fear commitment, feel that Annabel is dependent on them and so forth. Annabel may intend on one thing, where Bridgette may perceive another. As Bridgette interprets the disclosure and expressions from Annabel, a complex mixture of motives and goals is at work, {{g}} this is referred to as the interpretive filter. In this model {{g}} Reis and Shaver (1988) argue that Bridgette's response to Annabel's self-disclosure is just as important as Annabel's initial disclosure, and will determine whether the interaction will become emotionally intimate. Suitable responses will enhance feelings of connectedness between the individuals, and inappropriate responses or non-responsiveness will keep the individuals at a distance. However, what matters most is Annabel's interpretation and reaction to Bridgette's response, for example if Annabel interprets Bridgette's response as unsupportive or too intrusive, greater intimacy is unlikely to be achieved (Reis & Shaver, 1988). === The importance of emotional intimacy === Emotional intimacy is intrinsically gratifying, and involves feelings of safety, trust, understanding, validation, acceptance and overall feeling cared for {{f}}. Studies have found associations between emotional intimacy, communication and relationship satisfaction, such that partners' perception of their emotional intimacy predicts both their own relationship satisfaction as well as their partners (Yoo et al., 2014). Similarly, those with strong intimate relationships tend to have higher life satisfaction (Amati et al., 2018) and wellbeing (Greene et al., 2006). Overall, intimacy can result in reciprocal intimacy, mutual liking and relationship progress (Forgas, 2011). Emotional intimacy has been found to play a role in reducing stress responses (Phillips et al., 2009). It has also been found that loneliness and depression partly stem from a lack of emotional intimacy (Rokach & Ohiliber-Lignieres, 2015). This research highlights just how important emotional intimacy is in everyday life, therefore it is important to learn how to improve emotional intimacy. {{RoundBoxTop|theme=3}} ;Key points: [[File:Crystal Clear app ktip.svg|left|20px|]] *Intimacy is related to relationship and life satisfaction. *Emotional intimacy is related to higher wellbeing and reduced stress responses. *Emotional intimacy can facilitate reciprocal intimacy, mutual liking and relationship progression. {{RoundBoxBottom}} ==Fostering self-disclosure== Rewarding relationships are practically inconceivable without self-disclosure. Self-disclosure is important in initiating, maintaining and enhancing emotional intimacy within interpersonal relationships, which has been linked to relationship success and satisfaction, as well as improved mental health (Forgas, 2011). Therefore, learning to foster and encourage self-disclosure is essential for everyday life. === How to encourage and respond to self-disclosures === In order to encourage self-disclosures from others we must remember Social Exchange Theory (Thibaut & Kelley, 1959) and the cost-benefit analysis. If someone wanted to disclose something to you, think to yourself, would it be rewarding for them? Making a self-disclosure conversation rewarding can be done in a multitude of ways. Social cues such as enquiries of concern (asking if someone is okay for example) or initiating with a self-disclosure displays inviting cues to the other person to disclose (Greene et al., 2006). It has also been found that moods have an effect of self-disclosure, such that positive moods increase self-disclosure intimacy, abstractness and breadth, whereas negative moods reduce intimacy but improve reciprocity of self-disclosure (Forgas, 2011), in which is an important aspect to take into account. Additionally, adherence to the norm of reciprocity, in which you can respond with a similar self-disclosure. Similarly, the intimacy process model (Reis & Shaver, 1988) highlights the underlying cause of a self-disclosure as well as the importance of responsiveness to self-disclosures. It is helpful to remember that there is an underlying motive, goal, fear or reason for someone to self-disclose. In the case of responsiveness, High or good responsiveness is comprised of the content of the response (elaborating on or matching disclosure), the style of response (showing concern and care) as well as the timing of the response (immediate or delayed). However, it is extremely important to take into account boundaries and assess the appropriateness of the situation, such as the physical environment where the interaction is occurring (Greene et al., 2006). {{RoundBoxTop|theme=3}} ;Key points: [[File:Crystal Clear app ktip.svg|left|20px|]] How to encourage and respond to self-disclosure: *The cost-benefit analysis: make the interaction rewarding through listening and responsiveness. *Social cues: enquiries of concern and initiating self-disclosure. *The norm of reciprocity: use reciprocal self-disclosure. *Boundaries: assess the appropriateness of the situation. {{RoundBoxBottom}} ==Conclusion== Self-disclosure creates a sense of understanding, safety, and trust that is essential for emotionally intimate interpersonal relationships. This self-disclosure process is intrinsic and crucial to progress interpersonal relationships. Self-disclosure drives emotional intimacy by creating a loop of vulnerability, validation and mutual understanding. Emotional intimacy has effects on aspects of everyday life including on communication and relationship satisfaction, such that perception of emotional intimacy can predict both one's own relationship satisfaction as well as their partners. Strong emotionally intimate relationships tend to have higher life satisfaction and wellbeing. Additionally, intimacy can result in reciprocal intimacy, mutual liking and relationship progress. Emotional intimacy has also been found to play a role in reducing stress responses and a lack of emotional intimacy can play a role in loneliness and depression. Therefore, it is important to improve emotional intimacy through learning to foster and encourage self-disclosure. Self-disclosure can be fostered through social cues, making the interaction rewarding and validating, using the norm of reciprocity and taking responsiveness into account. {{RoundBoxTop|theme=11}} ;Take home messages [[File:Crystal Clear app ktip.svg|left|20px|]] *Self-disclosure is the process of divulging personal experiences, thoughts and feelings to others. *Emotional intimacy is a shared sense of closeness, trust and mutual understanding, that can have effects on life and relationship satisfaction, wellbeing, stress and loneliness. *Self-disclosure can be fostered to improve emotional intimacy in relationships through reciprocal self-disclosure, responsiveness, and social cues. {{RoundBoxBottom}} ==See also== * [[Motivation and emotion/Book/2026/Responsiveness and interpersonal trust|Responsiveness and interpersonal trust]] (Book chapter, 2026) * [[Motivation and emotion/Book/2025/Self-disclosure and well-being|Self-disclosure and well-being]] (Book chapter, 2025) * [[wikipedia:Social_exchange_theory|Social exchange theory]] (Wikipedia) * [[wikipedia:Social_penetration_theory|Social penetration theory]] (Wikipedia) ==References== {{Hanging indent|1= Altman, I., & Taylor, D. (1973). ''Social penetration: The development of interpersonal relationships''. Halsted Press. https://psycnet.apa.org/record/1973-28661-000 Amati, V., Meggiolaro, S., Rivellini, G., & Zaccarin. (2018). Social relations and life satisfaction: the role of friends. ''Genus, 74''(7). 1-18. https://doi.org/10.1186/s41118-018-0032-z Burger, J. M., Sanches, J., Imberi, J. E., & Grande L. R. (2009). The norm of reciprocity as an internalized social norm: Returning favors even when no one finds out. ''Social Influence, 4''(1). 11-17. https://doi.org/10.1080/15534510802131004 Carpenter, A., & Greene, K. (2016). Social penetration theory. In C. R. Berger & M. E. Roloff (Eds.), ''The International Encyclopedia of Interpersonal Communication'' (1st ed., pp. 1-4). John Wiley & Sons. https://doi.org/10.1002/9781118540190.wbeic0160 Derlega, V. J., Winstead, B. A., & Greene, K. (2008). Self-disclosure and starting a close relationship. In S. Sprecher, A. Wenzel, & J. Harvey (Eds.), ''Handbook of relationship initiation''. Psychology Press. 53–174. https://psycnet.apa.org/record/2008-09972-008 Forgas, J. P. (2011). Affective influences on self-disclosure: Mood effects on the intimacy and reciprocity of disclosing personal information. ''Journal of Personality and Social Psychology, 100''(3). 449-461. https://doi.org/10.1037/a0021129 Greene, K., Derlega, V. J., & Mathews, A. (2006). Self-disclosure in personal relationships. In A. L. Vangelisti & D. Perlman (Eds.), ''The Cambridge handbook of personal relationships.'' (pp. 409-427). Cambridge University Press. https://doi.org/10.1017/CBO9780511606632.023 Manning, J. (2019). Thinking about interpersonal relationships and social penetration theory: Is it the same for lesbian, gay or bisexual people? In C. J. Liberman, A. S. Rancer & T. A. Avtgis (Eds.), ''Casing Communication Theory'' (pp. 293-306). Kendall Hunt Publishing Company. Pecune, F. (2013). Toward a computational model of social relations for artificial companions. ''Proceedings - 2013 Humaine Association Conference on Affective Computing and Intelligent Interaction''. 677-682. [https://www.researchgate.net/publication/261271664%20Toward%20a%20Computational%20Model%20of%20Social%20Relations%20for%20Artificial%20Companions DOI:10.1109/ACII.2013.118] Phillips, A., Gallagher, S., & Carroll, D. (2009) Social support, social intimacy, and cardiovascular reactions to acute psychological stress. ''Annals of Behavioural Medicine, 37''. 38-45. http://dx.doi.org/10.1007/s12160-008-9077-0 Reis, H. T. (2018). Intimacy as an interpersonal process. In H. T. Reis (Ed.), ''Relationships, Well-Being and Behaviour: Selected works of Harry T. Reis'' (pp. 209-244). Routledge. https://doi.org/10.4324/9780203732496 Reis, H. T., & Shaver, P. (1988). Intimacy as an interpersonal process. In S. Duck, D. H. Fay, J. K. Simpson, & J. C. Murray (Eds.), ''Handbook of personal relationships: Theory, research and applications'' (p. 367–389). John Wiley & Sons. Rokach, A., & Philibert-Lignieres, G. (2015). Intimacy, loneliness & infidelity. ''The Open Psychology Journal, 8''. 71-77. DOI: 10.2174/1874350101508010071 Sprecher, S., Treger, S., Wondra, J. D., Hilaire, N., & Wallpe, K. (2013). Taking turns: Reciprocal self-disclosure promotes liking in initial interactions. ''Journal of Experimental Social Psychology, 49''(5). 860-866. https://doi.org/10.1016/j.jesp.2013.03.017. Tolstedt, B. E., & Stokes, J. P. (1984). Self-disclosure, intimacy, and the depenetration process. ''Journal of Personality and Social Psychology, 46''(1). 84-90. https://doi.org/10.1037/0022-3514.46.1.84 Utz, S. (2015). The function of self-disclosure on social network sites: Not only intimate, but also positive and entertaining self-disclosures increase the feeling of connection. ''Computers in Human Behaviour, 45''. 1-10. https://doi.org/10.1016/j.chb.2014.11.076. Yoo, H., Bartle-Haring, S., Day. R., & Gangamma, R. (2014). Couple communication, emotional and sexual intimacy, and relationship satisfaction. ''Journal of Sex & Marital Therapy, 40''(4), 275-293. https://doi.org/10.1080/0092623X.2012.751072 Yu, Y., & Li, W. (2026). A fNIRS hyperscanning study on enhancing intimacy through emotional self-disclosure from the disclosure-responder perspective. ''BMC Psychology, 14''. https://doi.org/10.1186/s40359-026-04257-3 }} ==External links== * [https://www.youtube.com/watch?v=Z4bRXozREKU&t=90s Self-disclosure in personal relationships: How to stop oversharing] (youtube.com) * [https://www.psychologytoday.com/au/blog/between-the-sheets/202011/why-romantic-intimacy-requires-self-disclosure Why romantic intimacy requires self-disclosure] (psychologytoday.com) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Relationships]] 0eoc8n2y4z9gb5vagtjnr4bzxzsxlnv 0 331111 2844513 2844348 2026-11-04T21:35:27Z U3143751 3106785 /* Emotional exhaustion */ 2844513 wikitext text/x-wiki {{title|Empathy fatigue and emotional exhaustion:<br>How does sustained empathic engagement contribute to emotional exhaustion?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}} [[File:Exhausted smartphone user.JPG|right|thumb|150px|'''Figure 1'''. An emotionally exhausted Aelin at the end of the work day]] ; Scenario Aelin, a 23-year-old social worker employed at headspace {{ic|Link to the Wikipedia article to help explain what this term means}}, works with approximately 20 {{ic|Explain what sort of}}Imagine you are only 23 years old and are working as a social worker at headspace. You see around 20 clients each week, many of whom have experienced rough living conditions. During your first six months in this role, you felt highly motivated and excited to help people in need. You are an excellent active listener and can easily remember details about your clients Live’s. You are also good at recognising signs of distress in your client’s body language and adjust yours in order to show warmth and understanding. As you progress in your role, you take on more cases that involve complex past traumas. The workload becomes too much for you to get done at work, but you don’t want your clients to miss out, so you take your work home to complete it in the evenings. Your workload increases again and you now start doing work on your weekends as well. After several months of working this way, you start to notice you feel exhausted every morning before you have even started work. You then start to notice you’re responding to people at work differently. Instead of your usual energised active listening response you feel frustrated, empathetic and detached. After work you are too drained to socialise with friends. This begins to worry you. What if you won’t be able to continue working as a social worker? {{RoundBoxBottom}} Aelin is experiencing '''emotional exhaustion''' from a sustained empathic response. Due to the nature of Aelin's work, they are constantly engaged in an empathic response. Empathic responses require cognitive and physical resources to sustain. Because Aelin is taking their work home and not resting on weekends, they do not have a chance to rest. It is important for Aelin to recognise what is happening and why. Not actioning the problem has led to empathy fatigue and emotional exhaustion. Emotional exhaustion is one of the key aspects of [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FEmpathy_fatigue_and_emotional_exhaustion&diff=2840411&oldid=2832859 burnout], a long-term condition of existing in a constant state of high stress. Psychological science can support Aelin in recognising and mapping the symptoms they are experiencing, understand coping strategies to minimise the current and future effects to improve their wellbeing. Understanding the symptoms will also support their role as a social worker, providing more tools for the people they support on a day-to-day basis. {{RoundBoxTop|theme=3}} '''Focus questions''' * Why do people become emotionally exhausted? * What is the relationship between empathy fatigue and emotional exhaustion? * What role does emotional labor play in emotional exhaustion? * How does understanding emotional labor and empathy fatigue work to reduce the effects of emotional exhaustion? {{RoundBoxBottom}} == Emotional exhaustion == To understand emotional exhaustion, it will be important to first understand burnout. Burnout is a long-term condition of existing in a constant state of high stress and affects millions of people annually. In the early 2000s, a survey from Australia found that more than 50% of social workers and occupational workers reported high levels of burnout symptoms (Llyod and King, 2004) and a recent report from 2026 found that more than one third of all workers in Australia were experiencing burnout (Chan, 2026). Being in a state of high stress for short periods of time is a useful adaptation for humans, so much so that it has been given a name, [[wikipedia:Eustress|Eustress]]. However, as a long-term condition it leads to individual health issues and can be linked to cardiovascular disease, increased vulnerability to upper respiratory diseases, and chronic inflammation (Schneiderman et al., 2005), and cognitive issues such as [[wikipedia:Inattentional_blindness|cognitive tunneling]] and impaired selective attention (Pooladvand and Hasanzadeh, 2023). Burnout can be defined across three dimensions: emotional exhaustion, [[wikipedia:Depersonalization|depersonalisation]], and reduced feelings of personal accomplishment (Maslach, 1993, as cited in Martingano et al., 2026). Emotional exhaustion may be considered the 'core' component of burnout and refers to the depletion of emotional resources, leading to feelings of overwhelm, overextension, and exhaustion (Martingano et al., 2026; De Beer et al., 2024; Seidler et al., 2014). As a major factor, its symptoms are similar to burnout and stress. Although antecedents of burnout are typically agreed upon (Bakker et al., 2014) and divided into individual (e.g. self-efficacy) and situational categories (e.g. work load), there are many theories that attempt to explain why emotional exhaustion occurs. Two of these theories with good evidence are the [[wikipedia:Conservation_of_resources_theory|Conservation of resources stress model]] (Hobfoll, 1989) and the [[wikipedia:Job_demands-resources_model|Job Demands-Resources model]] (Demerouti et al., 2001). === Conservation of resources stress model === The Conservation of resources stress model (CORT; Hobfall, 1989) is a theory that works to understand and explain why stress occurs. Under the model, humans are motivated to maintain, pursue, or attain resources. Stress, defined as "an individual's perceived pressure or strain" (Hollebeek et al., 2023 p.928), occurs when these resources are threatened or lost, or gained at a net loss. As noted above, resources aren't strictly tied to being physical objects. They can also be emotional states, conditions, or even ideas. For example, someone may place value on enjoying a calm, quiet Sunday morning at home. If their neighbour starts using power tools loudly next door, it would disrupt this state and induce stress in the individual. The CORT model focuses primarily on two principles. The first principle is that there is more harm in losing resources than there is in gaining them back (Halbesleben et al., 2014). This is known as loss salience, and some research suggests that the pain of loss is felt twice as strong as the pleasure of gain (Thaler, 2000). Looking at our previous example, imagine that the person who enjoys a quiet Sunday morning at home equally enjoys the sound of a loud car engine. Even though the person enjoys the sound of car engines, if the neighbour started revving their car loudly instead of using power tools, it would still disrupt their desired state of quiet and lead to an increase of stress. The second principle is that people must invest existing resources in order to gain or protect resources (Halbesleben et al., 2014). This resource investment includes recovering from resource loss. Coping is how people manage or overcome a difficult situation, such as protecting or recovering from resource loss. Coping can be broken down into different strategies that people use, but not all strategies are equal (Ito and Brotheridge, 2003). They are divided into two categories: control or escape strategies. Control strategies are considered better at reducing the strain from stress and consist of positive thinking, direct action, and help seeking (Havlovic & Keenan, 1991, as cited in Ito and Brotheridge, 2003). The most useful control strategies incorporate a person's [[locus of control]], emphasise seeking support from co-workers, and highlight the importance of autonomy in decision making. There is fairly robust evidence that supports the CORT model. === Job demands-resources model === Another model that attempts to understand why people become emotionally exhausted is the Job Demands-Resources model (JD-R; Demerouti et al., 2001). The JD-R model discusses two concepts: job demands and job resources. Demands and resources are both related to the physical, social, or psychological aspects of a job, but differ in their impact on a person. Job demands require physical or mental effort and exhaust a person's energy supply. Job resources typically support achieving goals, reduce the effort cost of job demands, and stimulate personal growth (Richter and Hacker, 1998, as cited in Demerouti et al., 2001). For example, an unsustainable workload would be considered a job demand, and receiving help and support from others would be considered a resource. The JD-R has two primary processes for how demands and resources interact with one another (Schaufeli, 2017). The stress process occurs when job demands are high and resources are low. As stress increases strain and pressure on an individual, it leads to the exhaustion of their energy eventually, eventually leading to emotional exhaustion. The second process is the motivational process. This process occurs when there is an excess of job resources. Crucially, the motivational process does not require low job demands. With enough job resources, a person will be energised and engaged, reframing job demands as a challenge to overcome. Although job resources seem like a cure-all for demands, it's important to note that they do not retroactively treat emotional exhaustion. Under the JD-R model, burnout must be reduced by lowering job demands. However, once someone has recovered, it is possible to increase job resources alongside job demands to prevent future burnout and support [[wikipedia:Work_engagement|work engagement]]. Work engagement (engagement) has been discussed as the opposite of burnout (Bakker et al., 2014). Engagement is how people express themselves when performing a role. A role being a pattern of behaviour that a person goes through based on the social expectations of that role, similar to how an actor takes on a role for a movie. For example, when attending a typical restaurant, it is socially expected that a person will engage with the waiter and provide them with their meal order to take to the kitchen. This person is performing the script of a restaurant patron. In the context of a workplace, an employee with high engagement may appear to be energised, enthusiastic, and become absorbed in their work losing track of time (Bakkar et al, 2014). <nowiki>**</nowiki>Update box below to relate back to Aelin. {{RoundBoxTop|theme=3}} '''Why do we feel emotionally exhausted''' The JD-R model and the CORT model do use different language and have slight differences, but are similar in how emotional exhaustion is considered, and share a solution to the problem. Emotional exhaustion occurs when the energy cost for completing tasks occur faster than a person can replenish it. If a person is emotionally exhausted, they will need to reduce their workload to recover and, as part of recovery, increasing autonomy (control) in decision making, and asking for help through social support, will lead to positive outcomes - reducing the likelihood of future exhaustion and increasing engagement (Bakker et al., 2014; Halbesleben et al., 2014). {{RoundBoxBottom}} == Empathy fatigue == You may be familiar with the term [[empathy]]. It is often described in terms of supporting others. For example, when someone close to you experiences a loss and is grieving, you may empathise with them and experience a negative emotion akin to grief too. However, empathy is not exclusive to so called negative emotions. For example, compersion is a positive empathic response when a loved one experiences joy. Parents may experience compersion when something joyful happens to their child. At its core, empathy is the individual ability to imagine or understand another person's feelings and/or emotions (Riess, 2017; Shao et al., 2024) and has four main components (Davis, 1983, as cited in Delgado et al., 2023): * Empathic concern is xyz * Fantasy is xyz * Perspective taking is xyz * Personal distress is xyz Much like physical or cognitive processes, such as running or problem solving, empathising will drain energy and resources. Further, and similar to the difference in energy expenditure between running and jogging, there are distinct types of empathy that each have different energy costs. === Types of empathy === Research makes a clear distinction between two types of empathy, affective and cognitive empathy. Affective empathy occurs when a person has an emotional reaction to someone else's mental state (Shao et al., 2024). This occurs because (unsure about this next part go through and update accordingly) ''unconscious processes interpret the social stimuli from another person leading to a physiological and cognitive response. These processes may interpret stimuli from any of the five senses.'' Cognitive empathy is when a person understands someone else's feelings or emotions (Cui et al., 2022). * Integrating the two - "affective component is the essence of empathy, and the cognitive component is a process through which that essence comes into being" (Bošnjaković & Radionov, 2018) ** (Author's note: change quote into paraphrasing) * === Empathy or compassion fatigue === * Distinction between empathy and compassion fatigue * Compassion fatigue is closely related to Secondary traumatic stress usually found in healthcare professionals (Figley, 2002) ** Compassion fatigue is to Empathy fatigue as emotional exhaustion is to burnout * Make it clear that Aelin is experiencing Compassion Fatigue vs Empathy fatigue (they are the same but different) (Author's note: include an update to the scenario here!) '''Empathy fatigue questions'''. Choose your answers and click "Submit": <quiz display=simple> {Surface acting is when the observer actually feels the emotion: |type="()"} - True + False {Compassion fatigue and empathy fatigue are synonymous: |type="()"} - True + False </quiz> {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' * Under the JD-R model, a sustained empathic response is a long-term job demand. * Under the CORT model, a sustained empathic response uses energy to protect existing resources. As resources become low and burnout sets in, cognitive focus narrows leading to empathy fatigue. {{RoundBoxBottom}} == Emotional labor as a mediating factor between empathy fatigue and emotional exhaustion == As a society there are expected [https://en.wikipedia.org/wiki/Social_norm social norms] in behaviour when expressing emotion, regardless if someone is conscious of the process of matching these norms or not. For example, laughing during an emotional speech at a funeral would be considered a rude or offensive expression from the other attendees. As social creatures, humans need a way to regulate socially appropriate emotional expression. This is the essence of Emotional labor: "the regulation of felt and expressed emotions (whether effortful or not) to match these emotional display rules with the objective of fulfilling organizational goals" (Vial & Cowgill, 2022) What is emotional labor (Hochschild, 2012). Theories behind emotional labor involve acting, and an extension of acting known as emotional contagion. === Types of acting === * Surface acting ** This is where emotive dissonance comes into play ** Leads to emotional exhaustion (Kenworthy et al., 2014) ** "Should" language - in this situation, this is what I ''should'' feel * Deep acting ** People who 'deep act', as in genuinely change the way they feel (match the feelings they have with the feelings that they need to display) often report positive relationships with emotional labor ** Actual language - in this situation, this is what I ''actually'' feel Discuss the differences in gender norms and expectations in relationships {| class="wikitable" style="margin: auto;" |- ! !! Deep acting !! Surface acting |- | '''Emotional dissonance''' || || X |- | '''Not known to others''' || Hidden area || Unknown |} === Emotional contagion === Emotional contagion is the idea that emotions are contagious. as the theory to moderate surface/deep acting * Emotional contagion (Panksepp and Lahvis, 2011) ** emotions can spread (expand, combines empathy and acting) ** “the tendency to automatically mimic and synchronize expressions, vocalizations, postures, and movements with those of another person’s and, consequently, to converge emotionally.” (Elaine Hatfield, 1993) * Affective process theory looks at unifying the mechanisms of emotional contagion into three types. Convergent linkage, Divergent linkage, and Complementary linkage (Elfenbein, 2014). Storytelling is a way to promote position emotions (Wei et al., 2024). === The mediating effect of emotional labor === Surface acting does not mediate the effect (high surface acting leads to emotional exhaustion) whereas the inverse is true for deep acting (Wrobel, 2013) Emotional Expression and Emotional Processing (Hoyt et al., 2024) {{RoundBoxTop|theme=3}} '''What role does emotional labor play in emotional exhaustion''' * In relation to the CORT and JD-R theories, emotional labor acts as a job demand and expenditure of resources to protect existing resources {{RoundBoxBottom}} == Strategies for combating empathy fatigue and emotional fatigue == * Talk about emotion regulation (Gross, 1998) and mood regulation strategies as a way to deal with coping (Wrobel, 2013) (Author's note: need more research/references for this section) * Talk about breaks ** Rest and downtime from not performing emotional labor regenerates * How do we reduce the amount of emotional labor we're performing (Jeung et al., 2018) ** Communication/boundaries ** Reducing surface acting (allowing ourselves to feel how we actually feel and reduce emotive dissonance) '''Table 1. Treatment phases of clinical burnout (based on Table 2 from van Dam, A. 2021)''' {| class="wikitable" style="margin: auto; |- ! Phase !! Treatment goal !! Therapeutic interventions |- | '''Phase 1: Crisis''' || Recognition of the patient that the problems are serious, and that serious action is required || Psychoeducation Sick leave Dropping domestic tasks and social obligations Inform social network |- | '''Phase 2: Recovery''' || Recovery of the (physiological) stress system to normal allostatic stress levels || Registration of stress and activities Relaxtion exercises Mindfulness healthy lifestyle advices Graded exercise Gradually resume activities Psychoeducation social network and employer Gradual return to work |- | '''Phase 3: Prevention, learning from the past''' || Acquiring insight and skills to prevent relapse in clinical burnout || Analysing stressful situations Analysing dysfunctional thought patterns Learning new coping skills Learning social skills Job crafting Time management Making choices regarding career and personal life |- | '''Phase 4: Post burnout growth''' || Improving sustainable quality of life. || Setting priorities regarding quality of life and interpersonal relationships. |} === Strategies for the workplace === * Discuss types of strategies that are available in the workplace (organisation models) * SCARF model (Aplin-Houtz, 2025) ** Link back to COR and JDR models e.g. praise could be a job resource vs criticism could be a job demand *Reduce workload to reduce emotional exhaustion, and by extension reduce emotional fatigue. {{RoundBoxTop|theme=3}} '''How does understanding emotional labor and empathy fatigue work to reduce the effects of emotional exhaustion''' *Psychoeducation has been shown to reduce the effects of emotional labor and empathy fatigue on emotional exhaustion {{RoundBoxBottom}} ==Conclusion== * Summarise what empathy fatigue is, and why people become emotionally exhausted (+ the role of emotional labor) * Summarise the symptoms and how to recognise them in yourself and someone else * Emphasise and summarise psychological strategies to support someone going through emotional exhaustion * Summarise ways to prevent emotional exhaustion from happening * Overall message is: *# What is emotional exhaustion *# how do you recognise it *# how do you treat it *# how do you prevent it * Link back to the focus questions ==See also== * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Metacognition_and_emotional_regulation&oldid=2756264 Metacognition and emotional regulation] (Book chapter, 2026) {{ic|Use an internal link style as shown in Tutorial 2}} * [[wikipedia:Conservation_of_resources_theory|Conservation of resources theory]] (Wikipedia) * [[wikipedia:Job_demands-resources_model|Job demands-resources model]] (Wikipedia) ==References== {{Hanging indent|1= Aplin-Houtz, M. 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The American Psychologist, 44(3), 513–524. https://doi.org/10.1037/0003-066X.44.3.513 Hochschild, A. R. (2012). The managed heart : commercialization of human feeling (Updated, with a new preface.). University of California Press. Jeung, D.-Y., Kim, C., & Chang, S.-J. (2018). Emotional Labor and Burnout: A Review of the Literature. Yonsei Medical Journal, 59(2), 187–193. https://doi.org/10.3349/ymj.2018.59.2.187 Lavelle, J. J., Rupp, D. E., Herda, D. N., Pandey, A., & Lauck, J. R. (2021). Customer Injustice and Employee Performance: Roles of Emotional Exhaustion, Surface Acting, and Emotional Demands–Abilities Fit. Journal of Management, 47(3), Article 0149206319869426. https://doi.org/10.1177/0149206319869426 Lloyd, C., & King, R. (2004). A survey of burnout among Australian mental health occupational therapists and social workers. Social Psychiatry and Psychiatric Epidemiology, 39(9), 752-757. https://doi.org/https://doi.org/10.1007/s00127-004-0808-7 Kenworthy, J., Fay, C., Frame, M., & Petree, R. (2014). A meta-analytic review of the relationship between emotional dissonance and emotional exhaustion. Journal of Applied Social Psychology, 44(2), 94-105. https://doi.org/https://doi.org/10.1111/jasp.12211 Martingano, A. J., Stosic, M. D., & Konrath, S. (2026). Different empathy types show opposing associations with burnout: systematic review and meta-analyses. Psychology, Health & Medicine, 31(3), 798-829. https://doi.org/10.1080/13548506.2025.2591859 Michielsen, H. J., Willemsen, T. M., Croon, M. A., de Vries, J., & Van Heck, G. L. (2004). Determinants of general fatigue and emotional exhaustion: a prospective study. Psychology & health, 19(2), 223-235. https://doi.org/10.1080/08870440310001627135 Panksepp, J. B., & Lahvis, G. P. (2011). Rodent empathy and affective neuroscience. Neuroscience and Biobehavioral Reviews, 35(9), 1864–1875. https://doi.org/10.1016/j.neubiorev.2011.05.013 Pooladvand, S., & Hasanzadeh, S. (2023). Impacts of Stress on Workers’ Risk-Taking Behaviors: Cognitive Tunneling and Impaired Selective Attention. Journal of Construction Engineering and Management, 149(8), 04023060. https://doi.org/doi:10.1061/JCEMD4.COENG-13339 Riess, H. (2017). The Science of Empathy. Journal of Patient Experience, 4(2), 74–77. https://doi.org/10.1177/2374373517699267 Schaufeli, W. B. (2017). Applying the job demands-resources model: a “how to” guide to measuring and tackling work engagement and burnout. Organizational Dynamics, 46(2), 120–132. https://doi.org/10.1016/j.orgdyn.2017.04.008 Schneiderman, N., Ironson, G., & Siegel, S. D. (2005). Stress and Health: Psychological, Behavioral, and Biological Determinants. Annual Review of Clinical Psychology, 1(Volume 1, 2005), 607-628. https://doi.org/https://doi.org/10.1146/annurev.clinpsy.1.102803.144141 Seidler, A., Thinschmidt, M., Deckert, S., Then, F., Hegewald, J., Nieuwenhuijsen, K., & Riedel-Heller, S. G. (2014). The role of psychosocial working conditions on burnout and its core component emotional exhaustion – a systematic review. Journal of Occupational Medicine and Toxicology, 9(1), 10. https://doi.org/10.1186/1745-6673-9-10 Shao, M., Li, L., Li, X., Wei, Z., Wang, J., Hong, M., Liu, X., & Meng, J. (2024). The effect of top-down attention on empathy fatigue. Cerebral Cortex (New York, N.Y. 1991), 34(1), Article bhad441. https://doi.org/10.1093/cercor/bhad441 van Dam, A. (2021). A clinical perspective on burnout: diagnosis, classification, and treatment of clinical burnout. European Journal of Work and Organizational Psychology, 30(5), 732-741. https://doi.org/10.1080/1359432X.2021.1948400 Vial, A. C., & Cowgill, C. M. (2022). Heavier Lies Her Crown: Gendered Patterns of Leader Emotional Labor and Their Downstream Effects. Frontiers in Psychology, 13, 849566. https://doi.org/https://doi.org/10.3389/fpsyg.2022.849566 Wharton, A. S. (2009). The Sociology of Emotional Labor. Annual Review of Sociology, 35(1), 147–165. https://doi.org/10.1146/annurev-soc-070308-115944 Wrobel, M. (2013). Can empathy lead to emotional exhaustion in teachers? The mediating role of emotional labor. International Journal of Occupational Medicine and Environmental Health, 26(4), 581–592. https://doi.org/10.2478/s13382-013-0123-1 }} ==External links== Provide [[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: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Burnout]] [[Category:Motivation and emotion/Book/Empathy]] 0s48xtvp4xvtpo4aeznm3pxkfomz5mu 2844515 2844513 2026-11-04T22:02:09Z U3143751 3106785 /* Job demands-resources model */ Included meta-analysis 2844515 wikitext text/x-wiki {{title|Empathy fatigue and emotional exhaustion:<br>How does sustained empathic engagement contribute to emotional exhaustion?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}} [[File:Exhausted smartphone user.JPG|right|thumb|150px|'''Figure 1'''. An emotionally exhausted Aelin at the end of the work day]] ; Scenario Aelin, a 23-year-old social worker employed at headspace {{ic|Link to the Wikipedia article to help explain what this term means}}, works with approximately 20 {{ic|Explain what sort of}}Imagine you are only 23 years old and are working as a social worker at headspace. You see around 20 clients each week, many of whom have experienced rough living conditions. During your first six months in this role, you felt highly motivated and excited to help people in need. You are an excellent active listener and can easily remember details about your clients Live’s. You are also good at recognising signs of distress in your client’s body language and adjust yours in order to show warmth and understanding. As you progress in your role, you take on more cases that involve complex past traumas. The workload becomes too much for you to get done at work, but you don’t want your clients to miss out, so you take your work home to complete it in the evenings. Your workload increases again and you now start doing work on your weekends as well. After several months of working this way, you start to notice you feel exhausted every morning before you have even started work. You then start to notice you’re responding to people at work differently. Instead of your usual energised active listening response you feel frustrated, empathetic and detached. After work you are too drained to socialise with friends. This begins to worry you. What if you won’t be able to continue working as a social worker? {{RoundBoxBottom}} Aelin is experiencing '''emotional exhaustion''' from a sustained empathic response. Due to the nature of Aelin's work, they are constantly engaged in an empathic response. Empathic responses require cognitive and physical resources to sustain. Because Aelin is taking their work home and not resting on weekends, they do not have a chance to rest. It is important for Aelin to recognise what is happening and why. Not actioning the problem has led to empathy fatigue and emotional exhaustion. Emotional exhaustion is one of the key aspects of [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FEmpathy_fatigue_and_emotional_exhaustion&diff=2840411&oldid=2832859 burnout], a long-term condition of existing in a constant state of high stress. Psychological science can support Aelin in recognising and mapping the symptoms they are experiencing, understand coping strategies to minimise the current and future effects to improve their wellbeing. Understanding the symptoms will also support their role as a social worker, providing more tools for the people they support on a day-to-day basis. {{RoundBoxTop|theme=3}} '''Focus questions''' * Why do people become emotionally exhausted? * What is the relationship between empathy fatigue and emotional exhaustion? * What role does emotional labor play in emotional exhaustion? * How does understanding emotional labor and empathy fatigue work to reduce the effects of emotional exhaustion? {{RoundBoxBottom}} == Emotional exhaustion == To understand emotional exhaustion, it will be important to first understand burnout. Burnout is a long-term condition of existing in a constant state of high stress and affects millions of people annually. In the early 2000s, a survey from Australia found that more than 50% of social workers and occupational workers reported high levels of burnout symptoms (Llyod and King, 2004) and a recent report from 2026 found that more than one third of all workers in Australia were experiencing burnout (Chan, 2026). Being in a state of high stress for short periods of time is a useful adaptation for humans, so much so that it has been given a name, [[wikipedia:Eustress|Eustress]]. However, as a long-term condition it leads to individual health issues and can be linked to cardiovascular disease, increased vulnerability to upper respiratory diseases, and chronic inflammation (Schneiderman et al., 2005), and cognitive issues such as [[wikipedia:Inattentional_blindness|cognitive tunneling]] and impaired selective attention (Pooladvand and Hasanzadeh, 2023). Burnout can be defined across three dimensions: emotional exhaustion, [[wikipedia:Depersonalization|depersonalisation]], and reduced feelings of personal accomplishment (Maslach, 1993, as cited in Martingano et al., 2026). Emotional exhaustion may be considered the 'core' component of burnout and refers to the depletion of emotional resources, leading to feelings of overwhelm, overextension, and exhaustion (Martingano et al., 2026; De Beer et al., 2024; Seidler et al., 2014). As a major factor, its symptoms are similar to burnout and stress. Although antecedents of burnout are typically agreed upon (Bakker et al., 2014) and divided into individual (e.g. self-efficacy) and situational categories (e.g. work load), there are many theories that attempt to explain why emotional exhaustion occurs. Two of these theories with good evidence are the [[wikipedia:Conservation_of_resources_theory|Conservation of resources stress model]] (Hobfoll, 1989) and the [[wikipedia:Job_demands-resources_model|Job Demands-Resources model]] (Demerouti et al., 2001). === Conservation of resources stress model === The Conservation of resources stress model (CORT; Hobfall, 1989) is a theory that works to understand and explain why stress occurs. Under the model, humans are motivated to maintain, pursue, or attain resources. Stress, defined as "an individual's perceived pressure or strain" (Hollebeek et al., 2023 p.928), occurs when these resources are threatened or lost, or gained at a net loss. As noted above, resources aren't strictly tied to being physical objects. They can also be emotional states, conditions, or even ideas. For example, someone may place value on enjoying a calm, quiet Sunday morning at home. If their neighbour starts using power tools loudly next door, it would disrupt this state and induce stress in the individual. The CORT model focuses primarily on two principles. The first principle is that there is more harm in losing resources than there is in gaining them back (Halbesleben et al., 2014). This is known as loss salience, and some research suggests that the pain of loss is felt twice as strong as the pleasure of gain (Thaler, 2000). Looking at our previous example, imagine that the person who enjoys a quiet Sunday morning at home equally enjoys the sound of a loud car engine. Even though the person enjoys the sound of car engines, if the neighbour started revving their car loudly instead of using power tools, it would still disrupt their desired state of quiet and lead to an increase of stress. The second principle is that people must invest existing resources in order to gain or protect resources (Halbesleben et al., 2014). This resource investment includes recovering from resource loss. Coping is how people manage or overcome a difficult situation, such as protecting or recovering from resource loss. Coping can be broken down into different strategies that people use, but not all strategies are equal (Ito and Brotheridge, 2003). They are divided into two categories: control or escape strategies. Control strategies are considered better at reducing the strain from stress and consist of positive thinking, direct action, and help seeking (Havlovic & Keenan, 1991, as cited in Ito and Brotheridge, 2003). The most useful control strategies incorporate a person's [[locus of control]], emphasise seeking support from co-workers, and highlight the importance of autonomy in decision making. There is fairly robust evidence that supports the CORT model. === Job demands-resources model === Another model that attempts to understand why people become emotionally exhausted is the Job Demands-Resources model (JD-R; Demerouti et al., 2001). The JD-R model discusses two concepts: job demands and job resources. Demands and resources are both related to the physical, social, or psychological aspects of a job, but differ in their impact on a person. Job demands require physical or mental effort and exhaust a person's energy supply. Job resources typically support achieving goals, reduce the effort cost of job demands, and stimulate personal growth (Richter and Hacker, 1998, as cited in Demerouti et al., 2001). For example, an unsustainable workload would be considered a job demand, and receiving help and support from others would be considered a resource. The JD-R has two primary processes for how demands and resources interact with one another (Schaufeli, 2017). The stress process occurs when job demands are high and resources are low. As stress increases strain and pressure on an individual, it leads to the exhaustion of their energy eventually, eventually leading to emotional exhaustion. The second process is the motivational process. This process occurs when there is an excess of job resources. Crucially, the motivational process does not require low job demands. With enough job resources, a person will be energised and engaged, reframing job demands as a challenge to overcome. The JD-R model, and Schaufeli's assumption has been studied to confirm its validity. A meta-analysis by Lesener et al. (2019) looked at 74 longitudinal studies of the JD-R with a strong average sample size (n=450). Although most of the studies appear to come from WEIRD countries (k=60) there were a good number from Asia and Africa (k=13) and a mix of genders . to confirm these assumptions of the JD-R model. Although job resources seem like a cure-all for demands, it's important to note that they do not retroactively treat emotional exhaustion. Under the JD-R model, burnout must be reduced by lowering job demands. However, once someone has recovered, it is possible to increase job resources alongside job demands to prevent future burnout and support [[wikipedia:Work_engagement|work engagement]]. Work engagement (engagement) has been discussed as the opposite of burnout (Bakker et al., 2014). Engagement is how people express themselves when performing a role. A role being a pattern of behaviour that a person goes through based on the social expectations of that role, similar to how an actor takes on a role for a movie. For example, when attending a typical restaurant, it is socially expected that a person will engage with the waiter and provide them with their meal order to take to the kitchen. This person is performing the script of a restaurant patron. In the context of a workplace, an employee with high engagement may appear to be energised, enthusiastic, and become absorbed in their work losing track of time (Bakkar et al, 2014). <nowiki>**</nowiki>Update box below to relate back to Aelin. {{RoundBoxTop|theme=3}} '''Why do we feel emotionally exhausted''' The JD-R model and the CORT model do use different language and have slight differences, but are similar in how emotional exhaustion is considered, and share a solution to the problem. Emotional exhaustion occurs when the energy cost for completing tasks occur faster than a person can replenish it. If a person is emotionally exhausted, they will need to reduce their workload to recover and, as part of recovery, increasing autonomy (control) in decision making, and asking for help through social support, will lead to positive outcomes - reducing the likelihood of future exhaustion and increasing engagement (Bakker et al., 2014; Halbesleben et al., 2014). {{RoundBoxBottom}} == Empathy fatigue == You may be familiar with the term [[empathy]]. It is often described in terms of supporting others. For example, when someone close to you experiences a loss and is grieving, you may empathise with them and experience a negative emotion akin to grief too. However, empathy is not exclusive to so called negative emotions. For example, compersion is a positive empathic response when a loved one experiences joy. Parents may experience compersion when something joyful happens to their child. At its core, empathy is the individual ability to imagine or understand another person's feelings and/or emotions (Riess, 2017; Shao et al., 2024) and has four main components (Davis, 1983, as cited in Delgado et al., 2023): * Empathic concern is xyz * Fantasy is xyz * Perspective taking is xyz * Personal distress is xyz Much like physical or cognitive processes, such as running or problem solving, empathising will drain energy and resources. Further, and similar to the difference in energy expenditure between running and jogging, there are distinct types of empathy that each have different energy costs. === Types of empathy === Research makes a clear distinction between two types of empathy, affective and cognitive empathy. Affective empathy occurs when a person has an emotional reaction to someone else's mental state (Shao et al., 2024). This occurs because (unsure about this next part go through and update accordingly) ''unconscious processes interpret the social stimuli from another person leading to a physiological and cognitive response. These processes may interpret stimuli from any of the five senses.'' Cognitive empathy is when a person understands someone else's feelings or emotions (Cui et al., 2022). * Integrating the two - "affective component is the essence of empathy, and the cognitive component is a process through which that essence comes into being" (Bošnjaković & Radionov, 2018) ** (Author's note: change quote into paraphrasing) * === Empathy or compassion fatigue === * Distinction between empathy and compassion fatigue * Compassion fatigue is closely related to Secondary traumatic stress usually found in healthcare professionals (Figley, 2002) ** Compassion fatigue is to Empathy fatigue as emotional exhaustion is to burnout * Make it clear that Aelin is experiencing Compassion Fatigue vs Empathy fatigue (they are the same but different) (Author's note: include an update to the scenario here!) '''Empathy fatigue questions'''. Choose your answers and click "Submit": <quiz display=simple> {Surface acting is when the observer actually feels the emotion: |type="()"} - True + False {Compassion fatigue and empathy fatigue are synonymous: |type="()"} - True + False </quiz> {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' * Under the JD-R model, a sustained empathic response is a long-term job demand. * Under the CORT model, a sustained empathic response uses energy to protect existing resources. As resources become low and burnout sets in, cognitive focus narrows leading to empathy fatigue. {{RoundBoxBottom}} == Emotional labor as a mediating factor between empathy fatigue and emotional exhaustion == As a society there are expected [https://en.wikipedia.org/wiki/Social_norm social norms] in behaviour when expressing emotion, regardless if someone is conscious of the process of matching these norms or not. For example, laughing during an emotional speech at a funeral would be considered a rude or offensive expression from the other attendees. As social creatures, humans need a way to regulate socially appropriate emotional expression. This is the essence of Emotional labor: "the regulation of felt and expressed emotions (whether effortful or not) to match these emotional display rules with the objective of fulfilling organizational goals" (Vial & Cowgill, 2022) What is emotional labor (Hochschild, 2012). Theories behind emotional labor involve acting, and an extension of acting known as emotional contagion. === Types of acting === * Surface acting ** This is where emotive dissonance comes into play ** Leads to emotional exhaustion (Kenworthy et al., 2014) ** "Should" language - in this situation, this is what I ''should'' feel * Deep acting ** People who 'deep act', as in genuinely change the way they feel (match the feelings they have with the feelings that they need to display) often report positive relationships with emotional labor ** Actual language - in this situation, this is what I ''actually'' feel Discuss the differences in gender norms and expectations in relationships {| class="wikitable" style="margin: auto;" |- ! !! Deep acting !! Surface acting |- | '''Emotional dissonance''' || || X |- | '''Not known to others''' || Hidden area || Unknown |} === Emotional contagion === Emotional contagion is the idea that emotions are contagious. as the theory to moderate surface/deep acting * Emotional contagion (Panksepp and Lahvis, 2011) ** emotions can spread (expand, combines empathy and acting) ** “the tendency to automatically mimic and synchronize expressions, vocalizations, postures, and movements with those of another person’s and, consequently, to converge emotionally.” (Elaine Hatfield, 1993) * Affective process theory looks at unifying the mechanisms of emotional contagion into three types. Convergent linkage, Divergent linkage, and Complementary linkage (Elfenbein, 2014). Storytelling is a way to promote position emotions (Wei et al., 2024). === The mediating effect of emotional labor === Surface acting does not mediate the effect (high surface acting leads to emotional exhaustion) whereas the inverse is true for deep acting (Wrobel, 2013) Emotional Expression and Emotional Processing (Hoyt et al., 2024) {{RoundBoxTop|theme=3}} '''What role does emotional labor play in emotional exhaustion''' * In relation to the CORT and JD-R theories, emotional labor acts as a job demand and expenditure of resources to protect existing resources {{RoundBoxBottom}} == Strategies for combating empathy fatigue and emotional fatigue == * Talk about emotion regulation (Gross, 1998) and mood regulation strategies as a way to deal with coping (Wrobel, 2013) (Author's note: need more research/references for this section) * Talk about breaks ** Rest and downtime from not performing emotional labor regenerates * How do we reduce the amount of emotional labor we're performing (Jeung et al., 2018) ** Communication/boundaries ** Reducing surface acting (allowing ourselves to feel how we actually feel and reduce emotive dissonance) '''Table 1. Treatment phases of clinical burnout (based on Table 2 from van Dam, A. 2021)''' {| class="wikitable" style="margin: auto; |- ! Phase !! Treatment goal !! Therapeutic interventions |- | '''Phase 1: Crisis''' || Recognition of the patient that the problems are serious, and that serious action is required || Psychoeducation Sick leave Dropping domestic tasks and social obligations Inform social network |- | '''Phase 2: Recovery''' || Recovery of the (physiological) stress system to normal allostatic stress levels || Registration of stress and activities Relaxtion exercises Mindfulness healthy lifestyle advices Graded exercise Gradually resume activities Psychoeducation social network and employer Gradual return to work |- | '''Phase 3: Prevention, learning from the past''' || Acquiring insight and skills to prevent relapse in clinical burnout || Analysing stressful situations Analysing dysfunctional thought patterns Learning new coping skills Learning social skills Job crafting Time management Making choices regarding career and personal life |- | '''Phase 4: Post burnout growth''' || Improving sustainable quality of life. || Setting priorities regarding quality of life and interpersonal relationships. |} === Strategies for the workplace === * Discuss types of strategies that are available in the workplace (organisation models) * SCARF model (Aplin-Houtz, 2025) ** Link back to COR and JDR models e.g. praise could be a job resource vs criticism could be a job demand *Reduce workload to reduce emotional exhaustion, and by extension reduce emotional fatigue. {{RoundBoxTop|theme=3}} '''How does understanding emotional labor and empathy fatigue work to reduce the effects of emotional exhaustion''' *Psychoeducation has been shown to reduce the effects of emotional labor and empathy fatigue on emotional exhaustion {{RoundBoxBottom}} ==Conclusion== * Summarise what empathy fatigue is, and why people become emotionally exhausted (+ the role of emotional labor) * Summarise the symptoms and how to recognise them in yourself and someone else * Emphasise and summarise psychological strategies to support someone going through emotional exhaustion * Summarise ways to prevent emotional exhaustion from happening * Overall message is: *# What is emotional exhaustion *# how do you recognise it *# how do you treat it *# how do you prevent it * Link back to the focus questions ==See also== * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Metacognition_and_emotional_regulation&oldid=2756264 Metacognition and emotional regulation] (Book chapter, 2026) {{ic|Use an internal link style as shown in Tutorial 2}} * [[wikipedia:Conservation_of_resources_theory|Conservation of resources theory]] (Wikipedia) * [[wikipedia:Job_demands-resources_model|Job demands-resources model]] (Wikipedia) ==References== {{Hanging indent|1= Aplin-Houtz, M. (2025). Unraveling the SCARF: How dimensions of the SCARF model influence the relationship between workplace exclusion and turnover intentions. Journal of Management & Organization, 31(6), 2699–2725. https://doi.org/10.1017/jmo.2025.10026 Bošnjaković, J., & Radionov, T. (2018). Empathy: Concepts, Theories and Neuroscientific Basis. Alcoholism (Zagreb), 54(2), 123–150. https://doi.org/10.20471/de..2018.54.02.04 Chan, F. M. (2026). The State of Workplace Burnout 2026: The Workplace Divide. Infinite Potential, Sydney, Australia. https://www.ahri.com.au/wp-content/uploads/The-State-of-Workplace-Burnout-2026-Final.pdf Cui, F., Huang, X., Jing, Y., Luo, Y., Liu, J., & Gu, R. (2022). How resource sharing resists scarcity: the role of cognitive empathy and its neurobiological mechanisms. Cerebral Cortex (New York, N.Y. 1991), 32(23), 5330–5342. https://doi.org/10.1093/cercor/bhac017 De Beer, L. T., van der Vaart, L., Escaffi-Schwarz, M., De Witte, H., & Schaufeli, W. B. (2024). Maslach Burnout Inventory - General Survey: A Systematic Review and Meta-Analysis of Measurement Properties. European journal of psychological assessment : official organ of the European Association of Psychological Assessment, 40(5), 360-375. https://doi.org/10.1027/1015-5759/a000797 Demerouti, E., Bakker, A. B., Nachreiner, F., & Schaufeli, W. B. (2001). The Job Demands-Resources Model of Burnout. Journal of Applied Psychology, 86(3), 499–512. https://doi.org/10.1037/0021-9010.86.3.499 Figley, C. R. (2002). Compassion fatigue: Psychotherapists’ chronic lack of self care. Journal of Clinical Psychology, 58(11), 1433–1441. https://doi.org/10.1002/jclp.10090 Grandey, A. A. (2003). When ‘the show must go on’: surface acting and deep acting as determinants of emotional exhaustion and peer-rated service delivery. Academy of Management Journal, 46(1), 86–96. https://doi.org/10.5465/30040678 Gross, J. J. (1998). The emerging field of emotion regulation: An integrative review. 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The American Psychologist, 44(3), 513–524. https://doi.org/10.1037/0003-066X.44.3.513 Hochschild, A. R. (2012). The managed heart : commercialization of human feeling (Updated, with a new preface.). University of California Press. Jeung, D.-Y., Kim, C., & Chang, S.-J. (2018). Emotional Labor and Burnout: A Review of the Literature. Yonsei Medical Journal, 59(2), 187–193. https://doi.org/10.3349/ymj.2018.59.2.187 Lavelle, J. J., Rupp, D. E., Herda, D. N., Pandey, A., & Lauck, J. R. (2021). Customer Injustice and Employee Performance: Roles of Emotional Exhaustion, Surface Acting, and Emotional Demands–Abilities Fit. Journal of Management, 47(3), Article 0149206319869426. https://doi.org/10.1177/0149206319869426 Lloyd, C., & King, R. (2004). A survey of burnout among Australian mental health occupational therapists and social workers. Social Psychiatry and Psychiatric Epidemiology, 39(9), 752-757. https://doi.org/https://doi.org/10.1007/s00127-004-0808-7 Kenworthy, J., Fay, C., Frame, M., & Petree, R. (2014). A meta-analytic review of the relationship between emotional dissonance and emotional exhaustion. Journal of Applied Social Psychology, 44(2), 94-105. https://doi.org/https://doi.org/10.1111/jasp.12211 Martingano, A. J., Stosic, M. D., & Konrath, S. (2026). Different empathy types show opposing associations with burnout: systematic review and meta-analyses. Psychology, Health & Medicine, 31(3), 798-829. https://doi.org/10.1080/13548506.2025.2591859 Michielsen, H. J., Willemsen, T. M., Croon, M. A., de Vries, J., & Van Heck, G. L. (2004). Determinants of general fatigue and emotional exhaustion: a prospective study. Psychology & health, 19(2), 223-235. https://doi.org/10.1080/08870440310001627135 Panksepp, J. B., & Lahvis, G. P. (2011). Rodent empathy and affective neuroscience. Neuroscience and Biobehavioral Reviews, 35(9), 1864–1875. https://doi.org/10.1016/j.neubiorev.2011.05.013 Pooladvand, S., & Hasanzadeh, S. (2023). Impacts of Stress on Workers’ Risk-Taking Behaviors: Cognitive Tunneling and Impaired Selective Attention. Journal of Construction Engineering and Management, 149(8), 04023060. https://doi.org/doi:10.1061/JCEMD4.COENG-13339 Riess, H. (2017). The Science of Empathy. Journal of Patient Experience, 4(2), 74–77. https://doi.org/10.1177/2374373517699267 Schaufeli, W. B. (2017). Applying the job demands-resources model: a “how to” guide to measuring and tackling work engagement and burnout. Organizational Dynamics, 46(2), 120–132. https://doi.org/10.1016/j.orgdyn.2017.04.008 Schneiderman, N., Ironson, G., & Siegel, S. D. (2005). Stress and Health: Psychological, Behavioral, and Biological Determinants. Annual Review of Clinical Psychology, 1(Volume 1, 2005), 607-628. https://doi.org/https://doi.org/10.1146/annurev.clinpsy.1.102803.144141 Seidler, A., Thinschmidt, M., Deckert, S., Then, F., Hegewald, J., Nieuwenhuijsen, K., & Riedel-Heller, S. G. (2014). The role of psychosocial working conditions on burnout and its core component emotional exhaustion – a systematic review. Journal of Occupational Medicine and Toxicology, 9(1), 10. https://doi.org/10.1186/1745-6673-9-10 Shao, M., Li, L., Li, X., Wei, Z., Wang, J., Hong, M., Liu, X., & Meng, J. (2024). The effect of top-down attention on empathy fatigue. Cerebral Cortex (New York, N.Y. 1991), 34(1), Article bhad441. https://doi.org/10.1093/cercor/bhad441 van Dam, A. (2021). A clinical perspective on burnout: diagnosis, classification, and treatment of clinical burnout. European Journal of Work and Organizational Psychology, 30(5), 732-741. https://doi.org/10.1080/1359432X.2021.1948400 Vial, A. C., & Cowgill, C. M. (2022). Heavier Lies Her Crown: Gendered Patterns of Leader Emotional Labor and Their Downstream Effects. Frontiers in Psychology, 13, 849566. https://doi.org/https://doi.org/10.3389/fpsyg.2022.849566 Wharton, A. S. (2009). The Sociology of Emotional Labor. Annual Review of Sociology, 35(1), 147–165. https://doi.org/10.1146/annurev-soc-070308-115944 Wrobel, M. (2013). Can empathy lead to emotional exhaustion in teachers? The mediating role of emotional labor. International Journal of Occupational Medicine and Environmental Health, 26(4), 581–592. https://doi.org/10.2478/s13382-013-0123-1 }} ==External links== Provide [[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: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Burnout]] [[Category:Motivation and emotion/Book/Empathy]] l5wr7ojje4pc1j09k0qiwj3q4fsu6m0 2844518 2844515 2026-11-04T22:07:51Z U3143751 3106785 /* Job demands-resources model */ updated meta-analysis 2844518 wikitext text/x-wiki {{title|Empathy fatigue and emotional exhaustion:<br>How does sustained empathic engagement contribute to emotional exhaustion?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}} [[File:Exhausted smartphone user.JPG|right|thumb|150px|'''Figure 1'''. An emotionally exhausted Aelin at the end of the work day]] ; Scenario Aelin, a 23-year-old social worker employed at headspace {{ic|Link to the Wikipedia article to help explain what this term means}}, works with approximately 20 {{ic|Explain what sort of}}Imagine you are only 23 years old and are working as a social worker at headspace. You see around 20 clients each week, many of whom have experienced rough living conditions. During your first six months in this role, you felt highly motivated and excited to help people in need. You are an excellent active listener and can easily remember details about your clients Live’s. You are also good at recognising signs of distress in your client’s body language and adjust yours in order to show warmth and understanding. As you progress in your role, you take on more cases that involve complex past traumas. The workload becomes too much for you to get done at work, but you don’t want your clients to miss out, so you take your work home to complete it in the evenings. Your workload increases again and you now start doing work on your weekends as well. After several months of working this way, you start to notice you feel exhausted every morning before you have even started work. You then start to notice you’re responding to people at work differently. Instead of your usual energised active listening response you feel frustrated, empathetic and detached. After work you are too drained to socialise with friends. This begins to worry you. What if you won’t be able to continue working as a social worker? {{RoundBoxBottom}} Aelin is experiencing '''emotional exhaustion''' from a sustained empathic response. Due to the nature of Aelin's work, they are constantly engaged in an empathic response. Empathic responses require cognitive and physical resources to sustain. Because Aelin is taking their work home and not resting on weekends, they do not have a chance to rest. It is important for Aelin to recognise what is happening and why. Not actioning the problem has led to empathy fatigue and emotional exhaustion. Emotional exhaustion is one of the key aspects of [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FEmpathy_fatigue_and_emotional_exhaustion&diff=2840411&oldid=2832859 burnout], a long-term condition of existing in a constant state of high stress. Psychological science can support Aelin in recognising and mapping the symptoms they are experiencing, understand coping strategies to minimise the current and future effects to improve their wellbeing. Understanding the symptoms will also support their role as a social worker, providing more tools for the people they support on a day-to-day basis. {{RoundBoxTop|theme=3}} '''Focus questions''' * Why do people become emotionally exhausted? * What is the relationship between empathy fatigue and emotional exhaustion? * What role does emotional labor play in emotional exhaustion? * How does understanding emotional labor and empathy fatigue work to reduce the effects of emotional exhaustion? {{RoundBoxBottom}} == Emotional exhaustion == To understand emotional exhaustion, it will be important to first understand burnout. Burnout is a long-term condition of existing in a constant state of high stress and affects millions of people annually. In the early 2000s, a survey from Australia found that more than 50% of social workers and occupational workers reported high levels of burnout symptoms (Llyod and King, 2004) and a recent report from 2026 found that more than one third of all workers in Australia were experiencing burnout (Chan, 2026). Being in a state of high stress for short periods of time is a useful adaptation for humans, so much so that it has been given a name, [[wikipedia:Eustress|Eustress]]. However, as a long-term condition it leads to individual health issues and can be linked to cardiovascular disease, increased vulnerability to upper respiratory diseases, and chronic inflammation (Schneiderman et al., 2005), and cognitive issues such as [[wikipedia:Inattentional_blindness|cognitive tunneling]] and impaired selective attention (Pooladvand and Hasanzadeh, 2023). Burnout can be defined across three dimensions: emotional exhaustion, [[wikipedia:Depersonalization|depersonalisation]], and reduced feelings of personal accomplishment (Maslach, 1993, as cited in Martingano et al., 2026). Emotional exhaustion may be considered the 'core' component of burnout and refers to the depletion of emotional resources, leading to feelings of overwhelm, overextension, and exhaustion (Martingano et al., 2026; De Beer et al., 2024; Seidler et al., 2014). As a major factor, its symptoms are similar to burnout and stress. Although antecedents of burnout are typically agreed upon (Bakker et al., 2014) and divided into individual (e.g. self-efficacy) and situational categories (e.g. work load), there are many theories that attempt to explain why emotional exhaustion occurs. Two of these theories with good evidence are the [[wikipedia:Conservation_of_resources_theory|Conservation of resources stress model]] (Hobfoll, 1989) and the [[wikipedia:Job_demands-resources_model|Job Demands-Resources model]] (Demerouti et al., 2001). === Conservation of resources stress model === The Conservation of resources stress model (CORT; Hobfall, 1989) is a theory that works to understand and explain why stress occurs. Under the model, humans are motivated to maintain, pursue, or attain resources. Stress, defined as "an individual's perceived pressure or strain" (Hollebeek et al., 2023 p.928), occurs when these resources are threatened or lost, or gained at a net loss. As noted above, resources aren't strictly tied to being physical objects. They can also be emotional states, conditions, or even ideas. For example, someone may place value on enjoying a calm, quiet Sunday morning at home. If their neighbour starts using power tools loudly next door, it would disrupt this state and induce stress in the individual. The CORT model focuses primarily on two principles. The first principle is that there is more harm in losing resources than there is in gaining them back (Halbesleben et al., 2014). This is known as loss salience, and some research suggests that the pain of loss is felt twice as strong as the pleasure of gain (Thaler, 2000). Looking at our previous example, imagine that the person who enjoys a quiet Sunday morning at home equally enjoys the sound of a loud car engine. Even though the person enjoys the sound of car engines, if the neighbour started revving their car loudly instead of using power tools, it would still disrupt their desired state of quiet and lead to an increase of stress. The second principle is that people must invest existing resources in order to gain or protect resources (Halbesleben et al., 2014). This resource investment includes recovering from resource loss. Coping is how people manage or overcome a difficult situation, such as protecting or recovering from resource loss. Coping can be broken down into different strategies that people use, but not all strategies are equal (Ito and Brotheridge, 2003). They are divided into two categories: control or escape strategies. Control strategies are considered better at reducing the strain from stress and consist of positive thinking, direct action, and help seeking (Havlovic & Keenan, 1991, as cited in Ito and Brotheridge, 2003). The most useful control strategies incorporate a person's [[locus of control]], emphasise seeking support from co-workers, and highlight the importance of autonomy in decision making. There is fairly robust evidence that supports the CORT model. === Job demands-resources model === Another model that attempts to understand why people become emotionally exhausted is the Job Demands-Resources model (JD-R; Demerouti et al., 2001). The JD-R model discusses two concepts: job demands and job resources. Demands and resources are both related to the physical, social, or psychological aspects of a job, but differ in their impact on a person. Job demands require physical or mental effort and exhaust a person's energy supply. Job resources typically support achieving goals, reduce the effort cost of job demands, and stimulate personal growth (Richter and Hacker, 1998, as cited in Demerouti et al., 2001). For example, an unsustainable workload would be considered a job demand, and receiving help and support from others would be considered a resource. The JD-R has two primary processes for how demands and resources interact with one another (Schaufeli, 2017). The stress process occurs when job demands are high and resources are low. As stress increases strain and pressure on an individual, it leads to the exhaustion of their energy eventually, eventually leading to emotional exhaustion. The second process is the motivational process. This process occurs when there is an excess of job resources. Crucially, the motivational process does not require low job demands. With enough job resources, a person will be energised and engaged, reframing job demands as a challenge to overcome. The JD-R model has had extensive research to confirm its validity. A meta-analysis by Lesener et al. (2019) looked at 74 longitudinal studies of the JD-R with a strong average sample size (n=450) against the assumptions as described by Schaufeli (2017). Although most of the studies appear to come from WEIRD countries (k=60) there were a good number from Asia and Africa (k=13) and a mix of genders. The study confirmed that across the literature, job resources predict burnout negatively (''β'' = −.17; ''p'' < .001) and job demands predict burnout over time (''β'' = .10; ''p'' < .001). Although job resources seem like a cure-all for demands, it's important to note that they do not retroactively treat emotional exhaustion. Under the JD-R model, burnout must be reduced by lowering job demands. However, once someone has recovered, it is possible to increase job resources alongside job demands to prevent future burnout and support [[wikipedia:Work_engagement|work engagement]]. Work engagement (engagement) has been discussed as the opposite of burnout (Bakker et al., 2014). Engagement is how people express themselves when performing a role. A role being a pattern of behaviour that a person goes through based on the social expectations of that role, similar to how an actor takes on a role for a movie. For example, when attending a typical restaurant, it is socially expected that a person will engage with the waiter and provide them with their meal order to take to the kitchen. This person is performing the script of a restaurant patron. In the context of a workplace, an employee with high engagement may appear to be energised, enthusiastic, and become absorbed in their work losing track of time (Bakkar et al, 2014). <nowiki>**</nowiki>Update box below to relate back to Aelin. {{RoundBoxTop|theme=3}} '''Why do we feel emotionally exhausted''' The JD-R model and the CORT model do use different language and have slight differences, but are similar in how emotional exhaustion is considered, and share a solution to the problem. Emotional exhaustion occurs when the energy cost for completing tasks occur faster than a person can replenish it. If a person is emotionally exhausted, they will need to reduce their workload to recover and, as part of recovery, increasing autonomy (control) in decision making, and asking for help through social support, will lead to positive outcomes - reducing the likelihood of future exhaustion and increasing engagement (Bakker et al., 2014; Halbesleben et al., 2014). {{RoundBoxBottom}} == Empathy fatigue == You may be familiar with the term [[empathy]]. It is often described in terms of supporting others. For example, when someone close to you experiences a loss and is grieving, you may empathise with them and experience a negative emotion akin to grief too. However, empathy is not exclusive to so called negative emotions. For example, compersion is a positive empathic response when a loved one experiences joy. Parents may experience compersion when something joyful happens to their child. At its core, empathy is the individual ability to imagine or understand another person's feelings and/or emotions (Riess, 2017; Shao et al., 2024) and has four main components (Davis, 1983, as cited in Delgado et al., 2023): * Empathic concern is xyz * Fantasy is xyz * Perspective taking is xyz * Personal distress is xyz Much like physical or cognitive processes, such as running or problem solving, empathising will drain energy and resources. Further, and similar to the difference in energy expenditure between running and jogging, there are distinct types of empathy that each have different energy costs. === Types of empathy === Research makes a clear distinction between two types of empathy, affective and cognitive empathy. Affective empathy occurs when a person has an emotional reaction to someone else's mental state (Shao et al., 2024). This occurs because (unsure about this next part go through and update accordingly) ''unconscious processes interpret the social stimuli from another person leading to a physiological and cognitive response. These processes may interpret stimuli from any of the five senses.'' Cognitive empathy is when a person understands someone else's feelings or emotions (Cui et al., 2022). * Integrating the two - "affective component is the essence of empathy, and the cognitive component is a process through which that essence comes into being" (Bošnjaković & Radionov, 2018) ** (Author's note: change quote into paraphrasing) * === Empathy or compassion fatigue === * Distinction between empathy and compassion fatigue * Compassion fatigue is closely related to Secondary traumatic stress usually found in healthcare professionals (Figley, 2002) ** Compassion fatigue is to Empathy fatigue as emotional exhaustion is to burnout * Make it clear that Aelin is experiencing Compassion Fatigue vs Empathy fatigue (they are the same but different) (Author's note: include an update to the scenario here!) '''Empathy fatigue questions'''. Choose your answers and click "Submit": <quiz display=simple> {Surface acting is when the observer actually feels the emotion: |type="()"} - True + False {Compassion fatigue and empathy fatigue are synonymous: |type="()"} - True + False </quiz> {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' * Under the JD-R model, a sustained empathic response is a long-term job demand. * Under the CORT model, a sustained empathic response uses energy to protect existing resources. As resources become low and burnout sets in, cognitive focus narrows leading to empathy fatigue. {{RoundBoxBottom}} == Emotional labor as a mediating factor between empathy fatigue and emotional exhaustion == As a society there are expected [https://en.wikipedia.org/wiki/Social_norm social norms] in behaviour when expressing emotion, regardless if someone is conscious of the process of matching these norms or not. For example, laughing during an emotional speech at a funeral would be considered a rude or offensive expression from the other attendees. As social creatures, humans need a way to regulate socially appropriate emotional expression. This is the essence of Emotional labor: "the regulation of felt and expressed emotions (whether effortful or not) to match these emotional display rules with the objective of fulfilling organizational goals" (Vial & Cowgill, 2022) What is emotional labor (Hochschild, 2012). Theories behind emotional labor involve acting, and an extension of acting known as emotional contagion. === Types of acting === * Surface acting ** This is where emotive dissonance comes into play ** Leads to emotional exhaustion (Kenworthy et al., 2014) ** "Should" language - in this situation, this is what I ''should'' feel * Deep acting ** People who 'deep act', as in genuinely change the way they feel (match the feelings they have with the feelings that they need to display) often report positive relationships with emotional labor ** Actual language - in this situation, this is what I ''actually'' feel Discuss the differences in gender norms and expectations in relationships {| class="wikitable" style="margin: auto;" |- ! !! Deep acting !! Surface acting |- | '''Emotional dissonance''' || || X |- | '''Not known to others''' || Hidden area || Unknown |} === Emotional contagion === Emotional contagion is the idea that emotions are contagious. as the theory to moderate surface/deep acting * Emotional contagion (Panksepp and Lahvis, 2011) ** emotions can spread (expand, combines empathy and acting) ** “the tendency to automatically mimic and synchronize expressions, vocalizations, postures, and movements with those of another person’s and, consequently, to converge emotionally.” (Elaine Hatfield, 1993) * Affective process theory looks at unifying the mechanisms of emotional contagion into three types. Convergent linkage, Divergent linkage, and Complementary linkage (Elfenbein, 2014). Storytelling is a way to promote position emotions (Wei et al., 2024). === The mediating effect of emotional labor === Surface acting does not mediate the effect (high surface acting leads to emotional exhaustion) whereas the inverse is true for deep acting (Wrobel, 2013) Emotional Expression and Emotional Processing (Hoyt et al., 2024) {{RoundBoxTop|theme=3}} '''What role does emotional labor play in emotional exhaustion''' * In relation to the CORT and JD-R theories, emotional labor acts as a job demand and expenditure of resources to protect existing resources {{RoundBoxBottom}} == Strategies for combating empathy fatigue and emotional fatigue == * Talk about emotion regulation (Gross, 1998) and mood regulation strategies as a way to deal with coping (Wrobel, 2013) (Author's note: need more research/references for this section) * Talk about breaks ** Rest and downtime from not performing emotional labor regenerates * How do we reduce the amount of emotional labor we're performing (Jeung et al., 2018) ** Communication/boundaries ** Reducing surface acting (allowing ourselves to feel how we actually feel and reduce emotive dissonance) '''Table 1. Treatment phases of clinical burnout (based on Table 2 from van Dam, A. 2021)''' {| class="wikitable" style="margin: auto; |- ! Phase !! Treatment goal !! Therapeutic interventions |- | '''Phase 1: Crisis''' || Recognition of the patient that the problems are serious, and that serious action is required || Psychoeducation Sick leave Dropping domestic tasks and social obligations Inform social network |- | '''Phase 2: Recovery''' || Recovery of the (physiological) stress system to normal allostatic stress levels || Registration of stress and activities Relaxtion exercises Mindfulness healthy lifestyle advices Graded exercise Gradually resume activities Psychoeducation social network and employer Gradual return to work |- | '''Phase 3: Prevention, learning from the past''' || Acquiring insight and skills to prevent relapse in clinical burnout || Analysing stressful situations Analysing dysfunctional thought patterns Learning new coping skills Learning social skills Job crafting Time management Making choices regarding career and personal life |- | '''Phase 4: Post burnout growth''' || Improving sustainable quality of life. || Setting priorities regarding quality of life and interpersonal relationships. |} === Strategies for the workplace === * Discuss types of strategies that are available in the workplace (organisation models) * SCARF model (Aplin-Houtz, 2025) ** Link back to COR and JDR models e.g. praise could be a job resource vs criticism could be a job demand *Reduce workload to reduce emotional exhaustion, and by extension reduce emotional fatigue. {{RoundBoxTop|theme=3}} '''How does understanding emotional labor and empathy fatigue work to reduce the effects of emotional exhaustion''' *Psychoeducation has been shown to reduce the effects of emotional labor and empathy fatigue on emotional exhaustion {{RoundBoxBottom}} ==Conclusion== * Summarise what empathy fatigue is, and why people become emotionally exhausted (+ the role of emotional labor) * Summarise the symptoms and how to recognise them in yourself and someone else * Emphasise and summarise psychological strategies to support someone going through emotional exhaustion * Summarise ways to prevent emotional exhaustion from happening * Overall message is: *# What is emotional exhaustion *# how do you recognise it *# how do you treat it *# how do you prevent it * Link back to the focus questions ==See also== * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Metacognition_and_emotional_regulation&oldid=2756264 Metacognition and emotional regulation] (Book chapter, 2026) {{ic|Use an internal link style as shown in Tutorial 2}} * [[wikipedia:Conservation_of_resources_theory|Conservation of resources theory]] (Wikipedia) * [[wikipedia:Job_demands-resources_model|Job demands-resources model]] (Wikipedia) ==References== {{Hanging indent|1= Aplin-Houtz, M. 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Neuroscience and Biobehavioral Reviews, 35(9), 1864–1875. https://doi.org/10.1016/j.neubiorev.2011.05.013 Pooladvand, S., & Hasanzadeh, S. (2023). Impacts of Stress on Workers’ Risk-Taking Behaviors: Cognitive Tunneling and Impaired Selective Attention. Journal of Construction Engineering and Management, 149(8), 04023060. https://doi.org/doi:10.1061/JCEMD4.COENG-13339 Riess, H. (2017). The Science of Empathy. Journal of Patient Experience, 4(2), 74–77. https://doi.org/10.1177/2374373517699267 Schaufeli, W. B. (2017). Applying the job demands-resources model: a “how to” guide to measuring and tackling work engagement and burnout. Organizational Dynamics, 46(2), 120–132. https://doi.org/10.1016/j.orgdyn.2017.04.008 Schneiderman, N., Ironson, G., & Siegel, S. D. (2005). Stress and Health: Psychological, Behavioral, and Biological Determinants. Annual Review of Clinical Psychology, 1(Volume 1, 2005), 607-628. https://doi.org/https://doi.org/10.1146/annurev.clinpsy.1.102803.144141 Seidler, A., Thinschmidt, M., Deckert, S., Then, F., Hegewald, J., Nieuwenhuijsen, K., & Riedel-Heller, S. G. (2014). The role of psychosocial working conditions on burnout and its core component emotional exhaustion – a systematic review. Journal of Occupational Medicine and Toxicology, 9(1), 10. https://doi.org/10.1186/1745-6673-9-10 Shao, M., Li, L., Li, X., Wei, Z., Wang, J., Hong, M., Liu, X., & Meng, J. (2024). The effect of top-down attention on empathy fatigue. Cerebral Cortex (New York, N.Y. 1991), 34(1), Article bhad441. https://doi.org/10.1093/cercor/bhad441 van Dam, A. (2021). A clinical perspective on burnout: diagnosis, classification, and treatment of clinical burnout. European Journal of Work and Organizational Psychology, 30(5), 732-741. https://doi.org/10.1080/1359432X.2021.1948400 Vial, A. C., & Cowgill, C. M. (2022). Heavier Lies Her Crown: Gendered Patterns of Leader Emotional Labor and Their Downstream Effects. Frontiers in Psychology, 13, 849566. https://doi.org/https://doi.org/10.3389/fpsyg.2022.849566 Wharton, A. S. (2009). The Sociology of Emotional Labor. Annual Review of Sociology, 35(1), 147–165. https://doi.org/10.1146/annurev-soc-070308-115944 Wrobel, M. (2013). Can empathy lead to emotional exhaustion in teachers? The mediating role of emotional labor. International Journal of Occupational Medicine and Environmental Health, 26(4), 581–592. https://doi.org/10.2478/s13382-013-0123-1 }} ==External links== Provide [[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: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Burnout]] [[Category:Motivation and emotion/Book/Empathy]] i4j06krhc484litdjebawjtwiv9418n 2844519 2844518 2026-11-04T22:22:09Z U3143751 3106785 /* Job demands-resources model */ 2844519 wikitext text/x-wiki {{title|Empathy fatigue and emotional exhaustion:<br>How does sustained empathic engagement contribute to emotional exhaustion?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}} [[File:Exhausted smartphone user.JPG|right|thumb|150px|'''Figure 1'''. An emotionally exhausted Aelin at the end of the work day]] ; Scenario Aelin, a 23-year-old social worker employed at headspace {{ic|Link to the Wikipedia article to help explain what this term means}}, works with approximately 20 {{ic|Explain what sort of}}Imagine you are only 23 years old and are working as a social worker at headspace. You see around 20 clients each week, many of whom have experienced rough living conditions. During your first six months in this role, you felt highly motivated and excited to help people in need. You are an excellent active listener and can easily remember details about your clients Live’s. You are also good at recognising signs of distress in your client’s body language and adjust yours in order to show warmth and understanding. As you progress in your role, you take on more cases that involve complex past traumas. The workload becomes too much for you to get done at work, but you don’t want your clients to miss out, so you take your work home to complete it in the evenings. Your workload increases again and you now start doing work on your weekends as well. After several months of working this way, you start to notice you feel exhausted every morning before you have even started work. You then start to notice you’re responding to people at work differently. Instead of your usual energised active listening response you feel frustrated, empathetic and detached. After work you are too drained to socialise with friends. This begins to worry you. What if you won’t be able to continue working as a social worker? {{RoundBoxBottom}} Aelin is experiencing '''emotional exhaustion''' from a sustained empathic response. Due to the nature of Aelin's work, they are constantly engaged in an empathic response. Empathic responses require cognitive and physical resources to sustain. Because Aelin is taking their work home and not resting on weekends, they do not have a chance to rest. It is important for Aelin to recognise what is happening and why. Not actioning the problem has led to empathy fatigue and emotional exhaustion. Emotional exhaustion is one of the key aspects of [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FEmpathy_fatigue_and_emotional_exhaustion&diff=2840411&oldid=2832859 burnout], a long-term condition of existing in a constant state of high stress. Psychological science can support Aelin in recognising and mapping the symptoms they are experiencing, understand coping strategies to minimise the current and future effects to improve their wellbeing. Understanding the symptoms will also support their role as a social worker, providing more tools for the people they support on a day-to-day basis. {{RoundBoxTop|theme=3}} '''Focus questions''' * Why do people become emotionally exhausted? * What is the relationship between empathy fatigue and emotional exhaustion? * What role does emotional labor play in emotional exhaustion? * How does understanding emotional labor and empathy fatigue work to reduce the effects of emotional exhaustion? {{RoundBoxBottom}} == Emotional exhaustion == To understand emotional exhaustion, it will be important to first understand burnout. Burnout is a long-term condition of existing in a constant state of high stress and affects millions of people annually. In the early 2000s, a survey from Australia found that more than 50% of social workers and occupational workers reported high levels of burnout symptoms (Llyod and King, 2004) and a recent report from 2026 found that more than one third of all workers in Australia were experiencing burnout (Chan, 2026). Being in a state of high stress for short periods of time is a useful adaptation for humans, so much so that it has been given a name, [[wikipedia:Eustress|Eustress]]. However, as a long-term condition it leads to individual health issues and can be linked to cardiovascular disease, increased vulnerability to upper respiratory diseases, and chronic inflammation (Schneiderman et al., 2005), and cognitive issues such as [[wikipedia:Inattentional_blindness|cognitive tunneling]] and impaired selective attention (Pooladvand and Hasanzadeh, 2023). Burnout can be defined across three dimensions: emotional exhaustion, [[wikipedia:Depersonalization|depersonalisation]], and reduced feelings of personal accomplishment (Maslach, 1993, as cited in Martingano et al., 2026). Emotional exhaustion may be considered the 'core' component of burnout and refers to the depletion of emotional resources, leading to feelings of overwhelm, overextension, and exhaustion (Martingano et al., 2026; De Beer et al., 2024; Seidler et al., 2014). As a major factor, its symptoms are similar to burnout and stress. Although antecedents of burnout are typically agreed upon (Bakker et al., 2014) and divided into individual (e.g. self-efficacy) and situational categories (e.g. work load), there are many theories that attempt to explain why emotional exhaustion occurs. Two of these theories with good evidence are the [[wikipedia:Conservation_of_resources_theory|Conservation of resources stress model]] (Hobfoll, 1989) and the [[wikipedia:Job_demands-resources_model|Job Demands-Resources model]] (Demerouti et al., 2001). === Conservation of resources stress model === The Conservation of resources stress model (CORT; Hobfall, 1989) is a theory that works to understand and explain why stress occurs. Under the model, humans are motivated to maintain, pursue, or attain resources. Stress, defined as "an individual's perceived pressure or strain" (Hollebeek et al., 2023 p.928), occurs when these resources are threatened or lost, or gained at a net loss. As noted above, resources aren't strictly tied to being physical objects. They can also be emotional states, conditions, or even ideas. For example, someone may place value on enjoying a calm, quiet Sunday morning at home. If their neighbour starts using power tools loudly next door, it would disrupt this state and induce stress in the individual. The CORT model focuses primarily on two principles. The first principle is that there is more harm in losing resources than there is in gaining them back (Halbesleben et al., 2014). This is known as loss salience, and some research suggests that the pain of loss is felt twice as strong as the pleasure of gain (Thaler, 2000). Looking at our previous example, imagine that the person who enjoys a quiet Sunday morning at home equally enjoys the sound of a loud car engine. Even though the person enjoys the sound of car engines, if the neighbour started revving their car loudly instead of using power tools, it would still disrupt their desired state of quiet and lead to an increase of stress. The second principle is that people must invest existing resources in order to gain or protect resources (Halbesleben et al., 2014). This resource investment includes recovering from resource loss. Coping is how people manage or overcome a difficult situation, such as protecting or recovering from resource loss. Coping can be broken down into different strategies that people use, but not all strategies are equal (Ito and Brotheridge, 2003). They are divided into two categories: control or escape strategies. Control strategies are considered better at reducing the strain from stress and consist of positive thinking, direct action, and help seeking (Havlovic & Keenan, 1991, as cited in Ito and Brotheridge, 2003). The most useful control strategies incorporate a person's [[locus of control]], emphasise seeking support from co-workers, and highlight the importance of autonomy in decision making. There is fairly robust evidence that supports the CORT model. === Job demands-resources model === Another model that attempts to understand why people become emotionally exhausted is the Job Demands-Resources model (JD-R; Demerouti et al., 2001). The JD-R model discusses two concepts: job demands and job resources. Demands and resources are both related to the physical, social, or psychological aspects of a job, but differ in their impact on a person. Job demands require physical or mental effort and exhaust a person's energy supply. Job resources typically support achieving goals, reduce the effort cost of job demands, and stimulate personal growth (Richter and Hacker, 1998, as cited in Demerouti et al., 2001). For example, an unsustainable workload would be considered a job demand, and receiving help and support from others would be considered a resource. The JD-R has two primary processes for how demands and resources interact with one another (Schaufeli, 2017). The stress process occurs when job demands are high and resources are low. As stress increases strain and pressure on an individual, it leads to the exhaustion of their energy eventually, eventually leading to emotional exhaustion. The second process is the motivational process. This process occurs when there is an excess of job resources. Crucially, the motivational process does not require low job demands. With enough job resources, a person will be energised and engaged, reframing job demands as a challenge to overcome. The JD-R model has had extensive research to confirm its validity. A meta-analysis by Lesener et al. (2019) looked at 74 longitudinal studies of the JD-R with a strong average sample size (n=450) against the assumptions as described by Schaufeli (2017). Although most of the studies appear to come from WEIRD countries (k=60) there were a good number from Asia and Africa (k=13) and a mix of genders. The study confirmed that across the literature, job resources predict burnout negatively (''β'' = −.17; ''p'' < .001) and job demands predict burnout over time (''β'' = .10; ''p'' < .001). Although job resources seem like a cure-all for demands, it's important to note that they do not retroactively treat emotional exhaustion. Under the JD-R model, burnout must be reduced by lowering job demands (Gynning et al., 2025). However, once someone has recovered, it is possible to increase job resources alongside job demands to prevent future burnout and support [[wikipedia:Work_engagement|work engagement]]. Work engagement (engagement) has been discussed as the opposite of burnout (Bakker et al., 2014). Engagement is how people express themselves when performing a role. A role being a pattern of behaviour that a person goes through based on the social expectations of that role, similar to how an actor takes on a role for a movie. For example, when attending a typical restaurant, it is socially expected that a person will engage with the waiter and provide them with their meal order to take to the kitchen. This person is performing the script of a restaurant patron. In the context of a workplace, an employee with high engagement may appear to be energised, enthusiastic, and become absorbed in their work losing track of time (Bakkar et al, 2014). <nowiki>**</nowiki>Update box below to relate back to Aelin. {{RoundBoxTop|theme=3}} '''Why do we feel emotionally exhausted''' The JD-R model and the CORT model do use different language and have slight differences, but are similar in how emotional exhaustion is considered, and share a solution to the problem. Emotional exhaustion occurs when the energy cost for completing tasks occur faster than a person can replenish it. If a person is emotionally exhausted, they will need to reduce their workload to recover and, as part of recovery, increasing autonomy (control) in decision making, and asking for help through social support, will lead to positive outcomes - reducing the likelihood of future exhaustion and increasing engagement (Bakker et al., 2014; Halbesleben et al., 2014). {{RoundBoxBottom}} == Empathy fatigue == You may be familiar with the term [[empathy]]. It is often described in terms of supporting others. For example, when someone close to you experiences a loss and is grieving, you may empathise with them and experience a negative emotion akin to grief too. However, empathy is not exclusive to so called negative emotions. For example, compersion is a positive empathic response when a loved one experiences joy. Parents may experience compersion when something joyful happens to their child. At its core, empathy is the individual ability to imagine or understand another person's feelings and/or emotions (Riess, 2017; Shao et al., 2024) and has four main components (Davis, 1983, as cited in Delgado et al., 2023): * Empathic concern is xyz * Fantasy is xyz * Perspective taking is xyz * Personal distress is xyz Much like physical or cognitive processes, such as running or problem solving, empathising will drain energy and resources. Further, and similar to the difference in energy expenditure between running and jogging, there are distinct types of empathy that each have different energy costs. === Types of empathy === Research makes a clear distinction between two types of empathy, affective and cognitive empathy. Affective empathy occurs when a person has an emotional reaction to someone else's mental state (Shao et al., 2024). This occurs because (unsure about this next part go through and update accordingly) ''unconscious processes interpret the social stimuli from another person leading to a physiological and cognitive response. These processes may interpret stimuli from any of the five senses.'' Cognitive empathy is when a person understands someone else's feelings or emotions (Cui et al., 2022). * Integrating the two - "affective component is the essence of empathy, and the cognitive component is a process through which that essence comes into being" (Bošnjaković & Radionov, 2018) ** (Author's note: change quote into paraphrasing) * === Empathy or compassion fatigue === * Distinction between empathy and compassion fatigue * Compassion fatigue is closely related to Secondary traumatic stress usually found in healthcare professionals (Figley, 2002) ** Compassion fatigue is to Empathy fatigue as emotional exhaustion is to burnout * Make it clear that Aelin is experiencing Compassion Fatigue vs Empathy fatigue (they are the same but different) (Author's note: include an update to the scenario here!) '''Empathy fatigue questions'''. Choose your answers and click "Submit": <quiz display=simple> {Surface acting is when the observer actually feels the emotion: |type="()"} - True + False {Compassion fatigue and empathy fatigue are synonymous: |type="()"} - True + False </quiz> {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' * Under the JD-R model, a sustained empathic response is a long-term job demand. * Under the CORT model, a sustained empathic response uses energy to protect existing resources. As resources become low and burnout sets in, cognitive focus narrows leading to empathy fatigue. {{RoundBoxBottom}} == Emotional labor as a mediating factor between empathy fatigue and emotional exhaustion == As a society there are expected [https://en.wikipedia.org/wiki/Social_norm social norms] in behaviour when expressing emotion, regardless if someone is conscious of the process of matching these norms or not. For example, laughing during an emotional speech at a funeral would be considered a rude or offensive expression from the other attendees. As social creatures, humans need a way to regulate socially appropriate emotional expression. This is the essence of Emotional labor: "the regulation of felt and expressed emotions (whether effortful or not) to match these emotional display rules with the objective of fulfilling organizational goals" (Vial & Cowgill, 2022) What is emotional labor (Hochschild, 2012). Theories behind emotional labor involve acting, and an extension of acting known as emotional contagion. === Types of acting === * Surface acting ** This is where emotive dissonance comes into play ** Leads to emotional exhaustion (Kenworthy et al., 2014) ** "Should" language - in this situation, this is what I ''should'' feel * Deep acting ** People who 'deep act', as in genuinely change the way they feel (match the feelings they have with the feelings that they need to display) often report positive relationships with emotional labor ** Actual language - in this situation, this is what I ''actually'' feel Discuss the differences in gender norms and expectations in relationships {| class="wikitable" style="margin: auto;" |- ! !! Deep acting !! Surface acting |- | '''Emotional dissonance''' || || X |- | '''Not known to others''' || Hidden area || Unknown |} === Emotional contagion === Emotional contagion is the idea that emotions are contagious. as the theory to moderate surface/deep acting * Emotional contagion (Panksepp and Lahvis, 2011) ** emotions can spread (expand, combines empathy and acting) ** “the tendency to automatically mimic and synchronize expressions, vocalizations, postures, and movements with those of another person’s and, consequently, to converge emotionally.” (Elaine Hatfield, 1993) * Affective process theory looks at unifying the mechanisms of emotional contagion into three types. Convergent linkage, Divergent linkage, and Complementary linkage (Elfenbein, 2014). Storytelling is a way to promote position emotions (Wei et al., 2024). === The mediating effect of emotional labor === Surface acting does not mediate the effect (high surface acting leads to emotional exhaustion) whereas the inverse is true for deep acting (Wrobel, 2013) Emotional Expression and Emotional Processing (Hoyt et al., 2024) {{RoundBoxTop|theme=3}} '''What role does emotional labor play in emotional exhaustion''' * In relation to the CORT and JD-R theories, emotional labor acts as a job demand and expenditure of resources to protect existing resources {{RoundBoxBottom}} == Strategies for combating empathy fatigue and emotional fatigue == * Talk about emotion regulation (Gross, 1998) and mood regulation strategies as a way to deal with coping (Wrobel, 2013) (Author's note: need more research/references for this section) * Talk about breaks ** Rest and downtime from not performing emotional labor regenerates * How do we reduce the amount of emotional labor we're performing (Jeung et al., 2018) ** Communication/boundaries ** Reducing surface acting (allowing ourselves to feel how we actually feel and reduce emotive dissonance) '''Table 1. Treatment phases of clinical burnout (based on Table 2 from van Dam, A. 2021)''' {| class="wikitable" style="margin: auto; |- ! Phase !! Treatment goal !! Therapeutic interventions |- | '''Phase 1: Crisis''' || Recognition of the patient that the problems are serious, and that serious action is required || Psychoeducation Sick leave Dropping domestic tasks and social obligations Inform social network |- | '''Phase 2: Recovery''' || Recovery of the (physiological) stress system to normal allostatic stress levels || Registration of stress and activities Relaxtion exercises Mindfulness healthy lifestyle advices Graded exercise Gradually resume activities Psychoeducation social network and employer Gradual return to work |- | '''Phase 3: Prevention, learning from the past''' || Acquiring insight and skills to prevent relapse in clinical burnout || Analysing stressful situations Analysing dysfunctional thought patterns Learning new coping skills Learning social skills Job crafting Time management Making choices regarding career and personal life |- | '''Phase 4: Post burnout growth''' || Improving sustainable quality of life. || Setting priorities regarding quality of life and interpersonal relationships. |} === Strategies for the workplace === * Discuss types of strategies that are available in the workplace (organisation models) * SCARF model (Aplin-Houtz, 2025) ** Link back to COR and JDR models e.g. praise could be a job resource vs criticism could be a job demand *Reduce workload to reduce emotional exhaustion, and by extension reduce emotional fatigue. {{RoundBoxTop|theme=3}} '''How does understanding emotional labor and empathy fatigue work to reduce the effects of emotional exhaustion''' *Psychoeducation has been shown to reduce the effects of emotional labor and empathy fatigue on emotional exhaustion {{RoundBoxBottom}} ==Conclusion== * Summarise what empathy fatigue is, and why people become emotionally exhausted (+ the role of emotional labor) * Summarise the symptoms and how to recognise them in yourself and someone else * Emphasise and summarise psychological strategies to support someone going through emotional exhaustion * Summarise ways to prevent emotional exhaustion from happening * Overall message is: *# What is emotional exhaustion *# how do you recognise it *# how do you treat it *# how do you prevent it * Link back to the focus questions ==See also== * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Metacognition_and_emotional_regulation&oldid=2756264 Metacognition and emotional regulation] (Book chapter, 2026) {{ic|Use an internal link style as shown in Tutorial 2}} * [[wikipedia:Conservation_of_resources_theory|Conservation of resources theory]] (Wikipedia) * [[wikipedia:Job_demands-resources_model|Job demands-resources model]] (Wikipedia) ==References== {{Hanging indent|1= Aplin-Houtz, M. (2025). Unraveling the SCARF: How dimensions of the SCARF model influence the relationship between workplace exclusion and turnover intentions. Journal of Management & Organization, 31(6), 2699–2725. https://doi.org/10.1017/jmo.2025.10026 Bošnjaković, J., & Radionov, T. (2018). Empathy: Concepts, Theories and Neuroscientific Basis. Alcoholism (Zagreb), 54(2), 123–150. https://doi.org/10.20471/de..2018.54.02.04 Chan, F. M. (2026). The State of Workplace Burnout 2026: The Workplace Divide. Infinite Potential, Sydney, Australia. https://www.ahri.com.au/wp-content/uploads/The-State-of-Workplace-Burnout-2026-Final.pdf Cui, F., Huang, X., Jing, Y., Luo, Y., Liu, J., & Gu, R. (2022). How resource sharing resists scarcity: the role of cognitive empathy and its neurobiological mechanisms. Cerebral Cortex (New York, N.Y. 1991), 32(23), 5330–5342. https://doi.org/10.1093/cercor/bhac017 De Beer, L. T., van der Vaart, L., Escaffi-Schwarz, M., De Witte, H., & Schaufeli, W. B. (2024). Maslach Burnout Inventory - General Survey: A Systematic Review and Meta-Analysis of Measurement Properties. European journal of psychological assessment : official organ of the European Association of Psychological Assessment, 40(5), 360-375. https://doi.org/10.1027/1015-5759/a000797 Demerouti, E., Bakker, A. B., Nachreiner, F., & Schaufeli, W. B. (2001). The Job Demands-Resources Model of Burnout. Journal of Applied Psychology, 86(3), 499–512. https://doi.org/10.1037/0021-9010.86.3.499 Figley, C. R. (2002). Compassion fatigue: Psychotherapists’ chronic lack of self care. Journal of Clinical Psychology, 58(11), 1433–1441. https://doi.org/10.1002/jclp.10090 Grandey, A. A. (2003). When ‘the show must go on’: surface acting and deep acting as determinants of emotional exhaustion and peer-rated service delivery. Academy of Management Journal, 46(1), 86–96. https://doi.org/10.5465/30040678 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 Halbesleben, J. R. B., Neveu, J.-P., Paustian-Underdahl, S. C., & Westman, M. (2014). Getting to the “COR”: Understanding the Role of Resources in Conservation of Resources Theory. Journal of Management, 40(5), 1334–1364. https://doi.org/10.1177/0149206314527130 Hatfield, E., Cacioppo, J. T., & Rapson, R. L. (1993). Emotional Contagion. Current Directions in Psychological Science : A Journal of the American Psychological Society, 2(3), 96–99. https://doi.org/10.1111/1467-8721.ep10770953 Hollebeek, L. D., Hammedi, W., & Sprott, D. E. (2023). Consumer engagement, stress, and conservation of resources theory: A review, conceptual development, and future research agenda. Psychology & marketing, 40(5), 926-937. https://doi.org/10.1002/mar.21807 Hobfoll, S. E. (1989). Conservation of Resources: A New Attempt at Conceptualizing Stress. The American Psychologist, 44(3), 513–524. https://doi.org/10.1037/0003-066X.44.3.513 Hochschild, A. R. (2012). The managed heart : commercialization of human feeling (Updated, with a new preface.). University of California Press. Jeung, D.-Y., Kim, C., & Chang, S.-J. (2018). Emotional Labor and Burnout: A Review of the Literature. Yonsei Medical Journal, 59(2), 187–193. https://doi.org/10.3349/ymj.2018.59.2.187 Lavelle, J. J., Rupp, D. E., Herda, D. N., Pandey, A., & Lauck, J. R. (2021). Customer Injustice and Employee Performance: Roles of Emotional Exhaustion, Surface Acting, and Emotional Demands–Abilities Fit. Journal of Management, 47(3), Article 0149206319869426. https://doi.org/10.1177/0149206319869426 Lloyd, C., & King, R. (2004). A survey of burnout among Australian mental health occupational therapists and social workers. Social Psychiatry and Psychiatric Epidemiology, 39(9), 752-757. https://doi.org/https://doi.org/10.1007/s00127-004-0808-7 Kenworthy, J., Fay, C., Frame, M., & Petree, R. (2014). A meta-analytic review of the relationship between emotional dissonance and emotional exhaustion. Journal of Applied Social Psychology, 44(2), 94-105. https://doi.org/https://doi.org/10.1111/jasp.12211 Martingano, A. J., Stosic, M. D., & Konrath, S. (2026). Different empathy types show opposing associations with burnout: systematic review and meta-analyses. Psychology, Health & Medicine, 31(3), 798-829. https://doi.org/10.1080/13548506.2025.2591859 Michielsen, H. J., Willemsen, T. M., Croon, M. A., de Vries, J., & Van Heck, G. L. (2004). Determinants of general fatigue and emotional exhaustion: a prospective study. Psychology & health, 19(2), 223-235. https://doi.org/10.1080/08870440310001627135 Panksepp, J. B., & Lahvis, G. P. (2011). Rodent empathy and affective neuroscience. Neuroscience and Biobehavioral Reviews, 35(9), 1864–1875. https://doi.org/10.1016/j.neubiorev.2011.05.013 Pooladvand, S., & Hasanzadeh, S. (2023). Impacts of Stress on Workers’ Risk-Taking Behaviors: Cognitive Tunneling and Impaired Selective Attention. Journal of Construction Engineering and Management, 149(8), 04023060. https://doi.org/doi:10.1061/JCEMD4.COENG-13339 Riess, H. (2017). The Science of Empathy. Journal of Patient Experience, 4(2), 74–77. https://doi.org/10.1177/2374373517699267 Schaufeli, W. B. (2017). Applying the job demands-resources model: a “how to” guide to measuring and tackling work engagement and burnout. Organizational Dynamics, 46(2), 120–132. https://doi.org/10.1016/j.orgdyn.2017.04.008 Schneiderman, N., Ironson, G., & Siegel, S. D. (2005). Stress and Health: Psychological, Behavioral, and Biological Determinants. Annual Review of Clinical Psychology, 1(Volume 1, 2005), 607-628. https://doi.org/https://doi.org/10.1146/annurev.clinpsy.1.102803.144141 Seidler, A., Thinschmidt, M., Deckert, S., Then, F., Hegewald, J., Nieuwenhuijsen, K., & Riedel-Heller, S. G. (2014). The role of psychosocial working conditions on burnout and its core component emotional exhaustion – a systematic review. Journal of Occupational Medicine and Toxicology, 9(1), 10. https://doi.org/10.1186/1745-6673-9-10 Shao, M., Li, L., Li, X., Wei, Z., Wang, J., Hong, M., Liu, X., & Meng, J. (2024). The effect of top-down attention on empathy fatigue. Cerebral Cortex (New York, N.Y. 1991), 34(1), Article bhad441. https://doi.org/10.1093/cercor/bhad441 van Dam, A. (2021). A clinical perspective on burnout: diagnosis, classification, and treatment of clinical burnout. European Journal of Work and Organizational Psychology, 30(5), 732-741. https://doi.org/10.1080/1359432X.2021.1948400 Vial, A. C., & Cowgill, C. M. (2022). Heavier Lies Her Crown: Gendered Patterns of Leader Emotional Labor and Their Downstream Effects. Frontiers in Psychology, 13, 849566. https://doi.org/https://doi.org/10.3389/fpsyg.2022.849566 Wharton, A. S. (2009). The Sociology of Emotional Labor. Annual Review of Sociology, 35(1), 147–165. https://doi.org/10.1146/annurev-soc-070308-115944 Wrobel, M. (2013). Can empathy lead to emotional exhaustion in teachers? The mediating role of emotional labor. International Journal of Occupational Medicine and Environmental Health, 26(4), 581–592. https://doi.org/10.2478/s13382-013-0123-1 }} ==External links== Provide [[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: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Burnout]] [[Category:Motivation and emotion/Book/Empathy]] plum7ngdmi10y92jvrvcn8jw3x8y4sf 2844527 2844519 2026-11-04T22:34:10Z U3143751 3106785 /* Job demands-resources model */ updated scenario focus question 2844527 wikitext text/x-wiki {{title|Empathy fatigue and emotional exhaustion:<br>How does sustained empathic engagement contribute to emotional exhaustion?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}} [[File:Exhausted smartphone user.JPG|right|thumb|150px|'''Figure 1'''. An emotionally exhausted Aelin at the end of the work day]] ; Scenario Aelin, a 23-year-old social worker employed at headspace {{ic|Link to the Wikipedia article to help explain what this term means}}, works with approximately 20 {{ic|Explain what sort of}}Imagine you are only 23 years old and are working as a social worker at headspace. You see around 20 clients each week, many of whom have experienced rough living conditions. During your first six months in this role, you felt highly motivated and excited to help people in need. You are an excellent active listener and can easily remember details about your clients Live’s. You are also good at recognising signs of distress in your client’s body language and adjust yours in order to show warmth and understanding. As you progress in your role, you take on more cases that involve complex past traumas. The workload becomes too much for you to get done at work, but you don’t want your clients to miss out, so you take your work home to complete it in the evenings. Your workload increases again and you now start doing work on your weekends as well. After several months of working this way, you start to notice you feel exhausted every morning before you have even started work. You then start to notice you’re responding to people at work differently. Instead of your usual energised active listening response you feel frustrated, empathetic and detached. After work you are too drained to socialise with friends. This begins to worry you. What if you won’t be able to continue working as a social worker? {{RoundBoxBottom}} Aelin is experiencing '''emotional exhaustion''' from a sustained empathic response. Due to the nature of Aelin's work, they are constantly engaged in an empathic response. Empathic responses require cognitive and physical resources to sustain. Because Aelin is taking their work home and not resting on weekends, they do not have a chance to rest. It is important for Aelin to recognise what is happening and why. Not actioning the problem has led to empathy fatigue and emotional exhaustion. Emotional exhaustion is one of the key aspects of [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FEmpathy_fatigue_and_emotional_exhaustion&diff=2840411&oldid=2832859 burnout], a long-term condition of existing in a constant state of high stress. Psychological science can support Aelin in recognising and mapping the symptoms they are experiencing, understand coping strategies to minimise the current and future effects to improve their wellbeing. Understanding the symptoms will also support their role as a social worker, providing more tools for the people they support on a day-to-day basis. {{RoundBoxTop|theme=3}} '''Focus questions''' * Why do people become emotionally exhausted? * What is the relationship between empathy fatigue and emotional exhaustion? * What role does emotional labor play in emotional exhaustion? * How does understanding emotional labor and empathy fatigue work to reduce the effects of emotional exhaustion? {{RoundBoxBottom}} == Emotional exhaustion == To understand emotional exhaustion, it will be important to first understand burnout. Burnout is a long-term condition of existing in a constant state of high stress and affects millions of people annually. In the early 2000s, a survey from Australia found that more than 50% of social workers and occupational workers reported high levels of burnout symptoms (Llyod and King, 2004) and a recent report from 2026 found that more than one third of all workers in Australia were experiencing burnout (Chan, 2026). Being in a state of high stress for short periods of time is a useful adaptation for humans, so much so that it has been given a name, [[wikipedia:Eustress|Eustress]]. However, as a long-term condition it leads to individual health issues and can be linked to cardiovascular disease, increased vulnerability to upper respiratory diseases, and chronic inflammation (Schneiderman et al., 2005), and cognitive issues such as [[wikipedia:Inattentional_blindness|cognitive tunneling]] and impaired selective attention (Pooladvand and Hasanzadeh, 2023). Burnout can be defined across three dimensions: emotional exhaustion, [[wikipedia:Depersonalization|depersonalisation]], and reduced feelings of personal accomplishment (Maslach, 1993, as cited in Martingano et al., 2026). Emotional exhaustion may be considered the 'core' component of burnout and refers to the depletion of emotional resources, leading to feelings of overwhelm, overextension, and exhaustion (Martingano et al., 2026; De Beer et al., 2024; Seidler et al., 2014). As a major factor, its symptoms are similar to burnout and stress. Although antecedents of burnout are typically agreed upon (Bakker et al., 2014) and divided into individual (e.g. self-efficacy) and situational categories (e.g. work load), there are many theories that attempt to explain why emotional exhaustion occurs. Two of these theories with good evidence are the [[wikipedia:Conservation_of_resources_theory|Conservation of resources stress model]] (Hobfoll, 1989) and the [[wikipedia:Job_demands-resources_model|Job Demands-Resources model]] (Demerouti et al., 2001). === Conservation of resources stress model === The Conservation of resources stress model (CORT; Hobfall, 1989) is a theory that works to understand and explain why stress occurs. Under the model, humans are motivated to maintain, pursue, or attain resources. Stress, defined as "an individual's perceived pressure or strain" (Hollebeek et al., 2023 p.928), occurs when these resources are threatened or lost, or gained at a net loss. As noted above, resources aren't strictly tied to being physical objects. They can also be emotional states, conditions, or even ideas. For example, someone may place value on enjoying a calm, quiet Sunday morning at home. If their neighbour starts using power tools loudly next door, it would disrupt this state and induce stress in the individual. The CORT model focuses primarily on two principles. The first principle is that there is more harm in losing resources than there is in gaining them back (Halbesleben et al., 2014). This is known as loss salience, and some research suggests that the pain of loss is felt twice as strong as the pleasure of gain (Thaler, 2000). Looking at our previous example, imagine that the person who enjoys a quiet Sunday morning at home equally enjoys the sound of a loud car engine. Even though the person enjoys the sound of car engines, if the neighbour started revving their car loudly instead of using power tools, it would still disrupt their desired state of quiet and lead to an increase of stress. The second principle is that people must invest existing resources in order to gain or protect resources (Halbesleben et al., 2014). This resource investment includes recovering from resource loss. Coping is how people manage or overcome a difficult situation, such as protecting or recovering from resource loss. Coping can be broken down into different strategies that people use, but not all strategies are equal (Ito and Brotheridge, 2003). They are divided into two categories: control or escape strategies. Control strategies are considered better at reducing the strain from stress and consist of positive thinking, direct action, and help seeking (Havlovic & Keenan, 1991, as cited in Ito and Brotheridge, 2003). The most useful control strategies incorporate a person's [[locus of control]], emphasise seeking support from co-workers, and highlight the importance of autonomy in decision making. There is fairly robust evidence that supports the CORT model. === Job demands-resources model === Another model that attempts to understand why people become emotionally exhausted is the Job Demands-Resources model (JD-R; Demerouti et al., 2001). The JD-R model discusses two concepts: job demands and job resources. Demands and resources are both related to the physical, social, or psychological aspects of a job, but differ in their impact on a person. Job demands require physical or mental effort and exhaust a person's energy supply. Job resources typically support achieving goals, reduce the effort cost of job demands, and stimulate personal growth (Richter and Hacker, 1998, as cited in Demerouti et al., 2001). For example, an unsustainable workload would be considered a job demand, and receiving help and support from others would be considered a resource. The JD-R has two primary processes for how demands and resources interact with one another (Schaufeli, 2017). The stress process occurs when job demands are high and resources are low. As stress increases strain and pressure on an individual, it leads to the exhaustion of their energy eventually, eventually leading to emotional exhaustion. The second process is the motivational process. This process occurs when there is an excess of job resources. Crucially, the motivational process does not require low job demands. With enough job resources, a person will be energised and engaged, reframing job demands as a challenge to overcome. The JD-R model has had extensive research to confirm its validity. A meta-analysis by Lesener et al. (2019) looked at 74 longitudinal studies of the JD-R with a strong average sample size (n=450) against the assumptions as described by Schaufeli (2017). Although most of the studies appear to come from WEIRD countries (k=60) there were a good number from Asia and Africa (k=13) and a mix of genders. The study confirmed that across the literature, job resources predict burnout negatively (''β'' = −.17; ''p'' < .001) and job demands predict burnout over time (''β'' = .10; ''p'' < .001). Although job resources seem like a cure-all for demands, it's important to note that they do not retroactively treat emotional exhaustion. Under the JD-R model, burnout must be reduced by lowering job demands (Gynning et al., 2025). However, once someone has recovered, it is possible to increase job resources alongside job demands to prevent future burnout and support [[wikipedia:Work_engagement|work engagement]]. Work engagement (engagement) has been discussed as the opposite of burnout (Bakker et al., 2014). Engagement is how people express themselves when performing a role. A role being a pattern of behaviour that a person goes through based on the social expectations of that role, similar to how an actor takes on a role for a movie. For example, when attending a typical restaurant, it is socially expected that a person will engage with the waiter and provide them with their meal order to take to the kitchen. This person is performing the script of a restaurant patron. In the context of a workplace, an employee with high engagement may appear to be energised, enthusiastic, and become absorbed in their work losing track of time (Bakkar et al, 2014). <nowiki>**</nowiki>Update box below to relate back to Aelin. {{RoundBoxTop|theme=3}} '''Why do we feel emotionally exhausted''' Under the JD-R and CORT models, emotional exhaustion occurs when the demand cost outweighs the resources available for an extended period of time, eventually depleting someone's energy reserves. Aelin is experiencing burnout because they increased their job demands by taking on more clients, and has experienced a loss in resources by reducing their free time and social support network. Under these models, Aelin should consider reducing their workload to begin recovering and, as part of recovery, increasing available resources such as asking for help through social support and considering ways to increase autonomy in their role. This will lead to positive outcomes - reducing the likelihood of future exhaustion and increasing engagement (Bakker et al., 2014; Halbesleben et al., 2014). {{RoundBoxBottom}} == Empathy fatigue == You may be familiar with the term [[empathy]]. It is often described in terms of supporting others. For example, when someone close to you experiences a loss and is grieving, you may empathise with them and experience a negative emotion akin to grief too. However, empathy is not exclusive to so called negative emotions. For example, compersion is a positive empathic response when a loved one experiences joy. Parents may experience compersion when something joyful happens to their child. At its core, empathy is the individual ability to imagine or understand another person's feelings and/or emotions (Riess, 2017; Shao et al., 2024) and has four main components (Davis, 1983, as cited in Delgado et al., 2023): * Empathic concern is xyz * Fantasy is xyz * Perspective taking is xyz * Personal distress is xyz Much like physical or cognitive processes, such as running or problem solving, empathising will drain energy and resources. Further, and similar to the difference in energy expenditure between running and jogging, there are distinct types of empathy that each have different energy costs. === Types of empathy === Research makes a clear distinction between two types of empathy, affective and cognitive empathy. Affective empathy occurs when a person has an emotional reaction to someone else's mental state (Shao et al., 2024). This occurs because (unsure about this next part go through and update accordingly) ''unconscious processes interpret the social stimuli from another person leading to a physiological and cognitive response. These processes may interpret stimuli from any of the five senses.'' Cognitive empathy is when a person understands someone else's feelings or emotions (Cui et al., 2022). * Integrating the two - "affective component is the essence of empathy, and the cognitive component is a process through which that essence comes into being" (Bošnjaković & Radionov, 2018) ** (Author's note: change quote into paraphrasing) * === Empathy or compassion fatigue === * Distinction between empathy and compassion fatigue * Compassion fatigue is closely related to Secondary traumatic stress usually found in healthcare professionals (Figley, 2002) ** Compassion fatigue is to Empathy fatigue as emotional exhaustion is to burnout * Make it clear that Aelin is experiencing Compassion Fatigue vs Empathy fatigue (they are the same but different) (Author's note: include an update to the scenario here!) '''Empathy fatigue questions'''. Choose your answers and click "Submit": <quiz display=simple> {Surface acting is when the observer actually feels the emotion: |type="()"} - True + False {Compassion fatigue and empathy fatigue are synonymous: |type="()"} - True + False </quiz> {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' * Under the JD-R model, a sustained empathic response is a long-term job demand. * Under the CORT model, a sustained empathic response uses energy to protect existing resources. As resources become low and burnout sets in, cognitive focus narrows leading to empathy fatigue. {{RoundBoxBottom}} == Emotional labor as a mediating factor between empathy fatigue and emotional exhaustion == As a society there are expected [https://en.wikipedia.org/wiki/Social_norm social norms] in behaviour when expressing emotion, regardless if someone is conscious of the process of matching these norms or not. For example, laughing during an emotional speech at a funeral would be considered a rude or offensive expression from the other attendees. As social creatures, humans need a way to regulate socially appropriate emotional expression. This is the essence of Emotional labor: "the regulation of felt and expressed emotions (whether effortful or not) to match these emotional display rules with the objective of fulfilling organizational goals" (Vial & Cowgill, 2022) What is emotional labor (Hochschild, 2012). Theories behind emotional labor involve acting, and an extension of acting known as emotional contagion. === Types of acting === * Surface acting ** This is where emotive dissonance comes into play ** Leads to emotional exhaustion (Kenworthy et al., 2014) ** "Should" language - in this situation, this is what I ''should'' feel * Deep acting ** People who 'deep act', as in genuinely change the way they feel (match the feelings they have with the feelings that they need to display) often report positive relationships with emotional labor ** Actual language - in this situation, this is what I ''actually'' feel Discuss the differences in gender norms and expectations in relationships {| class="wikitable" style="margin: auto;" |- ! !! Deep acting !! Surface acting |- | '''Emotional dissonance''' || || X |- | '''Not known to others''' || Hidden area || Unknown |} === Emotional contagion === Emotional contagion is the idea that emotions are contagious. as the theory to moderate surface/deep acting * Emotional contagion (Panksepp and Lahvis, 2011) ** emotions can spread (expand, combines empathy and acting) ** “the tendency to automatically mimic and synchronize expressions, vocalizations, postures, and movements with those of another person’s and, consequently, to converge emotionally.” (Elaine Hatfield, 1993) * Affective process theory looks at unifying the mechanisms of emotional contagion into three types. Convergent linkage, Divergent linkage, and Complementary linkage (Elfenbein, 2014). Storytelling is a way to promote position emotions (Wei et al., 2024). === The mediating effect of emotional labor === Surface acting does not mediate the effect (high surface acting leads to emotional exhaustion) whereas the inverse is true for deep acting (Wrobel, 2013) Emotional Expression and Emotional Processing (Hoyt et al., 2024) {{RoundBoxTop|theme=3}} '''What role does emotional labor play in emotional exhaustion''' * In relation to the CORT and JD-R theories, emotional labor acts as a job demand and expenditure of resources to protect existing resources {{RoundBoxBottom}} == Strategies for combating empathy fatigue and emotional fatigue == * Talk about emotion regulation (Gross, 1998) and mood regulation strategies as a way to deal with coping (Wrobel, 2013) (Author's note: need more research/references for this section) * Talk about breaks ** Rest and downtime from not performing emotional labor regenerates * How do we reduce the amount of emotional labor we're performing (Jeung et al., 2018) ** Communication/boundaries ** Reducing surface acting (allowing ourselves to feel how we actually feel and reduce emotive dissonance) '''Table 1. Treatment phases of clinical burnout (based on Table 2 from van Dam, A. 2021)''' {| class="wikitable" style="margin: auto; |- ! Phase !! Treatment goal !! Therapeutic interventions |- | '''Phase 1: Crisis''' || Recognition of the patient that the problems are serious, and that serious action is required || Psychoeducation Sick leave Dropping domestic tasks and social obligations Inform social network |- | '''Phase 2: Recovery''' || Recovery of the (physiological) stress system to normal allostatic stress levels || Registration of stress and activities Relaxtion exercises Mindfulness healthy lifestyle advices Graded exercise Gradually resume activities Psychoeducation social network and employer Gradual return to work |- | '''Phase 3: Prevention, learning from the past''' || Acquiring insight and skills to prevent relapse in clinical burnout || Analysing stressful situations Analysing dysfunctional thought patterns Learning new coping skills Learning social skills Job crafting Time management Making choices regarding career and personal life |- | '''Phase 4: Post burnout growth''' || Improving sustainable quality of life. || Setting priorities regarding quality of life and interpersonal relationships. |} === Strategies for the workplace === * Discuss types of strategies that are available in the workplace (organisation models) * SCARF model (Aplin-Houtz, 2025) ** Link back to COR and JDR models e.g. praise could be a job resource vs criticism could be a job demand *Reduce workload to reduce emotional exhaustion, and by extension reduce emotional fatigue. {{RoundBoxTop|theme=3}} '''How does understanding emotional labor and empathy fatigue work to reduce the effects of emotional exhaustion''' *Psychoeducation has been shown to reduce the effects of emotional labor and empathy fatigue on emotional exhaustion {{RoundBoxBottom}} ==Conclusion== * Summarise what empathy fatigue is, and why people become emotionally exhausted (+ the role of emotional labor) * Summarise the symptoms and how to recognise them in yourself and someone else * Emphasise and summarise psychological strategies to support someone going through emotional exhaustion * Summarise ways to prevent emotional exhaustion from happening * Overall message is: *# What is emotional exhaustion *# how do you recognise it *# how do you treat it *# how do you prevent it * Link back to the focus questions ==See also== * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Metacognition_and_emotional_regulation&oldid=2756264 Metacognition and emotional regulation] (Book chapter, 2026) {{ic|Use an internal link style as shown in Tutorial 2}} * [[wikipedia:Conservation_of_resources_theory|Conservation of resources theory]] (Wikipedia) * [[wikipedia:Job_demands-resources_model|Job demands-resources model]] (Wikipedia) ==References== {{Hanging indent|1= Aplin-Houtz, M. (2025). Unraveling the SCARF: How dimensions of the SCARF model influence the relationship between workplace exclusion and turnover intentions. Journal of Management & Organization, 31(6), 2699–2725. https://doi.org/10.1017/jmo.2025.10026 Bošnjaković, J., & Radionov, T. (2018). Empathy: Concepts, Theories and Neuroscientific Basis. Alcoholism (Zagreb), 54(2), 123–150. https://doi.org/10.20471/de..2018.54.02.04 Chan, F. M. (2026). The State of Workplace Burnout 2026: The Workplace Divide. Infinite Potential, Sydney, Australia. https://www.ahri.com.au/wp-content/uploads/The-State-of-Workplace-Burnout-2026-Final.pdf Cui, F., Huang, X., Jing, Y., Luo, Y., Liu, J., & Gu, R. (2022). How resource sharing resists scarcity: the role of cognitive empathy and its neurobiological mechanisms. Cerebral Cortex (New York, N.Y. 1991), 32(23), 5330–5342. https://doi.org/10.1093/cercor/bhac017 De Beer, L. T., van der Vaart, L., Escaffi-Schwarz, M., De Witte, H., & Schaufeli, W. B. (2024). Maslach Burnout Inventory - General Survey: A Systematic Review and Meta-Analysis of Measurement Properties. European journal of psychological assessment : official organ of the European Association of Psychological Assessment, 40(5), 360-375. https://doi.org/10.1027/1015-5759/a000797 Demerouti, E., Bakker, A. B., Nachreiner, F., & Schaufeli, W. B. (2001). The Job Demands-Resources Model of Burnout. Journal of Applied Psychology, 86(3), 499–512. https://doi.org/10.1037/0021-9010.86.3.499 Figley, C. R. (2002). Compassion fatigue: Psychotherapists’ chronic lack of self care. Journal of Clinical Psychology, 58(11), 1433–1441. https://doi.org/10.1002/jclp.10090 Grandey, A. A. (2003). When ‘the show must go on’: surface acting and deep acting as determinants of emotional exhaustion and peer-rated service delivery. Academy of Management Journal, 46(1), 86–96. https://doi.org/10.5465/30040678 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 Halbesleben, J. R. B., Neveu, J.-P., Paustian-Underdahl, S. C., & Westman, M. (2014). Getting to the “COR”: Understanding the Role of Resources in Conservation of Resources Theory. Journal of Management, 40(5), 1334–1364. https://doi.org/10.1177/0149206314527130 Hatfield, E., Cacioppo, J. T., & Rapson, R. L. (1993). Emotional Contagion. Current Directions in Psychological Science : A Journal of the American Psychological Society, 2(3), 96–99. https://doi.org/10.1111/1467-8721.ep10770953 Hollebeek, L. D., Hammedi, W., & Sprott, D. E. (2023). Consumer engagement, stress, and conservation of resources theory: A review, conceptual development, and future research agenda. Psychology & marketing, 40(5), 926-937. https://doi.org/10.1002/mar.21807 Hobfoll, S. E. (1989). Conservation of Resources: A New Attempt at Conceptualizing Stress. The American Psychologist, 44(3), 513–524. https://doi.org/10.1037/0003-066X.44.3.513 Hochschild, A. R. (2012). The managed heart : commercialization of human feeling (Updated, with a new preface.). University of California Press. Jeung, D.-Y., Kim, C., & Chang, S.-J. (2018). Emotional Labor and Burnout: A Review of the Literature. Yonsei Medical Journal, 59(2), 187–193. https://doi.org/10.3349/ymj.2018.59.2.187 Lavelle, J. J., Rupp, D. E., Herda, D. N., Pandey, A., & Lauck, J. R. (2021). Customer Injustice and Employee Performance: Roles of Emotional Exhaustion, Surface Acting, and Emotional Demands–Abilities Fit. Journal of Management, 47(3), Article 0149206319869426. https://doi.org/10.1177/0149206319869426 Lloyd, C., & King, R. (2004). A survey of burnout among Australian mental health occupational therapists and social workers. Social Psychiatry and Psychiatric Epidemiology, 39(9), 752-757. https://doi.org/https://doi.org/10.1007/s00127-004-0808-7 Kenworthy, J., Fay, C., Frame, M., & Petree, R. (2014). A meta-analytic review of the relationship between emotional dissonance and emotional exhaustion. Journal of Applied Social Psychology, 44(2), 94-105. https://doi.org/https://doi.org/10.1111/jasp.12211 Martingano, A. J., Stosic, M. D., & Konrath, S. (2026). Different empathy types show opposing associations with burnout: systematic review and meta-analyses. Psychology, Health & Medicine, 31(3), 798-829. https://doi.org/10.1080/13548506.2025.2591859 Michielsen, H. J., Willemsen, T. M., Croon, M. A., de Vries, J., & Van Heck, G. L. (2004). Determinants of general fatigue and emotional exhaustion: a prospective study. Psychology & health, 19(2), 223-235. https://doi.org/10.1080/08870440310001627135 Panksepp, J. B., & Lahvis, G. P. (2011). Rodent empathy and affective neuroscience. Neuroscience and Biobehavioral Reviews, 35(9), 1864–1875. https://doi.org/10.1016/j.neubiorev.2011.05.013 Pooladvand, S., & Hasanzadeh, S. (2023). Impacts of Stress on Workers’ Risk-Taking Behaviors: Cognitive Tunneling and Impaired Selective Attention. Journal of Construction Engineering and Management, 149(8), 04023060. https://doi.org/doi:10.1061/JCEMD4.COENG-13339 Riess, H. (2017). The Science of Empathy. Journal of Patient Experience, 4(2), 74–77. https://doi.org/10.1177/2374373517699267 Schaufeli, W. B. (2017). Applying the job demands-resources model: a “how to” guide to measuring and tackling work engagement and burnout. Organizational Dynamics, 46(2), 120–132. https://doi.org/10.1016/j.orgdyn.2017.04.008 Schneiderman, N., Ironson, G., & Siegel, S. D. (2005). Stress and Health: Psychological, Behavioral, and Biological Determinants. Annual Review of Clinical Psychology, 1(Volume 1, 2005), 607-628. https://doi.org/https://doi.org/10.1146/annurev.clinpsy.1.102803.144141 Seidler, A., Thinschmidt, M., Deckert, S., Then, F., Hegewald, J., Nieuwenhuijsen, K., & Riedel-Heller, S. G. (2014). The role of psychosocial working conditions on burnout and its core component emotional exhaustion – a systematic review. Journal of Occupational Medicine and Toxicology, 9(1), 10. https://doi.org/10.1186/1745-6673-9-10 Shao, M., Li, L., Li, X., Wei, Z., Wang, J., Hong, M., Liu, X., & Meng, J. (2024). The effect of top-down attention on empathy fatigue. Cerebral Cortex (New York, N.Y. 1991), 34(1), Article bhad441. https://doi.org/10.1093/cercor/bhad441 van Dam, A. (2021). A clinical perspective on burnout: diagnosis, classification, and treatment of clinical burnout. European Journal of Work and Organizational Psychology, 30(5), 732-741. https://doi.org/10.1080/1359432X.2021.1948400 Vial, A. C., & Cowgill, C. M. (2022). Heavier Lies Her Crown: Gendered Patterns of Leader Emotional Labor and Their Downstream Effects. Frontiers in Psychology, 13, 849566. https://doi.org/https://doi.org/10.3389/fpsyg.2022.849566 Wharton, A. S. (2009). The Sociology of Emotional Labor. Annual Review of Sociology, 35(1), 147–165. https://doi.org/10.1146/annurev-soc-070308-115944 Wrobel, M. (2013). Can empathy lead to emotional exhaustion in teachers? The mediating role of emotional labor. International Journal of Occupational Medicine and Environmental Health, 26(4), 581–592. https://doi.org/10.2478/s13382-013-0123-1 }} ==External links== Provide [[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: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Burnout]] [[Category:Motivation and emotion/Book/Empathy]] jjt6hcpkro4fpcm1s2qp449eaywie81 2844528 2844527 2026-11-04T23:08:01Z U3143751 3106785 /* Job demands-resources model */ included meta-analysis for CORT 2844528 wikitext text/x-wiki {{title|Empathy fatigue and emotional exhaustion:<br>How does sustained empathic engagement contribute to emotional exhaustion?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}} [[File:Exhausted smartphone user.JPG|right|thumb|150px|'''Figure 1'''. An emotionally exhausted Aelin at the end of the work day]] ; Scenario Aelin, a 23-year-old social worker employed at headspace {{ic|Link to the Wikipedia article to help explain what this term means}}, works with approximately 20 {{ic|Explain what sort of}}Imagine you are only 23 years old and are working as a social worker at headspace. You see around 20 clients each week, many of whom have experienced rough living conditions. During your first six months in this role, you felt highly motivated and excited to help people in need. You are an excellent active listener and can easily remember details about your clients Live’s. You are also good at recognising signs of distress in your client’s body language and adjust yours in order to show warmth and understanding. As you progress in your role, you take on more cases that involve complex past traumas. The workload becomes too much for you to get done at work, but you don’t want your clients to miss out, so you take your work home to complete it in the evenings. Your workload increases again and you now start doing work on your weekends as well. After several months of working this way, you start to notice you feel exhausted every morning before you have even started work. You then start to notice you’re responding to people at work differently. Instead of your usual energised active listening response you feel frustrated, empathetic and detached. After work you are too drained to socialise with friends. This begins to worry you. What if you won’t be able to continue working as a social worker? {{RoundBoxBottom}} Aelin is experiencing '''emotional exhaustion''' from a sustained empathic response. Due to the nature of Aelin's work, they are constantly engaged in an empathic response. Empathic responses require cognitive and physical resources to sustain. Because Aelin is taking their work home and not resting on weekends, they do not have a chance to rest. It is important for Aelin to recognise what is happening and why. Not actioning the problem has led to empathy fatigue and emotional exhaustion. Emotional exhaustion is one of the key aspects of [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FEmpathy_fatigue_and_emotional_exhaustion&diff=2840411&oldid=2832859 burnout], a long-term condition of existing in a constant state of high stress. Psychological science can support Aelin in recognising and mapping the symptoms they are experiencing, understand coping strategies to minimise the current and future effects to improve their wellbeing. Understanding the symptoms will also support their role as a social worker, providing more tools for the people they support on a day-to-day basis. {{RoundBoxTop|theme=3}} '''Focus questions''' * Why do people become emotionally exhausted? * What is the relationship between empathy fatigue and emotional exhaustion? * What role does emotional labor play in emotional exhaustion? * How does understanding emotional labor and empathy fatigue work to reduce the effects of emotional exhaustion? {{RoundBoxBottom}} == Emotional exhaustion == To understand emotional exhaustion, it will be important to first understand burnout. Burnout is a long-term condition of existing in a constant state of high stress and affects millions of people annually. In the early 2000s, a survey from Australia found that more than 50% of social workers and occupational workers reported high levels of burnout symptoms (Llyod and King, 2004) and a recent report from 2026 found that more than one third of all workers in Australia were experiencing burnout (Chan, 2026). Being in a state of high stress for short periods of time is a useful adaptation for humans, so much so that it has been given a name, [[wikipedia:Eustress|Eustress]]. However, as a long-term condition it leads to individual health issues and can be linked to cardiovascular disease, increased vulnerability to upper respiratory diseases, and chronic inflammation (Schneiderman et al., 2005), and cognitive issues such as [[wikipedia:Inattentional_blindness|cognitive tunneling]] and impaired selective attention (Pooladvand and Hasanzadeh, 2023). Burnout can be defined across three dimensions: emotional exhaustion, [[wikipedia:Depersonalization|depersonalisation]], and reduced feelings of personal accomplishment (Maslach, 1993, as cited in Martingano et al., 2026). Emotional exhaustion may be considered the 'core' component of burnout and refers to the depletion of emotional resources, leading to feelings of overwhelm, overextension, and exhaustion (Martingano et al., 2026; De Beer et al., 2024; Seidler et al., 2014). As a major factor, its symptoms are similar to burnout and stress. Although antecedents of burnout are typically agreed upon (Bakker et al., 2014) and divided into individual (e.g. self-efficacy) and situational categories (e.g. work load), there are many theories that attempt to explain why emotional exhaustion occurs. Two of these theories with good evidence are the [[wikipedia:Conservation_of_resources_theory|Conservation of resources stress model]] (Hobfoll, 1989) and the [[wikipedia:Job_demands-resources_model|Job Demands-Resources model]] (Demerouti et al., 2001). === Conservation of resources stress model === The Conservation of resources stress model (CORT; Hobfall, 1989) is a theory that works to understand and explain why stress occurs. Under the model, humans are motivated to maintain, pursue, or attain resources. Stress, defined as "an individual's perceived pressure or strain" (Hollebeek et al., 2023 p.928), occurs when these resources are threatened or lost, or gained at a net loss. As noted above, resources aren't strictly tied to being physical objects. They can also be emotional states, conditions, or even ideas. For example, someone may place value on enjoying a calm, quiet Sunday morning at home. If their neighbour starts using power tools loudly next door, it would disrupt this state and induce stress in the individual. The CORT model focuses primarily on two principles. The first principle is that there is more harm in losing resources than there is in gaining them back (Halbesleben et al., 2014). This is known as loss salience, and some research suggests that the pain of loss is felt twice as strong as the pleasure of gain (Thaler, 2000). Looking at our previous example, imagine that the person who enjoys a quiet Sunday morning at home equally enjoys the sound of a loud car engine. Even though the person enjoys the sound of car engines, if the neighbour started revving their car loudly instead of using power tools, it would still disrupt their desired state of quiet and lead to an increase of stress. The second principle is that people must invest existing resources in order to gain or protect resources (Halbesleben et al., 2014). This resource investment includes recovering from resource loss. Coping is how people manage or overcome a difficult situation, such as protecting or recovering from resource loss. Coping can be broken down into different strategies that people use, but not all strategies are equal (Ito and Brotheridge, 2003). They are divided into two categories: control or escape strategies. Control strategies are considered better at reducing the strain from stress and consist of positive thinking, direct action, and help seeking (Havlovic & Keenan, 1991, as cited in Ito and Brotheridge, 2003). The most useful control strategies incorporate a person's [[locus of control]], emphasise seeking support from co-workers, and highlight the importance of autonomy in decision making. Alarcon (2011) conducted a meta-analysis of 231 studies in a work setting that used the [[wikipedia:Maslach_Burnout_Inventory|Maslach Burnout Inventory]] (MBI) to review the relationship between these principles and burnout and confirm their validity. Although the study had no specific mention of demographics, it was noted that multiple translations of the MBI were used including Dutch (k=51), German (k=7), Italian (k=2), and French (k=2). The findings indicate that workload is positively related to emotional exhaustion (''ρ'' = .49, ''k'' = 86, ''N'' = 51,529) and that control (''ρ'' = − .26, ''k'' = 56, ''N'' = 33,297) and autonomy (''ρ'' = − .24, ''k'' = 34, ''N'' = 17,165) were each negatively associated with emotional exhaustion. Consistent with Thaler (2000), the negative association on emotional exhaustion of gaining resources is approximately half that of losing resources. === Job demands-resources model === Another model that attempts to understand why people become emotionally exhausted is the Job Demands-Resources model (JD-R; Demerouti et al., 2001). The JD-R model discusses two concepts: job demands and job resources. Demands and resources are both related to the physical, social, or psychological aspects of a job, but differ in their impact on a person. Job demands require physical or mental effort and exhaust a person's energy supply. Job resources typically support achieving goals, reduce the effort cost of job demands, and stimulate personal growth (Richter and Hacker, 1998, as cited in Demerouti et al., 2001). For example, an unsustainable workload would be considered a job demand, and receiving help and support from others would be considered a resource. The JD-R has two primary processes for how demands and resources interact with one another (Schaufeli, 2017). The stress process occurs when job demands are high and resources are low. As stress increases strain and pressure on an individual, it leads to the exhaustion of their energy eventually, eventually leading to emotional exhaustion. The second process is the motivational process. This process occurs when there is an excess of job resources. Crucially, the motivational process does not require low job demands. With enough job resources, a person will be energised and engaged, reframing job demands as a challenge to overcome. The JD-R model has had extensive research to confirm its validity. A meta-analysis by Lesener et al. (2019) looked at 74 longitudinal studies of the JD-R with a strong average sample size (n=450) against the assumptions as described by Schaufeli (2017). Although most of the studies appear to come from WEIRD countries (k=60) there were a good number from Asia and Africa (k=13) and a mix of genders. The study confirmed that across the literature, job resources predict burnout negatively (''β'' = −.17; ''p'' < .001) and job demands predict burnout over time (''β'' = .10; ''p'' < .001). Although job resources seem like a cure-all for demands, it's important to note that they do not retroactively treat emotional exhaustion. Under the JD-R model, burnout must be reduced by lowering job demands (Gynning et al., 2025). However, once someone has recovered, it is possible to increase job resources alongside job demands to prevent future burnout and support [[wikipedia:Work_engagement|work engagement]]. Work engagement (engagement) has been discussed as the opposite of burnout (Bakker et al., 2014). Engagement is how people express themselves when performing a role. A role being a pattern of behaviour that a person goes through based on the social expectations of that role, similar to how an actor takes on a role for a movie. For example, when attending a typical restaurant, it is socially expected that a person will engage with the waiter and provide them with their meal order to take to the kitchen. This person is performing the script of a restaurant patron. In the context of a workplace, an employee with high engagement may appear to be energised, enthusiastic, and become absorbed in their work losing track of time (Bakkar et al, 2014). {{RoundBoxTop|theme=3}} '''Why do we feel emotionally exhausted''' Under the JD-R and CORT models, emotional exhaustion occurs when the demand cost outweighs the resources available for an extended period of time, eventually depleting someone's energy reserves. Aelin is experiencing burnout because they increased their job demands by taking on more clients, and has experienced a loss in resources by reducing their free time and social support network. Under these models, Aelin should consider reducing their workload to begin recovering and, as part of recovery, increasing available resources such as asking for help through social support and considering ways to increase autonomy in their role. This will lead to positive outcomes - reducing the likelihood of future exhaustion and increasing engagement (Bakker et al., 2014; Halbesleben et al., 2014). {{RoundBoxBottom}} == Empathy fatigue == You may be familiar with the term [[empathy]]. It is often described in terms of supporting others. For example, when someone close to you experiences a loss and is grieving, you may empathise with them and experience a negative emotion akin to grief too. However, empathy is not exclusive to so called negative emotions. For example, compersion is a positive empathic response when a loved one experiences joy. Parents may experience compersion when something joyful happens to their child. At its core, empathy is the individual ability to imagine or understand another person's feelings and/or emotions (Riess, 2017; Shao et al., 2024) and has four main components (Davis, 1983, as cited in Delgado et al., 2023): * Empathic concern is xyz * Fantasy is xyz * Perspective taking is xyz * Personal distress is xyz Much like physical or cognitive processes, such as running or problem solving, empathising will drain energy and resources. Further, and similar to the difference in energy expenditure between running and jogging, there are distinct types of empathy that each have different energy costs. === Types of empathy === Research makes a clear distinction between two types of empathy, affective and cognitive empathy. Affective empathy occurs when a person has an emotional reaction to someone else's mental state (Shao et al., 2024). This occurs because (unsure about this next part go through and update accordingly) ''unconscious processes interpret the social stimuli from another person leading to a physiological and cognitive response. These processes may interpret stimuli from any of the five senses.'' Cognitive empathy is when a person understands someone else's feelings or emotions (Cui et al., 2022). * Integrating the two - "affective component is the essence of empathy, and the cognitive component is a process through which that essence comes into being" (Bošnjaković & Radionov, 2018) ** (Author's note: change quote into paraphrasing) * === Empathy or compassion fatigue === * Distinction between empathy and compassion fatigue * Compassion fatigue is closely related to Secondary traumatic stress usually found in healthcare professionals (Figley, 2002) ** Compassion fatigue is to Empathy fatigue as emotional exhaustion is to burnout * Make it clear that Aelin is experiencing Compassion Fatigue vs Empathy fatigue (they are the same but different) (Author's note: include an update to the scenario here!) '''Empathy fatigue questions'''. Choose your answers and click "Submit": <quiz display=simple> {Surface acting is when the observer actually feels the emotion: |type="()"} - True + False {Compassion fatigue and empathy fatigue are synonymous: |type="()"} - True + False </quiz> {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' * Under the JD-R model, a sustained empathic response is a long-term job demand. * Under the CORT model, a sustained empathic response uses energy to protect existing resources. As resources become low and burnout sets in, cognitive focus narrows leading to empathy fatigue. {{RoundBoxBottom}} == Emotional labor as a mediating factor between empathy fatigue and emotional exhaustion == As a society there are expected [https://en.wikipedia.org/wiki/Social_norm social norms] in behaviour when expressing emotion, regardless if someone is conscious of the process of matching these norms or not. For example, laughing during an emotional speech at a funeral would be considered a rude or offensive expression from the other attendees. As social creatures, humans need a way to regulate socially appropriate emotional expression. This is the essence of Emotional labor: "the regulation of felt and expressed emotions (whether effortful or not) to match these emotional display rules with the objective of fulfilling organizational goals" (Vial & Cowgill, 2022) What is emotional labor (Hochschild, 2012). Theories behind emotional labor involve acting, and an extension of acting known as emotional contagion. === Types of acting === * Surface acting ** This is where emotive dissonance comes into play ** Leads to emotional exhaustion (Kenworthy et al., 2014) ** "Should" language - in this situation, this is what I ''should'' feel * Deep acting ** People who 'deep act', as in genuinely change the way they feel (match the feelings they have with the feelings that they need to display) often report positive relationships with emotional labor ** Actual language - in this situation, this is what I ''actually'' feel Discuss the differences in gender norms and expectations in relationships {| class="wikitable" style="margin: auto;" |- ! !! Deep acting !! Surface acting |- | '''Emotional dissonance''' || || X |- | '''Not known to others''' || Hidden area || Unknown |} === Emotional contagion === Emotional contagion is the idea that emotions are contagious. as the theory to moderate surface/deep acting * Emotional contagion (Panksepp and Lahvis, 2011) ** emotions can spread (expand, combines empathy and acting) ** “the tendency to automatically mimic and synchronize expressions, vocalizations, postures, and movements with those of another person’s and, consequently, to converge emotionally.” (Elaine Hatfield, 1993) * Affective process theory looks at unifying the mechanisms of emotional contagion into three types. Convergent linkage, Divergent linkage, and Complementary linkage (Elfenbein, 2014). Storytelling is a way to promote position emotions (Wei et al., 2024). === The mediating effect of emotional labor === Surface acting does not mediate the effect (high surface acting leads to emotional exhaustion) whereas the inverse is true for deep acting (Wrobel, 2013) Emotional Expression and Emotional Processing (Hoyt et al., 2024) {{RoundBoxTop|theme=3}} '''What role does emotional labor play in emotional exhaustion''' * In relation to the CORT and JD-R theories, emotional labor acts as a job demand and expenditure of resources to protect existing resources {{RoundBoxBottom}} == Strategies for combating empathy fatigue and emotional fatigue == * Talk about emotion regulation (Gross, 1998) and mood regulation strategies as a way to deal with coping (Wrobel, 2013) (Author's note: need more research/references for this section) * Talk about breaks ** Rest and downtime from not performing emotional labor regenerates * How do we reduce the amount of emotional labor we're performing (Jeung et al., 2018) ** Communication/boundaries ** Reducing surface acting (allowing ourselves to feel how we actually feel and reduce emotive dissonance) '''Table 1. Treatment phases of clinical burnout (based on Table 2 from van Dam, A. 2021)''' {| class="wikitable" style="margin: auto; |- ! Phase !! Treatment goal !! Therapeutic interventions |- | '''Phase 1: Crisis''' || Recognition of the patient that the problems are serious, and that serious action is required || Psychoeducation Sick leave Dropping domestic tasks and social obligations Inform social network |- | '''Phase 2: Recovery''' || Recovery of the (physiological) stress system to normal allostatic stress levels || Registration of stress and activities Relaxtion exercises Mindfulness healthy lifestyle advices Graded exercise Gradually resume activities Psychoeducation social network and employer Gradual return to work |- | '''Phase 3: Prevention, learning from the past''' || Acquiring insight and skills to prevent relapse in clinical burnout || Analysing stressful situations Analysing dysfunctional thought patterns Learning new coping skills Learning social skills Job crafting Time management Making choices regarding career and personal life |- | '''Phase 4: Post burnout growth''' || Improving sustainable quality of life. || Setting priorities regarding quality of life and interpersonal relationships. |} === Strategies for the workplace === * Discuss types of strategies that are available in the workplace (organisation models) * SCARF model (Aplin-Houtz, 2025) ** Link back to COR and JDR models e.g. praise could be a job resource vs criticism could be a job demand *Reduce workload to reduce emotional exhaustion, and by extension reduce emotional fatigue. {{RoundBoxTop|theme=3}} '''How does understanding emotional labor and empathy fatigue work to reduce the effects of emotional exhaustion''' *Psychoeducation has been shown to reduce the effects of emotional labor and empathy fatigue on emotional exhaustion {{RoundBoxBottom}} ==Conclusion== * Summarise what empathy fatigue is, and why people become emotionally exhausted (+ the role of emotional labor) * Summarise the symptoms and how to recognise them in yourself and someone else * Emphasise and summarise psychological strategies to support someone going through emotional exhaustion * Summarise ways to prevent emotional exhaustion from happening * Overall message is: *# What is emotional exhaustion *# how do you recognise it *# how do you treat it *# how do you prevent it * Link back to the focus questions ==See also== * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Metacognition_and_emotional_regulation&oldid=2756264 Metacognition and emotional regulation] (Book chapter, 2026) {{ic|Use an internal link style as shown in Tutorial 2}} * [[wikipedia:Conservation_of_resources_theory|Conservation of resources theory]] (Wikipedia) * [[wikipedia:Job_demands-resources_model|Job demands-resources model]] (Wikipedia) ==References== {{Hanging indent|1= Aplin-Houtz, M. (2025). Unraveling the SCARF: How dimensions of the SCARF model influence the relationship between workplace exclusion and turnover intentions. Journal of Management & Organization, 31(6), 2699–2725. https://doi.org/10.1017/jmo.2025.10026 Bošnjaković, J., & Radionov, T. (2018). Empathy: Concepts, Theories and Neuroscientific Basis. Alcoholism (Zagreb), 54(2), 123–150. https://doi.org/10.20471/de..2018.54.02.04 Chan, F. M. (2026). The State of Workplace Burnout 2026: The Workplace Divide. Infinite Potential, Sydney, Australia. https://www.ahri.com.au/wp-content/uploads/The-State-of-Workplace-Burnout-2026-Final.pdf Cui, F., Huang, X., Jing, Y., Luo, Y., Liu, J., & Gu, R. (2022). How resource sharing resists scarcity: the role of cognitive empathy and its neurobiological mechanisms. Cerebral Cortex (New York, N.Y. 1991), 32(23), 5330–5342. https://doi.org/10.1093/cercor/bhac017 De Beer, L. T., van der Vaart, L., Escaffi-Schwarz, M., De Witte, H., & Schaufeli, W. B. (2024). Maslach Burnout Inventory - General Survey: A Systematic Review and Meta-Analysis of Measurement Properties. European journal of psychological assessment : official organ of the European Association of Psychological Assessment, 40(5), 360-375. https://doi.org/10.1027/1015-5759/a000797 Demerouti, E., Bakker, A. B., Nachreiner, F., & Schaufeli, W. B. (2001). The Job Demands-Resources Model of Burnout. Journal of Applied Psychology, 86(3), 499–512. https://doi.org/10.1037/0021-9010.86.3.499 Figley, C. R. (2002). Compassion fatigue: Psychotherapists’ chronic lack of self care. Journal of Clinical Psychology, 58(11), 1433–1441. https://doi.org/10.1002/jclp.10090 Grandey, A. A. (2003). When ‘the show must go on’: surface acting and deep acting as determinants of emotional exhaustion and peer-rated service delivery. Academy of Management Journal, 46(1), 86–96. https://doi.org/10.5465/30040678 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 Halbesleben, J. R. B., Neveu, J.-P., Paustian-Underdahl, S. C., & Westman, M. (2014). Getting to the “COR”: Understanding the Role of Resources in Conservation of Resources Theory. Journal of Management, 40(5), 1334–1364. https://doi.org/10.1177/0149206314527130 Hatfield, E., Cacioppo, J. T., & Rapson, R. L. (1993). Emotional Contagion. Current Directions in Psychological Science : A Journal of the American Psychological Society, 2(3), 96–99. https://doi.org/10.1111/1467-8721.ep10770953 Hollebeek, L. D., Hammedi, W., & Sprott, D. E. (2023). Consumer engagement, stress, and conservation of resources theory: A review, conceptual development, and future research agenda. Psychology & marketing, 40(5), 926-937. https://doi.org/10.1002/mar.21807 Hobfoll, S. E. (1989). Conservation of Resources: A New Attempt at Conceptualizing Stress. The American Psychologist, 44(3), 513–524. https://doi.org/10.1037/0003-066X.44.3.513 Hochschild, A. R. (2012). The managed heart : commercialization of human feeling (Updated, with a new preface.). University of California Press. Jeung, D.-Y., Kim, C., & Chang, S.-J. (2018). Emotional Labor and Burnout: A Review of the Literature. Yonsei Medical Journal, 59(2), 187–193. https://doi.org/10.3349/ymj.2018.59.2.187 Lavelle, J. J., Rupp, D. E., Herda, D. N., Pandey, A., & Lauck, J. R. (2021). Customer Injustice and Employee Performance: Roles of Emotional Exhaustion, Surface Acting, and Emotional Demands–Abilities Fit. Journal of Management, 47(3), Article 0149206319869426. https://doi.org/10.1177/0149206319869426 Lloyd, C., & King, R. (2004). A survey of burnout among Australian mental health occupational therapists and social workers. Social Psychiatry and Psychiatric Epidemiology, 39(9), 752-757. https://doi.org/https://doi.org/10.1007/s00127-004-0808-7 Kenworthy, J., Fay, C., Frame, M., & Petree, R. (2014). A meta-analytic review of the relationship between emotional dissonance and emotional exhaustion. Journal of Applied Social Psychology, 44(2), 94-105. https://doi.org/https://doi.org/10.1111/jasp.12211 Martingano, A. J., Stosic, M. D., & Konrath, S. (2026). Different empathy types show opposing associations with burnout: systematic review and meta-analyses. Psychology, Health & Medicine, 31(3), 798-829. https://doi.org/10.1080/13548506.2025.2591859 Michielsen, H. J., Willemsen, T. M., Croon, M. A., de Vries, J., & Van Heck, G. L. (2004). Determinants of general fatigue and emotional exhaustion: a prospective study. Psychology & health, 19(2), 223-235. https://doi.org/10.1080/08870440310001627135 Panksepp, J. B., & Lahvis, G. P. (2011). Rodent empathy and affective neuroscience. Neuroscience and Biobehavioral Reviews, 35(9), 1864–1875. https://doi.org/10.1016/j.neubiorev.2011.05.013 Pooladvand, S., & Hasanzadeh, S. (2023). Impacts of Stress on Workers’ Risk-Taking Behaviors: Cognitive Tunneling and Impaired Selective Attention. Journal of Construction Engineering and Management, 149(8), 04023060. https://doi.org/doi:10.1061/JCEMD4.COENG-13339 Riess, H. (2017). The Science of Empathy. Journal of Patient Experience, 4(2), 74–77. https://doi.org/10.1177/2374373517699267 Schaufeli, W. B. (2017). Applying the job demands-resources model: a “how to” guide to measuring and tackling work engagement and burnout. Organizational Dynamics, 46(2), 120–132. https://doi.org/10.1016/j.orgdyn.2017.04.008 Schneiderman, N., Ironson, G., & Siegel, S. D. (2005). Stress and Health: Psychological, Behavioral, and Biological Determinants. Annual Review of Clinical Psychology, 1(Volume 1, 2005), 607-628. https://doi.org/https://doi.org/10.1146/annurev.clinpsy.1.102803.144141 Seidler, A., Thinschmidt, M., Deckert, S., Then, F., Hegewald, J., Nieuwenhuijsen, K., & Riedel-Heller, S. G. (2014). The role of psychosocial working conditions on burnout and its core component emotional exhaustion – a systematic review. Journal of Occupational Medicine and Toxicology, 9(1), 10. https://doi.org/10.1186/1745-6673-9-10 Shao, M., Li, L., Li, X., Wei, Z., Wang, J., Hong, M., Liu, X., & Meng, J. (2024). The effect of top-down attention on empathy fatigue. Cerebral Cortex (New York, N.Y. 1991), 34(1), Article bhad441. https://doi.org/10.1093/cercor/bhad441 van Dam, A. (2021). A clinical perspective on burnout: diagnosis, classification, and treatment of clinical burnout. European Journal of Work and Organizational Psychology, 30(5), 732-741. https://doi.org/10.1080/1359432X.2021.1948400 Vial, A. C., & Cowgill, C. M. (2022). Heavier Lies Her Crown: Gendered Patterns of Leader Emotional Labor and Their Downstream Effects. Frontiers in Psychology, 13, 849566. https://doi.org/https://doi.org/10.3389/fpsyg.2022.849566 Wharton, A. S. (2009). The Sociology of Emotional Labor. Annual Review of Sociology, 35(1), 147–165. https://doi.org/10.1146/annurev-soc-070308-115944 Wrobel, M. (2013). Can empathy lead to emotional exhaustion in teachers? The mediating role of emotional labor. International Journal of Occupational Medicine and Environmental Health, 26(4), 581–592. https://doi.org/10.2478/s13382-013-0123-1 }} ==External links== Provide [[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: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Burnout]] [[Category:Motivation and emotion/Book/Empathy]] md8wuf3o2s2fhuubyfzs4lgy2svnquy 2844529 2844528 2026-11-04T23:10:24Z U3143751 3106785 /* Conservation of resources stress model */ 2844529 wikitext text/x-wiki {{title|Empathy fatigue and emotional exhaustion:<br>How does sustained empathic engagement contribute to emotional exhaustion?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}} [[File:Exhausted smartphone user.JPG|right|thumb|150px|'''Figure 1'''. An emotionally exhausted Aelin at the end of the work day]] ; Scenario Aelin, a 23-year-old social worker employed at headspace {{ic|Link to the Wikipedia article to help explain what this term means}}, works with approximately 20 {{ic|Explain what sort of}}Imagine you are only 23 years old and are working as a social worker at headspace. You see around 20 clients each week, many of whom have experienced rough living conditions. During your first six months in this role, you felt highly motivated and excited to help people in need. You are an excellent active listener and can easily remember details about your clients Live’s. You are also good at recognising signs of distress in your client’s body language and adjust yours in order to show warmth and understanding. As you progress in your role, you take on more cases that involve complex past traumas. The workload becomes too much for you to get done at work, but you don’t want your clients to miss out, so you take your work home to complete it in the evenings. Your workload increases again and you now start doing work on your weekends as well. After several months of working this way, you start to notice you feel exhausted every morning before you have even started work. You then start to notice you’re responding to people at work differently. Instead of your usual energised active listening response you feel frustrated, empathetic and detached. After work you are too drained to socialise with friends. This begins to worry you. What if you won’t be able to continue working as a social worker? {{RoundBoxBottom}} Aelin is experiencing '''emotional exhaustion''' from a sustained empathic response. Due to the nature of Aelin's work, they are constantly engaged in an empathic response. Empathic responses require cognitive and physical resources to sustain. Because Aelin is taking their work home and not resting on weekends, they do not have a chance to rest. It is important for Aelin to recognise what is happening and why. Not actioning the problem has led to empathy fatigue and emotional exhaustion. Emotional exhaustion is one of the key aspects of [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FEmpathy_fatigue_and_emotional_exhaustion&diff=2840411&oldid=2832859 burnout], a long-term condition of existing in a constant state of high stress. Psychological science can support Aelin in recognising and mapping the symptoms they are experiencing, understand coping strategies to minimise the current and future effects to improve their wellbeing. Understanding the symptoms will also support their role as a social worker, providing more tools for the people they support on a day-to-day basis. {{RoundBoxTop|theme=3}} '''Focus questions''' * Why do people become emotionally exhausted? * What is the relationship between empathy fatigue and emotional exhaustion? * What role does emotional labor play in emotional exhaustion? * How does understanding emotional labor and empathy fatigue work to reduce the effects of emotional exhaustion? {{RoundBoxBottom}} == Emotional exhaustion == To understand emotional exhaustion, it will be important to first understand burnout. Burnout is a long-term condition of existing in a constant state of high stress and affects millions of people annually. In the early 2000s, a survey from Australia found that more than 50% of social workers and occupational workers reported high levels of burnout symptoms (Llyod and King, 2004) and a recent report from 2026 found that more than one third of all workers in Australia were experiencing burnout (Chan, 2026). Being in a state of high stress for short periods of time is a useful adaptation for humans, so much so that it has been given a name, [[wikipedia:Eustress|Eustress]]. However, as a long-term condition it leads to individual health issues and can be linked to cardiovascular disease, increased vulnerability to upper respiratory diseases, and chronic inflammation (Schneiderman et al., 2005), and cognitive issues such as [[wikipedia:Inattentional_blindness|cognitive tunneling]] and impaired selective attention (Pooladvand and Hasanzadeh, 2023). Burnout can be defined across three dimensions: emotional exhaustion, [[wikipedia:Depersonalization|depersonalisation]], and reduced feelings of personal accomplishment (Maslach, 1993, as cited in Martingano et al., 2026). Emotional exhaustion may be considered the 'core' component of burnout and refers to the depletion of emotional resources, leading to feelings of overwhelm, overextension, and exhaustion (Martingano et al., 2026; De Beer et al., 2024; Seidler et al., 2014). As a major factor, its symptoms are similar to burnout and stress. Although antecedents of burnout are typically agreed upon (Bakker et al., 2014) and divided into individual (e.g. self-efficacy) and situational categories (e.g. work load), there are many theories that attempt to explain why emotional exhaustion occurs. Two of these theories with good evidence are the [[wikipedia:Conservation_of_resources_theory|Conservation of resources stress model]] (Hobfoll, 1989) and the [[wikipedia:Job_demands-resources_model|Job Demands-Resources model]] (Demerouti et al., 2001). === Conservation of resources stress model === The Conservation of resources stress model (CORT; Hobfall, 1989) is a theory that works to understand and explain why stress occurs. Under the model, humans are motivated to maintain, pursue, or attain resources. Stress, defined as "an individual's perceived pressure or strain" (Hollebeek et al., 2023 p.928), occurs when these resources are threatened or lost, or gained at a net loss. As noted above, resources aren't strictly tied to being physical objects. They can also be emotional states, conditions, or even ideas. For example, someone may place value on enjoying a calm, quiet Sunday morning at home. If their neighbour starts using power tools loudly next door, it would disrupt this state and induce stress in the individual. The CORT model focuses primarily on two principles. The first principle is that there is more harm in losing resources than there is in gaining them back (Halbesleben et al., 2014). This is known as loss salience, and some research suggests that the pain of loss is felt twice as strong as the pleasure of gain (Thaler, 2000). Looking at our previous example, imagine that the person who enjoys a quiet Sunday morning at home equally enjoys the sound of a loud car engine. Even though the person enjoys the sound of car engines, if the neighbour started revving their car loudly instead of using power tools, it would still disrupt their desired state of quiet and lead to an increase of stress. The second principle is that people must invest existing resources in order to gain or protect resources (Halbesleben et al., 2014). This resource investment includes recovering from resource loss. Coping is how people manage or overcome a difficult situation, such as protecting or recovering from resource loss. Coping can be broken down into different strategies that people use, but not all strategies are equal (Ito and Brotheridge, 2003). They are divided into two categories: control or escape strategies. Control strategies are considered better at reducing the strain from stress and consist of positive thinking, direct action, and help seeking (Havlovic & Keenan, 1991, as cited in Ito and Brotheridge, 2003). The most useful control strategies incorporate a person's [[locus of control]], emphasise seeking support from co-workers, and highlight the importance of autonomy in decision making. Alarcon (2011) conducted a meta-analysis of 231 studies in a work setting that used the [[wikipedia:Maslach_Burnout_Inventory|Maslach Burnout Inventory]] (MBI) to review the relationship between these principles and burnout to confirm their validity. Although the study had no specific mention of demographics, it was noted that multiple translations of the MBI were used including Dutch (k=51), German (k=7), Italian (k=2), and French (k=2). The findings indicate that workload is positively related to emotional exhaustion (''ρ'' = .49, ''k'' = 86, ''N'' = 51,529) and that control (''ρ'' = − .26, ''k'' = 56, ''N'' = 33,297) and autonomy (''ρ'' = − .24, ''k'' = 34, ''N'' = 17,165) were each negatively associated with emotional exhaustion. The difference of ''ρ'' values is consistent with Thaler (2000) and supports that the benefit of gaining is approximately half that of losing resources. === Job demands-resources model === Another model that attempts to understand why people become emotionally exhausted is the Job Demands-Resources model (JD-R; Demerouti et al., 2001). The JD-R model discusses two concepts: job demands and job resources. Demands and resources are both related to the physical, social, or psychological aspects of a job, but differ in their impact on a person. Job demands require physical or mental effort and exhaust a person's energy supply. Job resources typically support achieving goals, reduce the effort cost of job demands, and stimulate personal growth (Richter and Hacker, 1998, as cited in Demerouti et al., 2001). For example, an unsustainable workload would be considered a job demand, and receiving help and support from others would be considered a resource. The JD-R has two primary processes for how demands and resources interact with one another (Schaufeli, 2017). The stress process occurs when job demands are high and resources are low. As stress increases strain and pressure on an individual, it leads to the exhaustion of their energy eventually, eventually leading to emotional exhaustion. The second process is the motivational process. This process occurs when there is an excess of job resources. Crucially, the motivational process does not require low job demands. With enough job resources, a person will be energised and engaged, reframing job demands as a challenge to overcome. The JD-R model has had extensive research to confirm its validity. A meta-analysis by Lesener et al. (2019) looked at 74 longitudinal studies of the JD-R with a strong average sample size (n=450) against the assumptions as described by Schaufeli (2017). Although most of the studies appear to come from WEIRD countries (k=60) there were a good number from Asia and Africa (k=13) and a mix of genders. The study confirmed that across the literature, job resources predict burnout negatively (''β'' = −.17; ''p'' < .001) and job demands predict burnout over time (''β'' = .10; ''p'' < .001). Although job resources seem like a cure-all for demands, it's important to note that they do not retroactively treat emotional exhaustion. Under the JD-R model, burnout must be reduced by lowering job demands (Gynning et al., 2025). However, once someone has recovered, it is possible to increase job resources alongside job demands to prevent future burnout and support [[wikipedia:Work_engagement|work engagement]]. Work engagement (engagement) has been discussed as the opposite of burnout (Bakker et al., 2014). Engagement is how people express themselves when performing a role. A role being a pattern of behaviour that a person goes through based on the social expectations of that role, similar to how an actor takes on a role for a movie. For example, when attending a typical restaurant, it is socially expected that a person will engage with the waiter and provide them with their meal order to take to the kitchen. This person is performing the script of a restaurant patron. In the context of a workplace, an employee with high engagement may appear to be energised, enthusiastic, and become absorbed in their work losing track of time (Bakkar et al, 2014). {{RoundBoxTop|theme=3}} '''Why do we feel emotionally exhausted''' Under the JD-R and CORT models, emotional exhaustion occurs when the demand cost outweighs the resources available for an extended period of time, eventually depleting someone's energy reserves. Aelin is experiencing burnout because they increased their job demands by taking on more clients, and has experienced a loss in resources by reducing their free time and social support network. Under these models, Aelin should consider reducing their workload to begin recovering and, as part of recovery, increasing available resources such as asking for help through social support and considering ways to increase autonomy in their role. This will lead to positive outcomes - reducing the likelihood of future exhaustion and increasing engagement (Bakker et al., 2014; Halbesleben et al., 2014). {{RoundBoxBottom}} == Empathy fatigue == You may be familiar with the term [[empathy]]. It is often described in terms of supporting others. For example, when someone close to you experiences a loss and is grieving, you may empathise with them and experience a negative emotion akin to grief too. However, empathy is not exclusive to so called negative emotions. For example, compersion is a positive empathic response when a loved one experiences joy. Parents may experience compersion when something joyful happens to their child. At its core, empathy is the individual ability to imagine or understand another person's feelings and/or emotions (Riess, 2017; Shao et al., 2024) and has four main components (Davis, 1983, as cited in Delgado et al., 2023): * Empathic concern is xyz * Fantasy is xyz * Perspective taking is xyz * Personal distress is xyz Much like physical or cognitive processes, such as running or problem solving, empathising will drain energy and resources. Further, and similar to the difference in energy expenditure between running and jogging, there are distinct types of empathy that each have different energy costs. === Types of empathy === Research makes a clear distinction between two types of empathy, affective and cognitive empathy. Affective empathy occurs when a person has an emotional reaction to someone else's mental state (Shao et al., 2024). This occurs because (unsure about this next part go through and update accordingly) ''unconscious processes interpret the social stimuli from another person leading to a physiological and cognitive response. These processes may interpret stimuli from any of the five senses.'' Cognitive empathy is when a person understands someone else's feelings or emotions (Cui et al., 2022). * Integrating the two - "affective component is the essence of empathy, and the cognitive component is a process through which that essence comes into being" (Bošnjaković & Radionov, 2018) ** (Author's note: change quote into paraphrasing) * === Empathy or compassion fatigue === * Distinction between empathy and compassion fatigue * Compassion fatigue is closely related to Secondary traumatic stress usually found in healthcare professionals (Figley, 2002) ** Compassion fatigue is to Empathy fatigue as emotional exhaustion is to burnout * Make it clear that Aelin is experiencing Compassion Fatigue vs Empathy fatigue (they are the same but different) (Author's note: include an update to the scenario here!) '''Empathy fatigue questions'''. Choose your answers and click "Submit": <quiz display=simple> {Surface acting is when the observer actually feels the emotion: |type="()"} - True + False {Compassion fatigue and empathy fatigue are synonymous: |type="()"} - True + False </quiz> {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' * Under the JD-R model, a sustained empathic response is a long-term job demand. * Under the CORT model, a sustained empathic response uses energy to protect existing resources. As resources become low and burnout sets in, cognitive focus narrows leading to empathy fatigue. {{RoundBoxBottom}} == Emotional labor as a mediating factor between empathy fatigue and emotional exhaustion == As a society there are expected [https://en.wikipedia.org/wiki/Social_norm social norms] in behaviour when expressing emotion, regardless if someone is conscious of the process of matching these norms or not. For example, laughing during an emotional speech at a funeral would be considered a rude or offensive expression from the other attendees. As social creatures, humans need a way to regulate socially appropriate emotional expression. This is the essence of Emotional labor: "the regulation of felt and expressed emotions (whether effortful or not) to match these emotional display rules with the objective of fulfilling organizational goals" (Vial & Cowgill, 2022) What is emotional labor (Hochschild, 2012). Theories behind emotional labor involve acting, and an extension of acting known as emotional contagion. === Types of acting === * Surface acting ** This is where emotive dissonance comes into play ** Leads to emotional exhaustion (Kenworthy et al., 2014) ** "Should" language - in this situation, this is what I ''should'' feel * Deep acting ** People who 'deep act', as in genuinely change the way they feel (match the feelings they have with the feelings that they need to display) often report positive relationships with emotional labor ** Actual language - in this situation, this is what I ''actually'' feel Discuss the differences in gender norms and expectations in relationships {| class="wikitable" style="margin: auto;" |- ! !! Deep acting !! Surface acting |- | '''Emotional dissonance''' || || X |- | '''Not known to others''' || Hidden area || Unknown |} === Emotional contagion === Emotional contagion is the idea that emotions are contagious. as the theory to moderate surface/deep acting * Emotional contagion (Panksepp and Lahvis, 2011) ** emotions can spread (expand, combines empathy and acting) ** “the tendency to automatically mimic and synchronize expressions, vocalizations, postures, and movements with those of another person’s and, consequently, to converge emotionally.” (Elaine Hatfield, 1993) * Affective process theory looks at unifying the mechanisms of emotional contagion into three types. Convergent linkage, Divergent linkage, and Complementary linkage (Elfenbein, 2014). Storytelling is a way to promote position emotions (Wei et al., 2024). === The mediating effect of emotional labor === Surface acting does not mediate the effect (high surface acting leads to emotional exhaustion) whereas the inverse is true for deep acting (Wrobel, 2013) Emotional Expression and Emotional Processing (Hoyt et al., 2024) {{RoundBoxTop|theme=3}} '''What role does emotional labor play in emotional exhaustion''' * In relation to the CORT and JD-R theories, emotional labor acts as a job demand and expenditure of resources to protect existing resources {{RoundBoxBottom}} == Strategies for combating empathy fatigue and emotional fatigue == * Talk about emotion regulation (Gross, 1998) and mood regulation strategies as a way to deal with coping (Wrobel, 2013) (Author's note: need more research/references for this section) * Talk about breaks ** Rest and downtime from not performing emotional labor regenerates * How do we reduce the amount of emotional labor we're performing (Jeung et al., 2018) ** Communication/boundaries ** Reducing surface acting (allowing ourselves to feel how we actually feel and reduce emotive dissonance) '''Table 1. Treatment phases of clinical burnout (based on Table 2 from van Dam, A. 2021)''' {| class="wikitable" style="margin: auto; |- ! Phase !! Treatment goal !! Therapeutic interventions |- | '''Phase 1: Crisis''' || Recognition of the patient that the problems are serious, and that serious action is required || Psychoeducation Sick leave Dropping domestic tasks and social obligations Inform social network |- | '''Phase 2: Recovery''' || Recovery of the (physiological) stress system to normal allostatic stress levels || Registration of stress and activities Relaxtion exercises Mindfulness healthy lifestyle advices Graded exercise Gradually resume activities Psychoeducation social network and employer Gradual return to work |- | '''Phase 3: Prevention, learning from the past''' || Acquiring insight and skills to prevent relapse in clinical burnout || Analysing stressful situations Analysing dysfunctional thought patterns Learning new coping skills Learning social skills Job crafting Time management Making choices regarding career and personal life |- | '''Phase 4: Post burnout growth''' || Improving sustainable quality of life. || Setting priorities regarding quality of life and interpersonal relationships. |} === Strategies for the workplace === * Discuss types of strategies that are available in the workplace (organisation models) * SCARF model (Aplin-Houtz, 2025) ** Link back to COR and JDR models e.g. praise could be a job resource vs criticism could be a job demand *Reduce workload to reduce emotional exhaustion, and by extension reduce emotional fatigue. {{RoundBoxTop|theme=3}} '''How does understanding emotional labor and empathy fatigue work to reduce the effects of emotional exhaustion''' *Psychoeducation has been shown to reduce the effects of emotional labor and empathy fatigue on emotional exhaustion {{RoundBoxBottom}} ==Conclusion== * Summarise what empathy fatigue is, and why people become emotionally exhausted (+ the role of emotional labor) * Summarise the symptoms and how to recognise them in yourself and someone else * Emphasise and summarise psychological strategies to support someone going through emotional exhaustion * Summarise ways to prevent emotional exhaustion from happening * Overall message is: *# What is emotional exhaustion *# how do you recognise it *# how do you treat it *# how do you prevent it * Link back to the focus questions ==See also== * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Metacognition_and_emotional_regulation&oldid=2756264 Metacognition and emotional regulation] (Book chapter, 2026) {{ic|Use an internal link style as shown in Tutorial 2}} * [[wikipedia:Conservation_of_resources_theory|Conservation of resources theory]] (Wikipedia) * [[wikipedia:Job_demands-resources_model|Job demands-resources model]] (Wikipedia) ==References== {{Hanging indent|1= Aplin-Houtz, M. (2025). Unraveling the SCARF: How dimensions of the SCARF model influence the relationship between workplace exclusion and turnover intentions. Journal of Management & Organization, 31(6), 2699–2725. https://doi.org/10.1017/jmo.2025.10026 Bošnjaković, J., & Radionov, T. (2018). Empathy: Concepts, Theories and Neuroscientific Basis. Alcoholism (Zagreb), 54(2), 123–150. https://doi.org/10.20471/de..2018.54.02.04 Chan, F. M. (2026). The State of Workplace Burnout 2026: The Workplace Divide. Infinite Potential, Sydney, Australia. https://www.ahri.com.au/wp-content/uploads/The-State-of-Workplace-Burnout-2026-Final.pdf Cui, F., Huang, X., Jing, Y., Luo, Y., Liu, J., & Gu, R. (2022). How resource sharing resists scarcity: the role of cognitive empathy and its neurobiological mechanisms. Cerebral Cortex (New York, N.Y. 1991), 32(23), 5330–5342. https://doi.org/10.1093/cercor/bhac017 De Beer, L. T., van der Vaart, L., Escaffi-Schwarz, M., De Witte, H., & Schaufeli, W. B. (2024). Maslach Burnout Inventory - General Survey: A Systematic Review and Meta-Analysis of Measurement Properties. European journal of psychological assessment : official organ of the European Association of Psychological Assessment, 40(5), 360-375. https://doi.org/10.1027/1015-5759/a000797 Demerouti, E., Bakker, A. B., Nachreiner, F., & Schaufeli, W. B. (2001). The Job Demands-Resources Model of Burnout. Journal of Applied Psychology, 86(3), 499–512. https://doi.org/10.1037/0021-9010.86.3.499 Figley, C. R. (2002). Compassion fatigue: Psychotherapists’ chronic lack of self care. Journal of Clinical Psychology, 58(11), 1433–1441. https://doi.org/10.1002/jclp.10090 Grandey, A. A. (2003). When ‘the show must go on’: surface acting and deep acting as determinants of emotional exhaustion and peer-rated service delivery. Academy of Management Journal, 46(1), 86–96. https://doi.org/10.5465/30040678 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 Halbesleben, J. R. B., Neveu, J.-P., Paustian-Underdahl, S. C., & Westman, M. (2014). Getting to the “COR”: Understanding the Role of Resources in Conservation of Resources Theory. Journal of Management, 40(5), 1334–1364. https://doi.org/10.1177/0149206314527130 Hatfield, E., Cacioppo, J. T., & Rapson, R. L. (1993). Emotional Contagion. Current Directions in Psychological Science : A Journal of the American Psychological Society, 2(3), 96–99. https://doi.org/10.1111/1467-8721.ep10770953 Hollebeek, L. D., Hammedi, W., & Sprott, D. E. (2023). Consumer engagement, stress, and conservation of resources theory: A review, conceptual development, and future research agenda. Psychology & marketing, 40(5), 926-937. https://doi.org/10.1002/mar.21807 Hobfoll, S. E. (1989). Conservation of Resources: A New Attempt at Conceptualizing Stress. The American Psychologist, 44(3), 513–524. https://doi.org/10.1037/0003-066X.44.3.513 Hochschild, A. R. (2012). The managed heart : commercialization of human feeling (Updated, with a new preface.). University of California Press. Jeung, D.-Y., Kim, C., & Chang, S.-J. (2018). Emotional Labor and Burnout: A Review of the Literature. Yonsei Medical Journal, 59(2), 187–193. https://doi.org/10.3349/ymj.2018.59.2.187 Lavelle, J. J., Rupp, D. E., Herda, D. N., Pandey, A., & Lauck, J. R. (2021). Customer Injustice and Employee Performance: Roles of Emotional Exhaustion, Surface Acting, and Emotional Demands–Abilities Fit. Journal of Management, 47(3), Article 0149206319869426. https://doi.org/10.1177/0149206319869426 Lloyd, C., & King, R. (2004). A survey of burnout among Australian mental health occupational therapists and social workers. Social Psychiatry and Psychiatric Epidemiology, 39(9), 752-757. https://doi.org/https://doi.org/10.1007/s00127-004-0808-7 Kenworthy, J., Fay, C., Frame, M., & Petree, R. (2014). A meta-analytic review of the relationship between emotional dissonance and emotional exhaustion. Journal of Applied Social Psychology, 44(2), 94-105. https://doi.org/https://doi.org/10.1111/jasp.12211 Martingano, A. J., Stosic, M. D., & Konrath, S. (2026). Different empathy types show opposing associations with burnout: systematic review and meta-analyses. Psychology, Health & Medicine, 31(3), 798-829. https://doi.org/10.1080/13548506.2025.2591859 Michielsen, H. J., Willemsen, T. M., Croon, M. A., de Vries, J., & Van Heck, G. L. (2004). Determinants of general fatigue and emotional exhaustion: a prospective study. Psychology & health, 19(2), 223-235. https://doi.org/10.1080/08870440310001627135 Panksepp, J. B., & Lahvis, G. P. (2011). Rodent empathy and affective neuroscience. Neuroscience and Biobehavioral Reviews, 35(9), 1864–1875. https://doi.org/10.1016/j.neubiorev.2011.05.013 Pooladvand, S., & Hasanzadeh, S. (2023). Impacts of Stress on Workers’ Risk-Taking Behaviors: Cognitive Tunneling and Impaired Selective Attention. Journal of Construction Engineering and Management, 149(8), 04023060. https://doi.org/doi:10.1061/JCEMD4.COENG-13339 Riess, H. (2017). The Science of Empathy. Journal of Patient Experience, 4(2), 74–77. https://doi.org/10.1177/2374373517699267 Schaufeli, W. B. (2017). Applying the job demands-resources model: a “how to” guide to measuring and tackling work engagement and burnout. Organizational Dynamics, 46(2), 120–132. https://doi.org/10.1016/j.orgdyn.2017.04.008 Schneiderman, N., Ironson, G., & Siegel, S. D. (2005). Stress and Health: Psychological, Behavioral, and Biological Determinants. Annual Review of Clinical Psychology, 1(Volume 1, 2005), 607-628. https://doi.org/https://doi.org/10.1146/annurev.clinpsy.1.102803.144141 Seidler, A., Thinschmidt, M., Deckert, S., Then, F., Hegewald, J., Nieuwenhuijsen, K., & Riedel-Heller, S. G. (2014). The role of psychosocial working conditions on burnout and its core component emotional exhaustion – a systematic review. Journal of Occupational Medicine and Toxicology, 9(1), 10. https://doi.org/10.1186/1745-6673-9-10 Shao, M., Li, L., Li, X., Wei, Z., Wang, J., Hong, M., Liu, X., & Meng, J. (2024). The effect of top-down attention on empathy fatigue. Cerebral Cortex (New York, N.Y. 1991), 34(1), Article bhad441. https://doi.org/10.1093/cercor/bhad441 van Dam, A. (2021). A clinical perspective on burnout: diagnosis, classification, and treatment of clinical burnout. European Journal of Work and Organizational Psychology, 30(5), 732-741. https://doi.org/10.1080/1359432X.2021.1948400 Vial, A. C., & Cowgill, C. M. (2022). Heavier Lies Her Crown: Gendered Patterns of Leader Emotional Labor and Their Downstream Effects. Frontiers in Psychology, 13, 849566. https://doi.org/https://doi.org/10.3389/fpsyg.2022.849566 Wharton, A. S. (2009). The Sociology of Emotional Labor. Annual Review of Sociology, 35(1), 147–165. https://doi.org/10.1146/annurev-soc-070308-115944 Wrobel, M. (2013). Can empathy lead to emotional exhaustion in teachers? The mediating role of emotional labor. International Journal of Occupational Medicine and Environmental Health, 26(4), 581–592. https://doi.org/10.2478/s13382-013-0123-1 }} ==External links== Provide [[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: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Burnout]] [[Category:Motivation and emotion/Book/Empathy]] a4dupryp5lq8tfwpcjzfb2hwk03ediw 2844530 2844529 2026-11-04T23:11:50Z U3143751 3106785 /* Job demands-resources model */ moving scenario info to conclusion 2844530 wikitext text/x-wiki {{title|Empathy fatigue and emotional exhaustion:<br>How does sustained empathic engagement contribute to emotional exhaustion?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}} [[File:Exhausted smartphone user.JPG|right|thumb|150px|'''Figure 1'''. An emotionally exhausted Aelin at the end of the work day]] ; Scenario Aelin, a 23-year-old social worker employed at headspace {{ic|Link to the Wikipedia article to help explain what this term means}}, works with approximately 20 {{ic|Explain what sort of}}Imagine you are only 23 years old and are working as a social worker at headspace. You see around 20 clients each week, many of whom have experienced rough living conditions. During your first six months in this role, you felt highly motivated and excited to help people in need. You are an excellent active listener and can easily remember details about your clients Live’s. You are also good at recognising signs of distress in your client’s body language and adjust yours in order to show warmth and understanding. As you progress in your role, you take on more cases that involve complex past traumas. The workload becomes too much for you to get done at work, but you don’t want your clients to miss out, so you take your work home to complete it in the evenings. Your workload increases again and you now start doing work on your weekends as well. After several months of working this way, you start to notice you feel exhausted every morning before you have even started work. You then start to notice you’re responding to people at work differently. Instead of your usual energised active listening response you feel frustrated, empathetic and detached. After work you are too drained to socialise with friends. This begins to worry you. What if you won’t be able to continue working as a social worker? {{RoundBoxBottom}} Aelin is experiencing '''emotional exhaustion''' from a sustained empathic response. Due to the nature of Aelin's work, they are constantly engaged in an empathic response. Empathic responses require cognitive and physical resources to sustain. Because Aelin is taking their work home and not resting on weekends, they do not have a chance to rest. It is important for Aelin to recognise what is happening and why. Not actioning the problem has led to empathy fatigue and emotional exhaustion. Emotional exhaustion is one of the key aspects of [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FEmpathy_fatigue_and_emotional_exhaustion&diff=2840411&oldid=2832859 burnout], a long-term condition of existing in a constant state of high stress. Psychological science can support Aelin in recognising and mapping the symptoms they are experiencing, understand coping strategies to minimise the current and future effects to improve their wellbeing. Understanding the symptoms will also support their role as a social worker, providing more tools for the people they support on a day-to-day basis. {{RoundBoxTop|theme=3}} '''Focus questions''' * Why do people become emotionally exhausted? * What is the relationship between empathy fatigue and emotional exhaustion? * What role does emotional labor play in emotional exhaustion? * How does understanding emotional labor and empathy fatigue work to reduce the effects of emotional exhaustion? {{RoundBoxBottom}} == Emotional exhaustion == To understand emotional exhaustion, it will be important to first understand burnout. Burnout is a long-term condition of existing in a constant state of high stress and affects millions of people annually. In the early 2000s, a survey from Australia found that more than 50% of social workers and occupational workers reported high levels of burnout symptoms (Llyod and King, 2004) and a recent report from 2026 found that more than one third of all workers in Australia were experiencing burnout (Chan, 2026). Being in a state of high stress for short periods of time is a useful adaptation for humans, so much so that it has been given a name, [[wikipedia:Eustress|Eustress]]. However, as a long-term condition it leads to individual health issues and can be linked to cardiovascular disease, increased vulnerability to upper respiratory diseases, and chronic inflammation (Schneiderman et al., 2005), and cognitive issues such as [[wikipedia:Inattentional_blindness|cognitive tunneling]] and impaired selective attention (Pooladvand and Hasanzadeh, 2023). Burnout can be defined across three dimensions: emotional exhaustion, [[wikipedia:Depersonalization|depersonalisation]], and reduced feelings of personal accomplishment (Maslach, 1993, as cited in Martingano et al., 2026). Emotional exhaustion may be considered the 'core' component of burnout and refers to the depletion of emotional resources, leading to feelings of overwhelm, overextension, and exhaustion (Martingano et al., 2026; De Beer et al., 2024; Seidler et al., 2014). As a major factor, its symptoms are similar to burnout and stress. Although antecedents of burnout are typically agreed upon (Bakker et al., 2014) and divided into individual (e.g. self-efficacy) and situational categories (e.g. work load), there are many theories that attempt to explain why emotional exhaustion occurs. Two of these theories with good evidence are the [[wikipedia:Conservation_of_resources_theory|Conservation of resources stress model]] (Hobfoll, 1989) and the [[wikipedia:Job_demands-resources_model|Job Demands-Resources model]] (Demerouti et al., 2001). === Conservation of resources stress model === The Conservation of resources stress model (CORT; Hobfall, 1989) is a theory that works to understand and explain why stress occurs. Under the model, humans are motivated to maintain, pursue, or attain resources. Stress, defined as "an individual's perceived pressure or strain" (Hollebeek et al., 2023 p.928), occurs when these resources are threatened or lost, or gained at a net loss. As noted above, resources aren't strictly tied to being physical objects. They can also be emotional states, conditions, or even ideas. For example, someone may place value on enjoying a calm, quiet Sunday morning at home. If their neighbour starts using power tools loudly next door, it would disrupt this state and induce stress in the individual. The CORT model focuses primarily on two principles. The first principle is that there is more harm in losing resources than there is in gaining them back (Halbesleben et al., 2014). This is known as loss salience, and some research suggests that the pain of loss is felt twice as strong as the pleasure of gain (Thaler, 2000). Looking at our previous example, imagine that the person who enjoys a quiet Sunday morning at home equally enjoys the sound of a loud car engine. Even though the person enjoys the sound of car engines, if the neighbour started revving their car loudly instead of using power tools, it would still disrupt their desired state of quiet and lead to an increase of stress. The second principle is that people must invest existing resources in order to gain or protect resources (Halbesleben et al., 2014). This resource investment includes recovering from resource loss. Coping is how people manage or overcome a difficult situation, such as protecting or recovering from resource loss. Coping can be broken down into different strategies that people use, but not all strategies are equal (Ito and Brotheridge, 2003). They are divided into two categories: control or escape strategies. Control strategies are considered better at reducing the strain from stress and consist of positive thinking, direct action, and help seeking (Havlovic & Keenan, 1991, as cited in Ito and Brotheridge, 2003). The most useful control strategies incorporate a person's [[locus of control]], emphasise seeking support from co-workers, and highlight the importance of autonomy in decision making. Alarcon (2011) conducted a meta-analysis of 231 studies in a work setting that used the [[wikipedia:Maslach_Burnout_Inventory|Maslach Burnout Inventory]] (MBI) to review the relationship between these principles and burnout to confirm their validity. Although the study had no specific mention of demographics, it was noted that multiple translations of the MBI were used including Dutch (k=51), German (k=7), Italian (k=2), and French (k=2). The findings indicate that workload is positively related to emotional exhaustion (''ρ'' = .49, ''k'' = 86, ''N'' = 51,529) and that control (''ρ'' = − .26, ''k'' = 56, ''N'' = 33,297) and autonomy (''ρ'' = − .24, ''k'' = 34, ''N'' = 17,165) were each negatively associated with emotional exhaustion. The difference of ''ρ'' values is consistent with Thaler (2000) and supports that the benefit of gaining is approximately half that of losing resources. === Job demands-resources model === Another model that attempts to understand why people become emotionally exhausted is the Job Demands-Resources model (JD-R; Demerouti et al., 2001). The JD-R model discusses two concepts: job demands and job resources. Demands and resources are both related to the physical, social, or psychological aspects of a job, but differ in their impact on a person. Job demands require physical or mental effort and exhaust a person's energy supply. Job resources typically support achieving goals, reduce the effort cost of job demands, and stimulate personal growth (Richter and Hacker, 1998, as cited in Demerouti et al., 2001). For example, an unsustainable workload would be considered a job demand, and receiving help and support from others would be considered a resource. The JD-R has two primary processes for how demands and resources interact with one another (Schaufeli, 2017). The stress process occurs when job demands are high and resources are low. As stress increases strain and pressure on an individual, it leads to the exhaustion of their energy eventually, eventually leading to emotional exhaustion. The second process is the motivational process. This process occurs when there is an excess of job resources. Crucially, the motivational process does not require low job demands. With enough job resources, a person will be energised and engaged, reframing job demands as a challenge to overcome. The JD-R model has had extensive research to confirm its validity. A meta-analysis by Lesener et al. (2019) looked at 74 longitudinal studies of the JD-R with a strong average sample size (n=450) against the assumptions as described by Schaufeli (2017). Although most of the studies appear to come from WEIRD countries (k=60) there were a good number from Asia and Africa (k=13) and a mix of genders. The study confirmed that across the literature, job resources predict burnout negatively (''β'' = −.17; ''p'' < .001) and job demands predict burnout over time (''β'' = .10; ''p'' < .001). Although job resources seem like a cure-all for demands, it's important to note that they do not retroactively treat emotional exhaustion. Under the JD-R model, burnout must be reduced by lowering job demands (Gynning et al., 2025). However, once someone has recovered, it is possible to increase job resources alongside job demands to prevent future burnout and support [[wikipedia:Work_engagement|work engagement]]. Work engagement (engagement) has been discussed as the opposite of burnout (Bakker et al., 2014). Engagement is how people express themselves when performing a role. A role being a pattern of behaviour that a person goes through based on the social expectations of that role, similar to how an actor takes on a role for a movie. For example, when attending a typical restaurant, it is socially expected that a person will engage with the waiter and provide them with their meal order to take to the kitchen. This person is performing the script of a restaurant patron. In the context of a workplace, an employee with high engagement may appear to be energised, enthusiastic, and become absorbed in their work losing track of time (Bakkar et al, 2014). {{RoundBoxTop|theme=3}} '''Why do we feel emotionally exhausted''' Under the JD-R and CORT models, emotional exhaustion occurs when the demand cost outweighs the resources available for an extended period of time, eventually depleting someone's energy reserves. Aelin is experiencing burnout because they increased their job demands by taking on more clients, and has experienced a loss in resources by reducing their free time and social support network. {{RoundBoxBottom}} == Empathy fatigue == You may be familiar with the term [[empathy]]. It is often described in terms of supporting others. For example, when someone close to you experiences a loss and is grieving, you may empathise with them and experience a negative emotion akin to grief too. However, empathy is not exclusive to so called negative emotions. For example, compersion is a positive empathic response when a loved one experiences joy. Parents may experience compersion when something joyful happens to their child. At its core, empathy is the individual ability to imagine or understand another person's feelings and/or emotions (Riess, 2017; Shao et al., 2024) and has four main components (Davis, 1983, as cited in Delgado et al., 2023): * Empathic concern is xyz * Fantasy is xyz * Perspective taking is xyz * Personal distress is xyz Much like physical or cognitive processes, such as running or problem solving, empathising will drain energy and resources. Further, and similar to the difference in energy expenditure between running and jogging, there are distinct types of empathy that each have different energy costs. === Types of empathy === Research makes a clear distinction between two types of empathy, affective and cognitive empathy. Affective empathy occurs when a person has an emotional reaction to someone else's mental state (Shao et al., 2024). This occurs because (unsure about this next part go through and update accordingly) ''unconscious processes interpret the social stimuli from another person leading to a physiological and cognitive response. These processes may interpret stimuli from any of the five senses.'' Cognitive empathy is when a person understands someone else's feelings or emotions (Cui et al., 2022). * Integrating the two - "affective component is the essence of empathy, and the cognitive component is a process through which that essence comes into being" (Bošnjaković & Radionov, 2018) ** (Author's note: change quote into paraphrasing) * === Empathy or compassion fatigue === * Distinction between empathy and compassion fatigue * Compassion fatigue is closely related to Secondary traumatic stress usually found in healthcare professionals (Figley, 2002) ** Compassion fatigue is to Empathy fatigue as emotional exhaustion is to burnout * Make it clear that Aelin is experiencing Compassion Fatigue vs Empathy fatigue (they are the same but different) (Author's note: include an update to the scenario here!) '''Empathy fatigue questions'''. Choose your answers and click "Submit": <quiz display=simple> {Surface acting is when the observer actually feels the emotion: |type="()"} - True + False {Compassion fatigue and empathy fatigue are synonymous: |type="()"} - True + False </quiz> {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' * Under the JD-R model, a sustained empathic response is a long-term job demand. * Under the CORT model, a sustained empathic response uses energy to protect existing resources. As resources become low and burnout sets in, cognitive focus narrows leading to empathy fatigue. {{RoundBoxBottom}} == Emotional labor as a mediating factor between empathy fatigue and emotional exhaustion == As a society there are expected [https://en.wikipedia.org/wiki/Social_norm social norms] in behaviour when expressing emotion, regardless if someone is conscious of the process of matching these norms or not. For example, laughing during an emotional speech at a funeral would be considered a rude or offensive expression from the other attendees. As social creatures, humans need a way to regulate socially appropriate emotional expression. This is the essence of Emotional labor: "the regulation of felt and expressed emotions (whether effortful or not) to match these emotional display rules with the objective of fulfilling organizational goals" (Vial & Cowgill, 2022) What is emotional labor (Hochschild, 2012). Theories behind emotional labor involve acting, and an extension of acting known as emotional contagion. === Types of acting === * Surface acting ** This is where emotive dissonance comes into play ** Leads to emotional exhaustion (Kenworthy et al., 2014) ** "Should" language - in this situation, this is what I ''should'' feel * Deep acting ** People who 'deep act', as in genuinely change the way they feel (match the feelings they have with the feelings that they need to display) often report positive relationships with emotional labor ** Actual language - in this situation, this is what I ''actually'' feel Discuss the differences in gender norms and expectations in relationships {| class="wikitable" style="margin: auto;" |- ! !! Deep acting !! Surface acting |- | '''Emotional dissonance''' || || X |- | '''Not known to others''' || Hidden area || Unknown |} === Emotional contagion === Emotional contagion is the idea that emotions are contagious. as the theory to moderate surface/deep acting * Emotional contagion (Panksepp and Lahvis, 2011) ** emotions can spread (expand, combines empathy and acting) ** “the tendency to automatically mimic and synchronize expressions, vocalizations, postures, and movements with those of another person’s and, consequently, to converge emotionally.” (Elaine Hatfield, 1993) * Affective process theory looks at unifying the mechanisms of emotional contagion into three types. Convergent linkage, Divergent linkage, and Complementary linkage (Elfenbein, 2014). Storytelling is a way to promote position emotions (Wei et al., 2024). === The mediating effect of emotional labor === Surface acting does not mediate the effect (high surface acting leads to emotional exhaustion) whereas the inverse is true for deep acting (Wrobel, 2013) Emotional Expression and Emotional Processing (Hoyt et al., 2024) {{RoundBoxTop|theme=3}} '''What role does emotional labor play in emotional exhaustion''' * In relation to the CORT and JD-R theories, emotional labor acts as a job demand and expenditure of resources to protect existing resources {{RoundBoxBottom}} == Strategies for combating empathy fatigue and emotional fatigue == * Talk about emotion regulation (Gross, 1998) and mood regulation strategies as a way to deal with coping (Wrobel, 2013) (Author's note: need more research/references for this section) * Talk about breaks ** Rest and downtime from not performing emotional labor regenerates * How do we reduce the amount of emotional labor we're performing (Jeung et al., 2018) ** Communication/boundaries ** Reducing surface acting (allowing ourselves to feel how we actually feel and reduce emotive dissonance) '''Table 1. Treatment phases of clinical burnout (based on Table 2 from van Dam, A. 2021)''' {| class="wikitable" style="margin: auto; |- ! Phase !! Treatment goal !! Therapeutic interventions |- | '''Phase 1: Crisis''' || Recognition of the patient that the problems are serious, and that serious action is required || Psychoeducation Sick leave Dropping domestic tasks and social obligations Inform social network |- | '''Phase 2: Recovery''' || Recovery of the (physiological) stress system to normal allostatic stress levels || Registration of stress and activities Relaxtion exercises Mindfulness healthy lifestyle advices Graded exercise Gradually resume activities Psychoeducation social network and employer Gradual return to work |- | '''Phase 3: Prevention, learning from the past''' || Acquiring insight and skills to prevent relapse in clinical burnout || Analysing stressful situations Analysing dysfunctional thought patterns Learning new coping skills Learning social skills Job crafting Time management Making choices regarding career and personal life |- | '''Phase 4: Post burnout growth''' || Improving sustainable quality of life. || Setting priorities regarding quality of life and interpersonal relationships. |} === Strategies for the workplace === * Discuss types of strategies that are available in the workplace (organisation models) * SCARF model (Aplin-Houtz, 2025) ** Link back to COR and JDR models e.g. praise could be a job resource vs criticism could be a job demand *Reduce workload to reduce emotional exhaustion, and by extension reduce emotional fatigue. {{RoundBoxTop|theme=3}} '''How does understanding emotional labor and empathy fatigue work to reduce the effects of emotional exhaustion''' *Psychoeducation has been shown to reduce the effects of emotional labor and empathy fatigue on emotional exhaustion {{RoundBoxBottom}} ==Conclusion== * Summarise what empathy fatigue is, and why people become emotionally exhausted (+ the role of emotional labor) * Summarise the symptoms and how to recognise them in yourself and someone else * Emphasise and summarise psychological strategies to support someone going through emotional exhaustion * Summarise ways to prevent emotional exhaustion from happening * Overall message is: *# What is emotional exhaustion *# how do you recognise it *# how do you treat it *# how do you prevent it * Link back to the focus questions ==See also== * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Metacognition_and_emotional_regulation&oldid=2756264 Metacognition and emotional regulation] (Book chapter, 2026) {{ic|Use an internal link style as shown in Tutorial 2}} * [[wikipedia:Conservation_of_resources_theory|Conservation of resources theory]] (Wikipedia) * [[wikipedia:Job_demands-resources_model|Job demands-resources model]] (Wikipedia) ==References== {{Hanging indent|1= Aplin-Houtz, M. (2025). Unraveling the SCARF: How dimensions of the SCARF model influence the relationship between workplace exclusion and turnover intentions. Journal of Management & Organization, 31(6), 2699–2725. https://doi.org/10.1017/jmo.2025.10026 Bošnjaković, J., & Radionov, T. (2018). Empathy: Concepts, Theories and Neuroscientific Basis. Alcoholism (Zagreb), 54(2), 123–150. https://doi.org/10.20471/de..2018.54.02.04 Chan, F. M. (2026). The State of Workplace Burnout 2026: The Workplace Divide. Infinite Potential, Sydney, Australia. https://www.ahri.com.au/wp-content/uploads/The-State-of-Workplace-Burnout-2026-Final.pdf Cui, F., Huang, X., Jing, Y., Luo, Y., Liu, J., & Gu, R. (2022). How resource sharing resists scarcity: the role of cognitive empathy and its neurobiological mechanisms. Cerebral Cortex (New York, N.Y. 1991), 32(23), 5330–5342. https://doi.org/10.1093/cercor/bhac017 De Beer, L. T., van der Vaart, L., Escaffi-Schwarz, M., De Witte, H., & Schaufeli, W. B. (2024). Maslach Burnout Inventory - General Survey: A Systematic Review and Meta-Analysis of Measurement Properties. European journal of psychological assessment : official organ of the European Association of Psychological Assessment, 40(5), 360-375. https://doi.org/10.1027/1015-5759/a000797 Demerouti, E., Bakker, A. B., Nachreiner, F., & Schaufeli, W. B. (2001). The Job Demands-Resources Model of Burnout. Journal of Applied Psychology, 86(3), 499–512. https://doi.org/10.1037/0021-9010.86.3.499 Figley, C. R. (2002). Compassion fatigue: Psychotherapists’ chronic lack of self care. Journal of Clinical Psychology, 58(11), 1433–1441. https://doi.org/10.1002/jclp.10090 Grandey, A. A. (2003). When ‘the show must go on’: surface acting and deep acting as determinants of emotional exhaustion and peer-rated service delivery. Academy of Management Journal, 46(1), 86–96. https://doi.org/10.5465/30040678 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 Halbesleben, J. R. B., Neveu, J.-P., Paustian-Underdahl, S. C., & Westman, M. (2014). Getting to the “COR”: Understanding the Role of Resources in Conservation of Resources Theory. Journal of Management, 40(5), 1334–1364. https://doi.org/10.1177/0149206314527130 Hatfield, E., Cacioppo, J. T., & Rapson, R. L. (1993). Emotional Contagion. Current Directions in Psychological Science : A Journal of the American Psychological Society, 2(3), 96–99. https://doi.org/10.1111/1467-8721.ep10770953 Hollebeek, L. D., Hammedi, W., & Sprott, D. E. (2023). Consumer engagement, stress, and conservation of resources theory: A review, conceptual development, and future research agenda. Psychology & marketing, 40(5), 926-937. https://doi.org/10.1002/mar.21807 Hobfoll, S. E. (1989). Conservation of Resources: A New Attempt at Conceptualizing Stress. The American Psychologist, 44(3), 513–524. https://doi.org/10.1037/0003-066X.44.3.513 Hochschild, A. R. (2012). The managed heart : commercialization of human feeling (Updated, with a new preface.). University of California Press. Jeung, D.-Y., Kim, C., & Chang, S.-J. (2018). Emotional Labor and Burnout: A Review of the Literature. Yonsei Medical Journal, 59(2), 187–193. https://doi.org/10.3349/ymj.2018.59.2.187 Lavelle, J. J., Rupp, D. E., Herda, D. N., Pandey, A., & Lauck, J. R. (2021). Customer Injustice and Employee Performance: Roles of Emotional Exhaustion, Surface Acting, and Emotional Demands–Abilities Fit. Journal of Management, 47(3), Article 0149206319869426. https://doi.org/10.1177/0149206319869426 Lloyd, C., & King, R. (2004). A survey of burnout among Australian mental health occupational therapists and social workers. Social Psychiatry and Psychiatric Epidemiology, 39(9), 752-757. https://doi.org/https://doi.org/10.1007/s00127-004-0808-7 Kenworthy, J., Fay, C., Frame, M., & Petree, R. (2014). A meta-analytic review of the relationship between emotional dissonance and emotional exhaustion. Journal of Applied Social Psychology, 44(2), 94-105. https://doi.org/https://doi.org/10.1111/jasp.12211 Martingano, A. J., Stosic, M. D., & Konrath, S. (2026). Different empathy types show opposing associations with burnout: systematic review and meta-analyses. Psychology, Health & Medicine, 31(3), 798-829. https://doi.org/10.1080/13548506.2025.2591859 Michielsen, H. J., Willemsen, T. M., Croon, M. A., de Vries, J., & Van Heck, G. L. (2004). Determinants of general fatigue and emotional exhaustion: a prospective study. Psychology & health, 19(2), 223-235. https://doi.org/10.1080/08870440310001627135 Panksepp, J. B., & Lahvis, G. P. (2011). Rodent empathy and affective neuroscience. Neuroscience and Biobehavioral Reviews, 35(9), 1864–1875. https://doi.org/10.1016/j.neubiorev.2011.05.013 Pooladvand, S., & Hasanzadeh, S. (2023). Impacts of Stress on Workers’ Risk-Taking Behaviors: Cognitive Tunneling and Impaired Selective Attention. Journal of Construction Engineering and Management, 149(8), 04023060. https://doi.org/doi:10.1061/JCEMD4.COENG-13339 Riess, H. (2017). The Science of Empathy. Journal of Patient Experience, 4(2), 74–77. https://doi.org/10.1177/2374373517699267 Schaufeli, W. B. (2017). Applying the job demands-resources model: a “how to” guide to measuring and tackling work engagement and burnout. Organizational Dynamics, 46(2), 120–132. https://doi.org/10.1016/j.orgdyn.2017.04.008 Schneiderman, N., Ironson, G., & Siegel, S. D. (2005). Stress and Health: Psychological, Behavioral, and Biological Determinants. Annual Review of Clinical Psychology, 1(Volume 1, 2005), 607-628. https://doi.org/https://doi.org/10.1146/annurev.clinpsy.1.102803.144141 Seidler, A., Thinschmidt, M., Deckert, S., Then, F., Hegewald, J., Nieuwenhuijsen, K., & Riedel-Heller, S. G. (2014). The role of psychosocial working conditions on burnout and its core component emotional exhaustion – a systematic review. Journal of Occupational Medicine and Toxicology, 9(1), 10. https://doi.org/10.1186/1745-6673-9-10 Shao, M., Li, L., Li, X., Wei, Z., Wang, J., Hong, M., Liu, X., & Meng, J. (2024). The effect of top-down attention on empathy fatigue. Cerebral Cortex (New York, N.Y. 1991), 34(1), Article bhad441. https://doi.org/10.1093/cercor/bhad441 van Dam, A. (2021). A clinical perspective on burnout: diagnosis, classification, and treatment of clinical burnout. European Journal of Work and Organizational Psychology, 30(5), 732-741. https://doi.org/10.1080/1359432X.2021.1948400 Vial, A. C., & Cowgill, C. M. (2022). Heavier Lies Her Crown: Gendered Patterns of Leader Emotional Labor and Their Downstream Effects. Frontiers in Psychology, 13, 849566. https://doi.org/https://doi.org/10.3389/fpsyg.2022.849566 Wharton, A. S. (2009). The Sociology of Emotional Labor. Annual Review of Sociology, 35(1), 147–165. https://doi.org/10.1146/annurev-soc-070308-115944 Wrobel, M. (2013). Can empathy lead to emotional exhaustion in teachers? The mediating role of emotional labor. International Journal of Occupational Medicine and Environmental Health, 26(4), 581–592. https://doi.org/10.2478/s13382-013-0123-1 }} ==External links== Provide [[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: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Burnout]] [[Category:Motivation and emotion/Book/Empathy]] 61nwjkyd72jifz80f94ma62cpyfiwel 2844531 2844530 2026-11-04T23:13:06Z U3143751 3106785 /* Empathy fatigue */ 2844531 wikitext text/x-wiki {{title|Empathy fatigue and emotional exhaustion:<br>How does sustained empathic engagement contribute to emotional exhaustion?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}} [[File:Exhausted smartphone user.JPG|right|thumb|150px|'''Figure 1'''. An emotionally exhausted Aelin at the end of the work day]] ; Scenario Aelin, a 23-year-old social worker employed at headspace {{ic|Link to the Wikipedia article to help explain what this term means}}, works with approximately 20 {{ic|Explain what sort of}}Imagine you are only 23 years old and are working as a social worker at headspace. You see around 20 clients each week, many of whom have experienced rough living conditions. During your first six months in this role, you felt highly motivated and excited to help people in need. You are an excellent active listener and can easily remember details about your clients Live’s. You are also good at recognising signs of distress in your client’s body language and adjust yours in order to show warmth and understanding. As you progress in your role, you take on more cases that involve complex past traumas. The workload becomes too much for you to get done at work, but you don’t want your clients to miss out, so you take your work home to complete it in the evenings. Your workload increases again and you now start doing work on your weekends as well. After several months of working this way, you start to notice you feel exhausted every morning before you have even started work. You then start to notice you’re responding to people at work differently. Instead of your usual energised active listening response you feel frustrated, empathetic and detached. After work you are too drained to socialise with friends. This begins to worry you. What if you won’t be able to continue working as a social worker? {{RoundBoxBottom}} Aelin is experiencing '''emotional exhaustion''' from a sustained empathic response. Due to the nature of Aelin's work, they are constantly engaged in an empathic response. Empathic responses require cognitive and physical resources to sustain. Because Aelin is taking their work home and not resting on weekends, they do not have a chance to rest. It is important for Aelin to recognise what is happening and why. Not actioning the problem has led to empathy fatigue and emotional exhaustion. Emotional exhaustion is one of the key aspects of [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FEmpathy_fatigue_and_emotional_exhaustion&diff=2840411&oldid=2832859 burnout], a long-term condition of existing in a constant state of high stress. Psychological science can support Aelin in recognising and mapping the symptoms they are experiencing, understand coping strategies to minimise the current and future effects to improve their wellbeing. Understanding the symptoms will also support their role as a social worker, providing more tools for the people they support on a day-to-day basis. {{RoundBoxTop|theme=3}} '''Focus questions''' * Why do people become emotionally exhausted? * What is the relationship between empathy fatigue and emotional exhaustion? * What role does emotional labor play in emotional exhaustion? * How does understanding emotional labor and empathy fatigue work to reduce the effects of emotional exhaustion? {{RoundBoxBottom}} == Emotional exhaustion == To understand emotional exhaustion, it will be important to first understand burnout. Burnout is a long-term condition of existing in a constant state of high stress and affects millions of people annually. In the early 2000s, a survey from Australia found that more than 50% of social workers and occupational workers reported high levels of burnout symptoms (Llyod and King, 2004) and a recent report from 2026 found that more than one third of all workers in Australia were experiencing burnout (Chan, 2026). Being in a state of high stress for short periods of time is a useful adaptation for humans, so much so that it has been given a name, [[wikipedia:Eustress|Eustress]]. However, as a long-term condition it leads to individual health issues and can be linked to cardiovascular disease, increased vulnerability to upper respiratory diseases, and chronic inflammation (Schneiderman et al., 2005), and cognitive issues such as [[wikipedia:Inattentional_blindness|cognitive tunneling]] and impaired selective attention (Pooladvand and Hasanzadeh, 2023). Burnout can be defined across three dimensions: emotional exhaustion, [[wikipedia:Depersonalization|depersonalisation]], and reduced feelings of personal accomplishment (Maslach, 1993, as cited in Martingano et al., 2026). Emotional exhaustion may be considered the 'core' component of burnout and refers to the depletion of emotional resources, leading to feelings of overwhelm, overextension, and exhaustion (Martingano et al., 2026; De Beer et al., 2024; Seidler et al., 2014). As a major factor, its symptoms are similar to burnout and stress. Although antecedents of burnout are typically agreed upon (Bakker et al., 2014) and divided into individual (e.g. self-efficacy) and situational categories (e.g. work load), there are many theories that attempt to explain why emotional exhaustion occurs. Two of these theories with good evidence are the [[wikipedia:Conservation_of_resources_theory|Conservation of resources stress model]] (Hobfoll, 1989) and the [[wikipedia:Job_demands-resources_model|Job Demands-Resources model]] (Demerouti et al., 2001). === Conservation of resources stress model === The Conservation of resources stress model (CORT; Hobfall, 1989) is a theory that works to understand and explain why stress occurs. Under the model, humans are motivated to maintain, pursue, or attain resources. Stress, defined as "an individual's perceived pressure or strain" (Hollebeek et al., 2023 p.928), occurs when these resources are threatened or lost, or gained at a net loss. As noted above, resources aren't strictly tied to being physical objects. They can also be emotional states, conditions, or even ideas. For example, someone may place value on enjoying a calm, quiet Sunday morning at home. If their neighbour starts using power tools loudly next door, it would disrupt this state and induce stress in the individual. The CORT model focuses primarily on two principles. The first principle is that there is more harm in losing resources than there is in gaining them back (Halbesleben et al., 2014). This is known as loss salience, and some research suggests that the pain of loss is felt twice as strong as the pleasure of gain (Thaler, 2000). Looking at our previous example, imagine that the person who enjoys a quiet Sunday morning at home equally enjoys the sound of a loud car engine. Even though the person enjoys the sound of car engines, if the neighbour started revving their car loudly instead of using power tools, it would still disrupt their desired state of quiet and lead to an increase of stress. The second principle is that people must invest existing resources in order to gain or protect resources (Halbesleben et al., 2014). This resource investment includes recovering from resource loss. Coping is how people manage or overcome a difficult situation, such as protecting or recovering from resource loss. Coping can be broken down into different strategies that people use, but not all strategies are equal (Ito and Brotheridge, 2003). They are divided into two categories: control or escape strategies. Control strategies are considered better at reducing the strain from stress and consist of positive thinking, direct action, and help seeking (Havlovic & Keenan, 1991, as cited in Ito and Brotheridge, 2003). The most useful control strategies incorporate a person's [[locus of control]], emphasise seeking support from co-workers, and highlight the importance of autonomy in decision making. Alarcon (2011) conducted a meta-analysis of 231 studies in a work setting that used the [[wikipedia:Maslach_Burnout_Inventory|Maslach Burnout Inventory]] (MBI) to review the relationship between these principles and burnout to confirm their validity. Although the study had no specific mention of demographics, it was noted that multiple translations of the MBI were used including Dutch (k=51), German (k=7), Italian (k=2), and French (k=2). The findings indicate that workload is positively related to emotional exhaustion (''ρ'' = .49, ''k'' = 86, ''N'' = 51,529) and that control (''ρ'' = − .26, ''k'' = 56, ''N'' = 33,297) and autonomy (''ρ'' = − .24, ''k'' = 34, ''N'' = 17,165) were each negatively associated with emotional exhaustion. The difference of ''ρ'' values is consistent with Thaler (2000) and supports that the benefit of gaining is approximately half that of losing resources. === Job demands-resources model === Another model that attempts to understand why people become emotionally exhausted is the Job Demands-Resources model (JD-R; Demerouti et al., 2001). The JD-R model discusses two concepts: job demands and job resources. Demands and resources are both related to the physical, social, or psychological aspects of a job, but differ in their impact on a person. Job demands require physical or mental effort and exhaust a person's energy supply. Job resources typically support achieving goals, reduce the effort cost of job demands, and stimulate personal growth (Richter and Hacker, 1998, as cited in Demerouti et al., 2001). For example, an unsustainable workload would be considered a job demand, and receiving help and support from others would be considered a resource. The JD-R has two primary processes for how demands and resources interact with one another (Schaufeli, 2017). The stress process occurs when job demands are high and resources are low. As stress increases strain and pressure on an individual, it leads to the exhaustion of their energy eventually, eventually leading to emotional exhaustion. The second process is the motivational process. This process occurs when there is an excess of job resources. Crucially, the motivational process does not require low job demands. With enough job resources, a person will be energised and engaged, reframing job demands as a challenge to overcome. The JD-R model has had extensive research to confirm its validity. A meta-analysis by Lesener et al. (2019) looked at 74 longitudinal studies of the JD-R with a strong average sample size (n=450) against the assumptions as described by Schaufeli (2017). Although most of the studies appear to come from WEIRD countries (k=60) there were a good number from Asia and Africa (k=13) and a mix of genders. The study confirmed that across the literature, job resources predict burnout negatively (''β'' = −.17; ''p'' < .001) and job demands predict burnout over time (''β'' = .10; ''p'' < .001). Although job resources seem like a cure-all for demands, it's important to note that they do not retroactively treat emotional exhaustion. Under the JD-R model, burnout must be reduced by lowering job demands (Gynning et al., 2025). However, once someone has recovered, it is possible to increase job resources alongside job demands to prevent future burnout and support [[wikipedia:Work_engagement|work engagement]]. Work engagement (engagement) has been discussed as the opposite of burnout (Bakker et al., 2014). Engagement is how people express themselves when performing a role. A role being a pattern of behaviour that a person goes through based on the social expectations of that role, similar to how an actor takes on a role for a movie. For example, when attending a typical restaurant, it is socially expected that a person will engage with the waiter and provide them with their meal order to take to the kitchen. This person is performing the script of a restaurant patron. In the context of a workplace, an employee with high engagement may appear to be energised, enthusiastic, and become absorbed in their work losing track of time (Bakkar et al, 2014). {{RoundBoxTop|theme=3}} '''Why do we feel emotionally exhausted''' Under the JD-R and CORT models, emotional exhaustion occurs when the demand cost outweighs the resources available for an extended period of time, eventually depleting someone's energy reserves. Aelin is experiencing burnout because they increased their job demands by taking on more clients, and has experienced a loss in resources by reducing their free time and social support network. {{RoundBoxBottom}} == Empathy fatigue == You may be familiar with the term [[empathy]]. It is often described in terms of supporting others. For example, when someone close to you experiences a loss and is grieving, you may empathise with them and experience a negative emotion akin to grief too. However, empathy is not exclusive to so called negative emotions. For example, compersion is a positive empathic response when a loved one experiences joy. Parents may experience compersion when something joyful happens to their child. At its core, empathy is the individual ability to imagine or understand another person's feelings and/or emotions (Riess, 2017; Shao et al., 2024). Much like physical or cognitive processes, such as running or problem solving, empathising will drain energy and resources. Further, and similar to the difference in energy expenditure between running and jogging, there are distinct types of empathy that each have different energy costs. === Types of empathy === Research makes a clear distinction between two types of empathy: affective and cognitive empathy. Affective empathy occurs when a person has an emotional reaction to someone else's mental state (Shao et al., 2024). This occurs because (unsure about this next part go through and update accordingly) ''unconscious processes interpret the social stimuli from another person leading to a physiological and cognitive response. These processes may interpret stimuli from any of the five senses.'' Cognitive empathy is when a person understands someone else's feelings or emotions (Cui et al., 2022). * Integrating the two - "affective component is the essence of empathy, and the cognitive component is a process through which that essence comes into being" (Bošnjaković & Radionov, 2018) ** (Author's note: change quote into paraphrasing) * === Empathy or compassion fatigue === * Distinction between empathy and compassion fatigue * Compassion fatigue is closely related to Secondary traumatic stress usually found in healthcare professionals (Figley, 2002) ** Compassion fatigue is to Empathy fatigue as emotional exhaustion is to burnout * Make it clear that Aelin is experiencing Compassion Fatigue vs Empathy fatigue (they are the same but different) (Author's note: include an update to the scenario here!) '''Empathy fatigue questions'''. Choose your answers and click "Submit": <quiz display=simple> {Surface acting is when the observer actually feels the emotion: |type="()"} - True + False {Compassion fatigue and empathy fatigue are synonymous: |type="()"} - True + False </quiz> {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' * Under the JD-R model, a sustained empathic response is a long-term job demand. * Under the CORT model, a sustained empathic response uses energy to protect existing resources. As resources become low and burnout sets in, cognitive focus narrows leading to empathy fatigue. {{RoundBoxBottom}} == Emotional labor as a mediating factor between empathy fatigue and emotional exhaustion == As a society there are expected [https://en.wikipedia.org/wiki/Social_norm social norms] in behaviour when expressing emotion, regardless if someone is conscious of the process of matching these norms or not. For example, laughing during an emotional speech at a funeral would be considered a rude or offensive expression from the other attendees. As social creatures, humans need a way to regulate socially appropriate emotional expression. This is the essence of Emotional labor: "the regulation of felt and expressed emotions (whether effortful or not) to match these emotional display rules with the objective of fulfilling organizational goals" (Vial & Cowgill, 2022) What is emotional labor (Hochschild, 2012). Theories behind emotional labor involve acting, and an extension of acting known as emotional contagion. === Types of acting === * Surface acting ** This is where emotive dissonance comes into play ** Leads to emotional exhaustion (Kenworthy et al., 2014) ** "Should" language - in this situation, this is what I ''should'' feel * Deep acting ** People who 'deep act', as in genuinely change the way they feel (match the feelings they have with the feelings that they need to display) often report positive relationships with emotional labor ** Actual language - in this situation, this is what I ''actually'' feel Discuss the differences in gender norms and expectations in relationships {| class="wikitable" style="margin: auto;" |- ! !! Deep acting !! Surface acting |- | '''Emotional dissonance''' || || X |- | '''Not known to others''' || Hidden area || Unknown |} === Emotional contagion === Emotional contagion is the idea that emotions are contagious. as the theory to moderate surface/deep acting * Emotional contagion (Panksepp and Lahvis, 2011) ** emotions can spread (expand, combines empathy and acting) ** “the tendency to automatically mimic and synchronize expressions, vocalizations, postures, and movements with those of another person’s and, consequently, to converge emotionally.” (Elaine Hatfield, 1993) * Affective process theory looks at unifying the mechanisms of emotional contagion into three types. Convergent linkage, Divergent linkage, and Complementary linkage (Elfenbein, 2014). Storytelling is a way to promote position emotions (Wei et al., 2024). === The mediating effect of emotional labor === Surface acting does not mediate the effect (high surface acting leads to emotional exhaustion) whereas the inverse is true for deep acting (Wrobel, 2013) Emotional Expression and Emotional Processing (Hoyt et al., 2024) {{RoundBoxTop|theme=3}} '''What role does emotional labor play in emotional exhaustion''' * In relation to the CORT and JD-R theories, emotional labor acts as a job demand and expenditure of resources to protect existing resources {{RoundBoxBottom}} == Strategies for combating empathy fatigue and emotional fatigue == * Talk about emotion regulation (Gross, 1998) and mood regulation strategies as a way to deal with coping (Wrobel, 2013) (Author's note: need more research/references for this section) * Talk about breaks ** Rest and downtime from not performing emotional labor regenerates * How do we reduce the amount of emotional labor we're performing (Jeung et al., 2018) ** Communication/boundaries ** Reducing surface acting (allowing ourselves to feel how we actually feel and reduce emotive dissonance) '''Table 1. Treatment phases of clinical burnout (based on Table 2 from van Dam, A. 2021)''' {| class="wikitable" style="margin: auto; |- ! Phase !! Treatment goal !! Therapeutic interventions |- | '''Phase 1: Crisis''' || Recognition of the patient that the problems are serious, and that serious action is required || Psychoeducation Sick leave Dropping domestic tasks and social obligations Inform social network |- | '''Phase 2: Recovery''' || Recovery of the (physiological) stress system to normal allostatic stress levels || Registration of stress and activities Relaxtion exercises Mindfulness healthy lifestyle advices Graded exercise Gradually resume activities Psychoeducation social network and employer Gradual return to work |- | '''Phase 3: Prevention, learning from the past''' || Acquiring insight and skills to prevent relapse in clinical burnout || Analysing stressful situations Analysing dysfunctional thought patterns Learning new coping skills Learning social skills Job crafting Time management Making choices regarding career and personal life |- | '''Phase 4: Post burnout growth''' || Improving sustainable quality of life. || Setting priorities regarding quality of life and interpersonal relationships. |} === Strategies for the workplace === * Discuss types of strategies that are available in the workplace (organisation models) * SCARF model (Aplin-Houtz, 2025) ** Link back to COR and JDR models e.g. praise could be a job resource vs criticism could be a job demand *Reduce workload to reduce emotional exhaustion, and by extension reduce emotional fatigue. {{RoundBoxTop|theme=3}} '''How does understanding emotional labor and empathy fatigue work to reduce the effects of emotional exhaustion''' *Psychoeducation has been shown to reduce the effects of emotional labor and empathy fatigue on emotional exhaustion {{RoundBoxBottom}} ==Conclusion== * Summarise what empathy fatigue is, and why people become emotionally exhausted (+ the role of emotional labor) * Summarise the symptoms and how to recognise them in yourself and someone else * Emphasise and summarise psychological strategies to support someone going through emotional exhaustion * Summarise ways to prevent emotional exhaustion from happening * Overall message is: *# What is emotional exhaustion *# how do you recognise it *# how do you treat it *# how do you prevent it * Link back to the focus questions ==See also== * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Metacognition_and_emotional_regulation&oldid=2756264 Metacognition and emotional regulation] (Book chapter, 2026) {{ic|Use an internal link style as shown in Tutorial 2}} * [[wikipedia:Conservation_of_resources_theory|Conservation of resources theory]] (Wikipedia) * [[wikipedia:Job_demands-resources_model|Job demands-resources model]] (Wikipedia) ==References== {{Hanging indent|1= Aplin-Houtz, M. (2025). Unraveling the SCARF: How dimensions of the SCARF model influence the relationship between workplace exclusion and turnover intentions. Journal of Management & Organization, 31(6), 2699–2725. https://doi.org/10.1017/jmo.2025.10026 Bošnjaković, J., & Radionov, T. (2018). Empathy: Concepts, Theories and Neuroscientific Basis. Alcoholism (Zagreb), 54(2), 123–150. https://doi.org/10.20471/de..2018.54.02.04 Chan, F. M. (2026). The State of Workplace Burnout 2026: The Workplace Divide. Infinite Potential, Sydney, Australia. https://www.ahri.com.au/wp-content/uploads/The-State-of-Workplace-Burnout-2026-Final.pdf Cui, F., Huang, X., Jing, Y., Luo, Y., Liu, J., & Gu, R. (2022). How resource sharing resists scarcity: the role of cognitive empathy and its neurobiological mechanisms. Cerebral Cortex (New York, N.Y. 1991), 32(23), 5330–5342. https://doi.org/10.1093/cercor/bhac017 De Beer, L. T., van der Vaart, L., Escaffi-Schwarz, M., De Witte, H., & Schaufeli, W. B. (2024). Maslach Burnout Inventory - General Survey: A Systematic Review and Meta-Analysis of Measurement Properties. European journal of psychological assessment : official organ of the European Association of Psychological Assessment, 40(5), 360-375. https://doi.org/10.1027/1015-5759/a000797 Demerouti, E., Bakker, A. B., Nachreiner, F., & Schaufeli, W. B. (2001). The Job Demands-Resources Model of Burnout. Journal of Applied Psychology, 86(3), 499–512. https://doi.org/10.1037/0021-9010.86.3.499 Figley, C. R. (2002). Compassion fatigue: Psychotherapists’ chronic lack of self care. Journal of Clinical Psychology, 58(11), 1433–1441. https://doi.org/10.1002/jclp.10090 Grandey, A. A. (2003). When ‘the show must go on’: surface acting and deep acting as determinants of emotional exhaustion and peer-rated service delivery. Academy of Management Journal, 46(1), 86–96. https://doi.org/10.5465/30040678 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 Halbesleben, J. R. B., Neveu, J.-P., Paustian-Underdahl, S. C., & Westman, M. (2014). Getting to the “COR”: Understanding the Role of Resources in Conservation of Resources Theory. Journal of Management, 40(5), 1334–1364. https://doi.org/10.1177/0149206314527130 Hatfield, E., Cacioppo, J. T., & Rapson, R. L. (1993). Emotional Contagion. Current Directions in Psychological Science : A Journal of the American Psychological Society, 2(3), 96–99. https://doi.org/10.1111/1467-8721.ep10770953 Hollebeek, L. D., Hammedi, W., & Sprott, D. E. (2023). Consumer engagement, stress, and conservation of resources theory: A review, conceptual development, and future research agenda. Psychology & marketing, 40(5), 926-937. https://doi.org/10.1002/mar.21807 Hobfoll, S. E. (1989). Conservation of Resources: A New Attempt at Conceptualizing Stress. The American Psychologist, 44(3), 513–524. https://doi.org/10.1037/0003-066X.44.3.513 Hochschild, A. R. (2012). The managed heart : commercialization of human feeling (Updated, with a new preface.). University of California Press. Jeung, D.-Y., Kim, C., & Chang, S.-J. (2018). Emotional Labor and Burnout: A Review of the Literature. Yonsei Medical Journal, 59(2), 187–193. https://doi.org/10.3349/ymj.2018.59.2.187 Lavelle, J. J., Rupp, D. E., Herda, D. N., Pandey, A., & Lauck, J. R. (2021). Customer Injustice and Employee Performance: Roles of Emotional Exhaustion, Surface Acting, and Emotional Demands–Abilities Fit. Journal of Management, 47(3), Article 0149206319869426. https://doi.org/10.1177/0149206319869426 Lloyd, C., & King, R. (2004). A survey of burnout among Australian mental health occupational therapists and social workers. Social Psychiatry and Psychiatric Epidemiology, 39(9), 752-757. https://doi.org/https://doi.org/10.1007/s00127-004-0808-7 Kenworthy, J., Fay, C., Frame, M., & Petree, R. (2014). A meta-analytic review of the relationship between emotional dissonance and emotional exhaustion. Journal of Applied Social Psychology, 44(2), 94-105. https://doi.org/https://doi.org/10.1111/jasp.12211 Martingano, A. J., Stosic, M. D., & Konrath, S. (2026). Different empathy types show opposing associations with burnout: systematic review and meta-analyses. Psychology, Health & Medicine, 31(3), 798-829. https://doi.org/10.1080/13548506.2025.2591859 Michielsen, H. J., Willemsen, T. M., Croon, M. A., de Vries, J., & Van Heck, G. L. (2004). Determinants of general fatigue and emotional exhaustion: a prospective study. Psychology & health, 19(2), 223-235. https://doi.org/10.1080/08870440310001627135 Panksepp, J. B., & Lahvis, G. P. (2011). Rodent empathy and affective neuroscience. Neuroscience and Biobehavioral Reviews, 35(9), 1864–1875. https://doi.org/10.1016/j.neubiorev.2011.05.013 Pooladvand, S., & Hasanzadeh, S. (2023). Impacts of Stress on Workers’ Risk-Taking Behaviors: Cognitive Tunneling and Impaired Selective Attention. Journal of Construction Engineering and Management, 149(8), 04023060. https://doi.org/doi:10.1061/JCEMD4.COENG-13339 Riess, H. (2017). The Science of Empathy. Journal of Patient Experience, 4(2), 74–77. https://doi.org/10.1177/2374373517699267 Schaufeli, W. B. (2017). Applying the job demands-resources model: a “how to” guide to measuring and tackling work engagement and burnout. Organizational Dynamics, 46(2), 120–132. https://doi.org/10.1016/j.orgdyn.2017.04.008 Schneiderman, N., Ironson, G., & Siegel, S. D. (2005). Stress and Health: Psychological, Behavioral, and Biological Determinants. Annual Review of Clinical Psychology, 1(Volume 1, 2005), 607-628. https://doi.org/https://doi.org/10.1146/annurev.clinpsy.1.102803.144141 Seidler, A., Thinschmidt, M., Deckert, S., Then, F., Hegewald, J., Nieuwenhuijsen, K., & Riedel-Heller, S. G. (2014). The role of psychosocial working conditions on burnout and its core component emotional exhaustion – a systematic review. Journal of Occupational Medicine and Toxicology, 9(1), 10. https://doi.org/10.1186/1745-6673-9-10 Shao, M., Li, L., Li, X., Wei, Z., Wang, J., Hong, M., Liu, X., & Meng, J. (2024). The effect of top-down attention on empathy fatigue. Cerebral Cortex (New York, N.Y. 1991), 34(1), Article bhad441. https://doi.org/10.1093/cercor/bhad441 van Dam, A. (2021). A clinical perspective on burnout: diagnosis, classification, and treatment of clinical burnout. European Journal of Work and Organizational Psychology, 30(5), 732-741. https://doi.org/10.1080/1359432X.2021.1948400 Vial, A. C., & Cowgill, C. M. (2022). Heavier Lies Her Crown: Gendered Patterns of Leader Emotional Labor and Their Downstream Effects. Frontiers in Psychology, 13, 849566. https://doi.org/https://doi.org/10.3389/fpsyg.2022.849566 Wharton, A. S. (2009). The Sociology of Emotional Labor. Annual Review of Sociology, 35(1), 147–165. https://doi.org/10.1146/annurev-soc-070308-115944 Wrobel, M. (2013). Can empathy lead to emotional exhaustion in teachers? The mediating role of emotional labor. International Journal of Occupational Medicine and Environmental Health, 26(4), 581–592. https://doi.org/10.2478/s13382-013-0123-1 }} ==External links== Provide [[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: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Burnout]] [[Category:Motivation and emotion/Book/Empathy]] e26qy27b88diql9wktadl2b1h17e1a5 2844532 2844531 2026-11-04T23:15:58Z U3143751 3106785 /* Empathy fatigue */ 2844532 wikitext text/x-wiki {{title|Empathy fatigue and emotional exhaustion:<br>How does sustained empathic engagement contribute to emotional exhaustion?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}} [[File:Exhausted smartphone user.JPG|right|thumb|150px|'''Figure 1'''. An emotionally exhausted Aelin at the end of the work day]] ; Scenario Aelin, a 23-year-old social worker employed at headspace {{ic|Link to the Wikipedia article to help explain what this term means}}, works with approximately 20 {{ic|Explain what sort of}}Imagine you are only 23 years old and are working as a social worker at headspace. You see around 20 clients each week, many of whom have experienced rough living conditions. During your first six months in this role, you felt highly motivated and excited to help people in need. You are an excellent active listener and can easily remember details about your clients Live’s. You are also good at recognising signs of distress in your client’s body language and adjust yours in order to show warmth and understanding. As you progress in your role, you take on more cases that involve complex past traumas. The workload becomes too much for you to get done at work, but you don’t want your clients to miss out, so you take your work home to complete it in the evenings. Your workload increases again and you now start doing work on your weekends as well. After several months of working this way, you start to notice you feel exhausted every morning before you have even started work. You then start to notice you’re responding to people at work differently. Instead of your usual energised active listening response you feel frustrated, empathetic and detached. After work you are too drained to socialise with friends. This begins to worry you. What if you won’t be able to continue working as a social worker? {{RoundBoxBottom}} Aelin is experiencing '''emotional exhaustion''' from a sustained empathic response. Due to the nature of Aelin's work, they are constantly engaged in an empathic response. Empathic responses require cognitive and physical resources to sustain. Because Aelin is taking their work home and not resting on weekends, they do not have a chance to rest. It is important for Aelin to recognise what is happening and why. Not actioning the problem has led to empathy fatigue and emotional exhaustion. Emotional exhaustion is one of the key aspects of [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FEmpathy_fatigue_and_emotional_exhaustion&diff=2840411&oldid=2832859 burnout], a long-term condition of existing in a constant state of high stress. Psychological science can support Aelin in recognising and mapping the symptoms they are experiencing, understand coping strategies to minimise the current and future effects to improve their wellbeing. Understanding the symptoms will also support their role as a social worker, providing more tools for the people they support on a day-to-day basis. {{RoundBoxTop|theme=3}} '''Focus questions''' * Why do people become emotionally exhausted? * What is the relationship between empathy fatigue and emotional exhaustion? * What role does emotional labor play in emotional exhaustion? * How does understanding emotional labor and empathy fatigue work to reduce the effects of emotional exhaustion? {{RoundBoxBottom}} == Emotional exhaustion == To understand emotional exhaustion, it will be important to first understand burnout. Burnout is a long-term condition of existing in a constant state of high stress and affects millions of people annually. In the early 2000s, a survey from Australia found that more than 50% of social workers and occupational workers reported high levels of burnout symptoms (Llyod and King, 2004) and a recent report from 2026 found that more than one third of all workers in Australia were experiencing burnout (Chan, 2026). Being in a state of high stress for short periods of time is a useful adaptation for humans, so much so that it has been given a name, [[wikipedia:Eustress|Eustress]]. However, as a long-term condition it leads to individual health issues and can be linked to cardiovascular disease, increased vulnerability to upper respiratory diseases, and chronic inflammation (Schneiderman et al., 2005), and cognitive issues such as [[wikipedia:Inattentional_blindness|cognitive tunneling]] and impaired selective attention (Pooladvand and Hasanzadeh, 2023). Burnout can be defined across three dimensions: emotional exhaustion, [[wikipedia:Depersonalization|depersonalisation]], and reduced feelings of personal accomplishment (Maslach, 1993, as cited in Martingano et al., 2026). Emotional exhaustion may be considered the 'core' component of burnout and refers to the depletion of emotional resources, leading to feelings of overwhelm, overextension, and exhaustion (Martingano et al., 2026; De Beer et al., 2024; Seidler et al., 2014). As a major factor, its symptoms are similar to burnout and stress. Although antecedents of burnout are typically agreed upon (Bakker et al., 2014) and divided into individual (e.g. self-efficacy) and situational categories (e.g. work load), there are many theories that attempt to explain why emotional exhaustion occurs. Two of these theories with good evidence are the [[wikipedia:Conservation_of_resources_theory|Conservation of resources stress model]] (Hobfoll, 1989) and the [[wikipedia:Job_demands-resources_model|Job Demands-Resources model]] (Demerouti et al., 2001). === Conservation of resources stress model === The Conservation of resources stress model (CORT; Hobfall, 1989) is a theory that works to understand and explain why stress occurs. Under the model, humans are motivated to maintain, pursue, or attain resources. Stress, defined as "an individual's perceived pressure or strain" (Hollebeek et al., 2023 p.928), occurs when these resources are threatened or lost, or gained at a net loss. As noted above, resources aren't strictly tied to being physical objects. They can also be emotional states, conditions, or even ideas. For example, someone may place value on enjoying a calm, quiet Sunday morning at home. If their neighbour starts using power tools loudly next door, it would disrupt this state and induce stress in the individual. The CORT model focuses primarily on two principles. The first principle is that there is more harm in losing resources than there is in gaining them back (Halbesleben et al., 2014). This is known as loss salience, and some research suggests that the pain of loss is felt twice as strong as the pleasure of gain (Thaler, 2000). Looking at our previous example, imagine that the person who enjoys a quiet Sunday morning at home equally enjoys the sound of a loud car engine. Even though the person enjoys the sound of car engines, if the neighbour started revving their car loudly instead of using power tools, it would still disrupt their desired state of quiet and lead to an increase of stress. The second principle is that people must invest existing resources in order to gain or protect resources (Halbesleben et al., 2014). This resource investment includes recovering from resource loss. Coping is how people manage or overcome a difficult situation, such as protecting or recovering from resource loss. Coping can be broken down into different strategies that people use, but not all strategies are equal (Ito and Brotheridge, 2003). They are divided into two categories: control or escape strategies. Control strategies are considered better at reducing the strain from stress and consist of positive thinking, direct action, and help seeking (Havlovic & Keenan, 1991, as cited in Ito and Brotheridge, 2003). The most useful control strategies incorporate a person's [[locus of control]], emphasise seeking support from co-workers, and highlight the importance of autonomy in decision making. Alarcon (2011) conducted a meta-analysis of 231 studies in a work setting that used the [[wikipedia:Maslach_Burnout_Inventory|Maslach Burnout Inventory]] (MBI) to review the relationship between these principles and burnout to confirm their validity. Although the study had no specific mention of demographics, it was noted that multiple translations of the MBI were used including Dutch (k=51), German (k=7), Italian (k=2), and French (k=2). The findings indicate that workload is positively related to emotional exhaustion (''ρ'' = .49, ''k'' = 86, ''N'' = 51,529) and that control (''ρ'' = − .26, ''k'' = 56, ''N'' = 33,297) and autonomy (''ρ'' = − .24, ''k'' = 34, ''N'' = 17,165) were each negatively associated with emotional exhaustion. The difference of ''ρ'' values is consistent with Thaler (2000) and supports that the benefit of gaining is approximately half that of losing resources. === Job demands-resources model === Another model that attempts to understand why people become emotionally exhausted is the Job Demands-Resources model (JD-R; Demerouti et al., 2001). The JD-R model discusses two concepts: job demands and job resources. Demands and resources are both related to the physical, social, or psychological aspects of a job, but differ in their impact on a person. Job demands require physical or mental effort and exhaust a person's energy supply. Job resources typically support achieving goals, reduce the effort cost of job demands, and stimulate personal growth (Richter and Hacker, 1998, as cited in Demerouti et al., 2001). For example, an unsustainable workload would be considered a job demand, and receiving help and support from others would be considered a resource. The JD-R has two primary processes for how demands and resources interact with one another (Schaufeli, 2017). The stress process occurs when job demands are high and resources are low. As stress increases strain and pressure on an individual, it leads to the exhaustion of their energy eventually, eventually leading to emotional exhaustion. The second process is the motivational process. This process occurs when there is an excess of job resources. Crucially, the motivational process does not require low job demands. With enough job resources, a person will be energised and engaged, reframing job demands as a challenge to overcome. The JD-R model has had extensive research to confirm its validity. A meta-analysis by Lesener et al. (2019) looked at 74 longitudinal studies of the JD-R with a strong average sample size (n=450) against the assumptions as described by Schaufeli (2017). Although most of the studies appear to come from WEIRD countries (k=60) there were a good number from Asia and Africa (k=13) and a mix of genders. The study confirmed that across the literature, job resources predict burnout negatively (''β'' = −.17; ''p'' < .001) and job demands predict burnout over time (''β'' = .10; ''p'' < .001). Although job resources seem like a cure-all for demands, it's important to note that they do not retroactively treat emotional exhaustion. Under the JD-R model, burnout must be reduced by lowering job demands (Gynning et al., 2025). However, once someone has recovered, it is possible to increase job resources alongside job demands to prevent future burnout and support [[wikipedia:Work_engagement|work engagement]]. Work engagement (engagement) has been discussed as the opposite of burnout (Bakker et al., 2014). Engagement is how people express themselves when performing a role. A role being a pattern of behaviour that a person goes through based on the social expectations of that role, similar to how an actor takes on a role for a movie. For example, when attending a typical restaurant, it is socially expected that a person will engage with the waiter and provide them with their meal order to take to the kitchen. This person is performing the script of a restaurant patron. In the context of a workplace, an employee with high engagement may appear to be energised, enthusiastic, and become absorbed in their work losing track of time (Bakkar et al, 2014). {{RoundBoxTop|theme=3}} '''Why do we feel emotionally exhausted''' Under the JD-R and CORT models, emotional exhaustion occurs when the demand cost outweighs the resources available for an extended period of time, eventually depleting someone's energy reserves. Aelin is experiencing burnout because they increased their job demands by taking on more clients, and has experienced a loss in resources by reducing their free time and social support network. {{RoundBoxBottom}} == Empathy fatigue == You may be familiar with the term [[empathy]]. It is often described in terms of supporting others. For example, when someone close to you experiences a loss and is grieving, you may empathise with them and experience a negative emotion akin to grief too. However, empathy is not exclusive to so called negative emotions. For example, compersion is a positive empathic response when a loved one experiences joy. Parents may experience compersion when something joyful happens to their child. At its core, empathy is the individual ability to imagine or understand another person's feelings and/or emotions (Riess, 2017; Shao et al., 2024). Much like physical or cognitive processes, such as running or problem solving, empathising will drain energy and resources. Further, and similar to the difference in energy expenditure between running and jogging, there are distinct types of empathy that each have different energy costs. === Types of empathy === Research makes a clear distinction between two types of empathy: affective and cognitive empathy. Affective empathy occurs when a person has an emotional reaction to someone else's mental state (Shao et al., 2024). This occurs because (unsure about this next part go through and update accordingly) ''unconscious processes interpret the social stimuli from another person leading to a physiological and cognitive response. These processes may interpret stimuli from any of the five senses.'' Cognitive empathy is when a person understands someone else's feelings or emotions (Cui et al., 2022). * Integrating the two - "affective component is the essence of empathy, and the cognitive component is a process through which that essence comes into being" (Bošnjaković & Radionov, 2018) ** (Author's note: change quote into paraphrasing) * === Empathy or compassion fatigue === * Distinction between empathy and compassion fatigue * Compassion fatigue is closely related to Secondary traumatic stress usually found in healthcare professionals (Figley, 2002) ** Compassion fatigue is to Empathy fatigue as emotional exhaustion is to burnout * Make it clear that Aelin is experiencing Compassion Fatigue vs Empathy fatigue (they are the same but different) (Author's note: include an update to the scenario here!) '''Empathy fatigue questions'''. Choose your answers and click "Submit": <quiz display=simple> {Surface acting is when the observer actually feels the emotion: |type="()"} - True + False {Compassion fatigue and empathy fatigue are synonymous: |type="()"} - True + False </quiz> {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' Under the JD-R and CORT models, a sustained empathic response is a long-term job demand that uses energy to protect existing resources. As Aelin has had their resources is experiencing a * Under the CORT model, a sustained empathic response uses energy to protect existing resources. As resources become low and burnout sets in, cognitive focus narrows leading to empathy fatigue. {{RoundBoxBottom}} == Emotional labor as a mediating factor between empathy fatigue and emotional exhaustion == As a society there are expected [https://en.wikipedia.org/wiki/Social_norm social norms] in behaviour when expressing emotion, regardless if someone is conscious of the process of matching these norms or not. For example, laughing during an emotional speech at a funeral would be considered a rude or offensive expression from the other attendees. As social creatures, humans need a way to regulate socially appropriate emotional expression. This is the essence of Emotional labor: "the regulation of felt and expressed emotions (whether effortful or not) to match these emotional display rules with the objective of fulfilling organizational goals" (Vial & Cowgill, 2022) What is emotional labor (Hochschild, 2012). Theories behind emotional labor involve acting, and an extension of acting known as emotional contagion. === Types of acting === * Surface acting ** This is where emotive dissonance comes into play ** Leads to emotional exhaustion (Kenworthy et al., 2014) ** "Should" language - in this situation, this is what I ''should'' feel * Deep acting ** People who 'deep act', as in genuinely change the way they feel (match the feelings they have with the feelings that they need to display) often report positive relationships with emotional labor ** Actual language - in this situation, this is what I ''actually'' feel Discuss the differences in gender norms and expectations in relationships {| class="wikitable" style="margin: auto;" |- ! !! Deep acting !! Surface acting |- | '''Emotional dissonance''' || || X |- | '''Not known to others''' || Hidden area || Unknown |} === Emotional contagion === Emotional contagion is the idea that emotions are contagious. as the theory to moderate surface/deep acting * Emotional contagion (Panksepp and Lahvis, 2011) ** emotions can spread (expand, combines empathy and acting) ** “the tendency to automatically mimic and synchronize expressions, vocalizations, postures, and movements with those of another person’s and, consequently, to converge emotionally.” (Elaine Hatfield, 1993) * Affective process theory looks at unifying the mechanisms of emotional contagion into three types. Convergent linkage, Divergent linkage, and Complementary linkage (Elfenbein, 2014). Storytelling is a way to promote position emotions (Wei et al., 2024). === The mediating effect of emotional labor === Surface acting does not mediate the effect (high surface acting leads to emotional exhaustion) whereas the inverse is true for deep acting (Wrobel, 2013) Emotional Expression and Emotional Processing (Hoyt et al., 2024) {{RoundBoxTop|theme=3}} '''What role does emotional labor play in emotional exhaustion''' * In relation to the CORT and JD-R theories, emotional labor acts as a job demand and expenditure of resources to protect existing resources {{RoundBoxBottom}} == Strategies for combating empathy fatigue and emotional fatigue == * Talk about emotion regulation (Gross, 1998) and mood regulation strategies as a way to deal with coping (Wrobel, 2013) (Author's note: need more research/references for this section) * Talk about breaks ** Rest and downtime from not performing emotional labor regenerates * How do we reduce the amount of emotional labor we're performing (Jeung et al., 2018) ** Communication/boundaries ** Reducing surface acting (allowing ourselves to feel how we actually feel and reduce emotive dissonance) '''Table 1. Treatment phases of clinical burnout (based on Table 2 from van Dam, A. 2021)''' {| class="wikitable" style="margin: auto; |- ! Phase !! Treatment goal !! Therapeutic interventions |- | '''Phase 1: Crisis''' || Recognition of the patient that the problems are serious, and that serious action is required || Psychoeducation Sick leave Dropping domestic tasks and social obligations Inform social network |- | '''Phase 2: Recovery''' || Recovery of the (physiological) stress system to normal allostatic stress levels || Registration of stress and activities Relaxtion exercises Mindfulness healthy lifestyle advices Graded exercise Gradually resume activities Psychoeducation social network and employer Gradual return to work |- | '''Phase 3: Prevention, learning from the past''' || Acquiring insight and skills to prevent relapse in clinical burnout || Analysing stressful situations Analysing dysfunctional thought patterns Learning new coping skills Learning social skills Job crafting Time management Making choices regarding career and personal life |- | '''Phase 4: Post burnout growth''' || Improving sustainable quality of life. || Setting priorities regarding quality of life and interpersonal relationships. |} === Strategies for the workplace === * Discuss types of strategies that are available in the workplace (organisation models) * SCARF model (Aplin-Houtz, 2025) ** Link back to COR and JDR models e.g. praise could be a job resource vs criticism could be a job demand *Reduce workload to reduce emotional exhaustion, and by extension reduce emotional fatigue. {{RoundBoxTop|theme=3}} '''How does understanding emotional labor and empathy fatigue work to reduce the effects of emotional exhaustion''' *Psychoeducation has been shown to reduce the effects of emotional labor and empathy fatigue on emotional exhaustion {{RoundBoxBottom}} ==Conclusion== * Summarise what empathy fatigue is, and why people become emotionally exhausted (+ the role of emotional labor) * Summarise the symptoms and how to recognise them in yourself and someone else * Emphasise and summarise psychological strategies to support someone going through emotional exhaustion * Summarise ways to prevent emotional exhaustion from happening * Overall message is: *# What is emotional exhaustion *# how do you recognise it *# how do you treat it *# how do you prevent it * Link back to the focus questions ==See also== * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Metacognition_and_emotional_regulation&oldid=2756264 Metacognition and emotional regulation] (Book chapter, 2026) {{ic|Use an internal link style as shown in Tutorial 2}} * [[wikipedia:Conservation_of_resources_theory|Conservation of resources theory]] (Wikipedia) * [[wikipedia:Job_demands-resources_model|Job demands-resources model]] (Wikipedia) ==References== {{Hanging indent|1= Aplin-Houtz, M. (2025). Unraveling the SCARF: How dimensions of the SCARF model influence the relationship between workplace exclusion and turnover intentions. Journal of Management & Organization, 31(6), 2699–2725. https://doi.org/10.1017/jmo.2025.10026 Bošnjaković, J., & Radionov, T. (2018). Empathy: Concepts, Theories and Neuroscientific Basis. Alcoholism (Zagreb), 54(2), 123–150. https://doi.org/10.20471/de..2018.54.02.04 Chan, F. M. (2026). The State of Workplace Burnout 2026: The Workplace Divide. Infinite Potential, Sydney, Australia. https://www.ahri.com.au/wp-content/uploads/The-State-of-Workplace-Burnout-2026-Final.pdf Cui, F., Huang, X., Jing, Y., Luo, Y., Liu, J., & Gu, R. (2022). How resource sharing resists scarcity: the role of cognitive empathy and its neurobiological mechanisms. Cerebral Cortex (New York, N.Y. 1991), 32(23), 5330–5342. https://doi.org/10.1093/cercor/bhac017 De Beer, L. T., van der Vaart, L., Escaffi-Schwarz, M., De Witte, H., & Schaufeli, W. B. (2024). Maslach Burnout Inventory - General Survey: A Systematic Review and Meta-Analysis of Measurement Properties. European journal of psychological assessment : official organ of the European Association of Psychological Assessment, 40(5), 360-375. https://doi.org/10.1027/1015-5759/a000797 Demerouti, E., Bakker, A. B., Nachreiner, F., & Schaufeli, W. B. (2001). The Job Demands-Resources Model of Burnout. Journal of Applied Psychology, 86(3), 499–512. https://doi.org/10.1037/0021-9010.86.3.499 Figley, C. R. (2002). Compassion fatigue: Psychotherapists’ chronic lack of self care. Journal of Clinical Psychology, 58(11), 1433–1441. https://doi.org/10.1002/jclp.10090 Grandey, A. A. (2003). When ‘the show must go on’: surface acting and deep acting as determinants of emotional exhaustion and peer-rated service delivery. Academy of Management Journal, 46(1), 86–96. https://doi.org/10.5465/30040678 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 Halbesleben, J. R. B., Neveu, J.-P., Paustian-Underdahl, S. C., & Westman, M. (2014). Getting to the “COR”: Understanding the Role of Resources in Conservation of Resources Theory. Journal of Management, 40(5), 1334–1364. https://doi.org/10.1177/0149206314527130 Hatfield, E., Cacioppo, J. T., & Rapson, R. L. (1993). Emotional Contagion. Current Directions in Psychological Science : A Journal of the American Psychological Society, 2(3), 96–99. https://doi.org/10.1111/1467-8721.ep10770953 Hollebeek, L. D., Hammedi, W., & Sprott, D. E. (2023). Consumer engagement, stress, and conservation of resources theory: A review, conceptual development, and future research agenda. Psychology & marketing, 40(5), 926-937. https://doi.org/10.1002/mar.21807 Hobfoll, S. E. (1989). Conservation of Resources: A New Attempt at Conceptualizing Stress. The American Psychologist, 44(3), 513–524. https://doi.org/10.1037/0003-066X.44.3.513 Hochschild, A. R. (2012). The managed heart : commercialization of human feeling (Updated, with a new preface.). University of California Press. Jeung, D.-Y., Kim, C., & Chang, S.-J. (2018). Emotional Labor and Burnout: A Review of the Literature. Yonsei Medical Journal, 59(2), 187–193. https://doi.org/10.3349/ymj.2018.59.2.187 Lavelle, J. J., Rupp, D. E., Herda, D. N., Pandey, A., & Lauck, J. R. (2021). Customer Injustice and Employee Performance: Roles of Emotional Exhaustion, Surface Acting, and Emotional Demands–Abilities Fit. Journal of Management, 47(3), Article 0149206319869426. https://doi.org/10.1177/0149206319869426 Lloyd, C., & King, R. (2004). A survey of burnout among Australian mental health occupational therapists and social workers. Social Psychiatry and Psychiatric Epidemiology, 39(9), 752-757. https://doi.org/https://doi.org/10.1007/s00127-004-0808-7 Kenworthy, J., Fay, C., Frame, M., & Petree, R. (2014). A meta-analytic review of the relationship between emotional dissonance and emotional exhaustion. Journal of Applied Social Psychology, 44(2), 94-105. https://doi.org/https://doi.org/10.1111/jasp.12211 Martingano, A. J., Stosic, M. D., & Konrath, S. (2026). Different empathy types show opposing associations with burnout: systematic review and meta-analyses. Psychology, Health & Medicine, 31(3), 798-829. https://doi.org/10.1080/13548506.2025.2591859 Michielsen, H. J., Willemsen, T. M., Croon, M. A., de Vries, J., & Van Heck, G. L. (2004). Determinants of general fatigue and emotional exhaustion: a prospective study. Psychology & health, 19(2), 223-235. https://doi.org/10.1080/08870440310001627135 Panksepp, J. B., & Lahvis, G. P. (2011). Rodent empathy and affective neuroscience. Neuroscience and Biobehavioral Reviews, 35(9), 1864–1875. https://doi.org/10.1016/j.neubiorev.2011.05.013 Pooladvand, S., & Hasanzadeh, S. (2023). Impacts of Stress on Workers’ Risk-Taking Behaviors: Cognitive Tunneling and Impaired Selective Attention. Journal of Construction Engineering and Management, 149(8), 04023060. https://doi.org/doi:10.1061/JCEMD4.COENG-13339 Riess, H. (2017). The Science of Empathy. Journal of Patient Experience, 4(2), 74–77. https://doi.org/10.1177/2374373517699267 Schaufeli, W. B. (2017). Applying the job demands-resources model: a “how to” guide to measuring and tackling work engagement and burnout. Organizational Dynamics, 46(2), 120–132. https://doi.org/10.1016/j.orgdyn.2017.04.008 Schneiderman, N., Ironson, G., & Siegel, S. D. (2005). Stress and Health: Psychological, Behavioral, and Biological Determinants. Annual Review of Clinical Psychology, 1(Volume 1, 2005), 607-628. https://doi.org/https://doi.org/10.1146/annurev.clinpsy.1.102803.144141 Seidler, A., Thinschmidt, M., Deckert, S., Then, F., Hegewald, J., Nieuwenhuijsen, K., & Riedel-Heller, S. G. (2014). The role of psychosocial working conditions on burnout and its core component emotional exhaustion – a systematic review. Journal of Occupational Medicine and Toxicology, 9(1), 10. https://doi.org/10.1186/1745-6673-9-10 Shao, M., Li, L., Li, X., Wei, Z., Wang, J., Hong, M., Liu, X., & Meng, J. (2024). The effect of top-down attention on empathy fatigue. Cerebral Cortex (New York, N.Y. 1991), 34(1), Article bhad441. https://doi.org/10.1093/cercor/bhad441 van Dam, A. (2021). A clinical perspective on burnout: diagnosis, classification, and treatment of clinical burnout. European Journal of Work and Organizational Psychology, 30(5), 732-741. https://doi.org/10.1080/1359432X.2021.1948400 Vial, A. C., & Cowgill, C. M. (2022). Heavier Lies Her Crown: Gendered Patterns of Leader Emotional Labor and Their Downstream Effects. Frontiers in Psychology, 13, 849566. https://doi.org/https://doi.org/10.3389/fpsyg.2022.849566 Wharton, A. S. (2009). The Sociology of Emotional Labor. Annual Review of Sociology, 35(1), 147–165. https://doi.org/10.1146/annurev-soc-070308-115944 Wrobel, M. (2013). Can empathy lead to emotional exhaustion in teachers? The mediating role of emotional labor. International Journal of Occupational Medicine and Environmental Health, 26(4), 581–592. https://doi.org/10.2478/s13382-013-0123-1 }} ==External links== Provide [[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: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Burnout]] [[Category:Motivation and emotion/Book/Empathy]] 50sp8413myfkcr15wdmj4wzj41c6kyd 2844536 2844532 2026-11-04T23:38:30Z U3143751 3106785 /* Job demands-resources model */ 2844536 wikitext text/x-wiki {{title|Empathy fatigue and emotional exhaustion:<br>How does sustained empathic engagement contribute to emotional exhaustion?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}} [[File:Exhausted smartphone user.JPG|right|thumb|150px|'''Figure 1'''. An emotionally exhausted Aelin at the end of the work day]] ; Scenario Aelin, a 23-year-old social worker employed at headspace {{ic|Link to the Wikipedia article to help explain what this term means}}, works with approximately 20 {{ic|Explain what sort of}}Imagine you are only 23 years old and are working as a social worker at headspace. You see around 20 clients each week, many of whom have experienced rough living conditions. During your first six months in this role, you felt highly motivated and excited to help people in need. You are an excellent active listener and can easily remember details about your clients Live’s. You are also good at recognising signs of distress in your client’s body language and adjust yours in order to show warmth and understanding. As you progress in your role, you take on more cases that involve complex past traumas. The workload becomes too much for you to get done at work, but you don’t want your clients to miss out, so you take your work home to complete it in the evenings. Your workload increases again and you now start doing work on your weekends as well. After several months of working this way, you start to notice you feel exhausted every morning before you have even started work. You then start to notice you’re responding to people at work differently. Instead of your usual energised active listening response you feel frustrated, empathetic and detached. After work you are too drained to socialise with friends. This begins to worry you. What if you won’t be able to continue working as a social worker? {{RoundBoxBottom}} Aelin is experiencing '''emotional exhaustion''' from a sustained empathic response. Due to the nature of Aelin's work, they are constantly engaged in an empathic response. Empathic responses require cognitive and physical resources to sustain. Because Aelin is taking their work home and not resting on weekends, they do not have a chance to rest. It is important for Aelin to recognise what is happening and why. Not actioning the problem has led to empathy fatigue and emotional exhaustion. Emotional exhaustion is one of the key aspects of [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FEmpathy_fatigue_and_emotional_exhaustion&diff=2840411&oldid=2832859 burnout], a long-term condition of existing in a constant state of high stress. Psychological science can support Aelin in recognising and mapping the symptoms they are experiencing, understand coping strategies to minimise the current and future effects to improve their wellbeing. Understanding the symptoms will also support their role as a social worker, providing more tools for the people they support on a day-to-day basis. {{RoundBoxTop|theme=3}} '''Focus questions''' * Why do people become emotionally exhausted? * What is the relationship between empathy fatigue and emotional exhaustion? * What role does emotional labor play in emotional exhaustion? * How does understanding emotional labor and empathy fatigue work to reduce the effects of emotional exhaustion? {{RoundBoxBottom}} == Emotional exhaustion == To understand emotional exhaustion, it will be important to first understand burnout. Burnout is a long-term condition of existing in a constant state of high stress and affects millions of people annually. In the early 2000s, a survey from Australia found that more than 50% of social workers and occupational workers reported high levels of burnout symptoms (Llyod and King, 2004) and a recent report from 2026 found that more than one third of all workers in Australia were experiencing burnout (Chan, 2026). Being in a state of high stress for short periods of time is a useful adaptation for humans, so much so that it has been given a name, [[wikipedia:Eustress|Eustress]]. However, as a long-term condition it leads to individual health issues and can be linked to cardiovascular disease, increased vulnerability to upper respiratory diseases, and chronic inflammation (Schneiderman et al., 2005), and cognitive issues such as [[wikipedia:Inattentional_blindness|cognitive tunneling]] and impaired selective attention (Pooladvand and Hasanzadeh, 2023). Burnout can be defined across three dimensions: emotional exhaustion, [[wikipedia:Depersonalization|depersonalisation]], and reduced feelings of personal accomplishment (Maslach, 1993, as cited in Martingano et al., 2026). Emotional exhaustion may be considered the 'core' component of burnout and refers to the depletion of emotional resources, leading to feelings of overwhelm, overextension, and exhaustion (Martingano et al., 2026; De Beer et al., 2024; Seidler et al., 2014). As a major factor, its symptoms are similar to burnout and stress. Although antecedents of burnout are typically agreed upon (Bakker et al., 2014) and divided into individual (e.g. self-efficacy) and situational categories (e.g. work load), there are many theories that attempt to explain why emotional exhaustion occurs. Two of these theories with good evidence are the [[wikipedia:Conservation_of_resources_theory|Conservation of resources stress model]] (Hobfoll, 1989) and the [[wikipedia:Job_demands-resources_model|Job Demands-Resources model]] (Demerouti et al., 2001). === Conservation of resources stress model === The Conservation of resources stress model (CORT; Hobfall, 1989) is a theory that works to understand and explain why stress occurs. Under the model, humans are motivated to maintain, pursue, or attain resources. Stress, defined as "an individual's perceived pressure or strain" (Hollebeek et al., 2023 p.928), occurs when these resources are threatened or lost, or gained at a net loss. As noted above, resources aren't strictly tied to being physical objects. They can also be emotional states, conditions, or even ideas. For example, someone may place value on enjoying a calm, quiet Sunday morning at home. If their neighbour starts using power tools loudly next door, it would disrupt this state and induce stress in the individual. The CORT model focuses primarily on two principles. The first principle is that there is more harm in losing resources than there is in gaining them back (Halbesleben et al., 2014). This is known as loss salience, and some research suggests that the pain of loss is felt twice as strong as the pleasure of gain (Thaler, 2000). Looking at our previous example, imagine that the person who enjoys a quiet Sunday morning at home equally enjoys the sound of a loud car engine. Even though the person enjoys the sound of car engines, if the neighbour started revving their car loudly instead of using power tools, it would still disrupt their desired state of quiet and lead to an increase of stress. The second principle is that people must invest existing resources in order to gain or protect resources (Halbesleben et al., 2014). This resource investment includes recovering from resource loss. Coping is how people manage or overcome a difficult situation, such as protecting or recovering from resource loss. Coping can be broken down into different strategies that people use, but not all strategies are equal (Ito and Brotheridge, 2003). They are divided into two categories: control or escape strategies. Control strategies are considered better at reducing the strain from stress and consist of positive thinking, direct action, and help seeking (Havlovic & Keenan, 1991, as cited in Ito and Brotheridge, 2003). The most useful control strategies incorporate a person's [[locus of control]], emphasise seeking support from co-workers, and highlight the importance of autonomy in decision making. Alarcon (2011) conducted a meta-analysis of 231 studies in a work setting that used the [[wikipedia:Maslach_Burnout_Inventory|Maslach Burnout Inventory]] (MBI) to review the relationship between these principles and burnout to confirm their validity. Although the study had no specific mention of demographics, it was noted that multiple translations of the MBI were used including Dutch (k=51), German (k=7), Italian (k=2), and French (k=2). The findings indicate that workload is positively related to emotional exhaustion (''ρ'' = .49, ''k'' = 86, ''N'' = 51,529) and that control (''ρ'' = − .26, ''k'' = 56, ''N'' = 33,297) and autonomy (''ρ'' = − .24, ''k'' = 34, ''N'' = 17,165) were each negatively associated with emotional exhaustion. The difference of ''ρ'' values is consistent with Thaler (2000) and supports that the benefit of gaining is approximately half that of losing resources. === Job demands-resources model === Another model that attempts to understand why people become emotionally exhausted is the Job Demands-Resources model (JD-R; Demerouti et al., 2001). The JD-R model discusses two concepts: job demands and job resources. Demands and resources are both related to the physical, social, or psychological aspects of a job, but differ in their impact on a person. Job demands require physical or mental effort and exhaust a person's energy supply. Job resources typically support achieving goals, reduce the effort cost of job demands, and stimulate personal growth (Richter and Hacker, 1998, as cited in Demerouti et al., 2001). For example, an unsustainable workload would be considered a job demand, and receiving help and support from others would be considered a resource. The JD-R has two primary processes for how demands and resources interact with one another (Schaufeli, 2017). The stress process occurs when job demands are high and resources are low. As stress increases strain and pressure on an individual, it leads to the exhaustion of their energy eventually, eventually leading to emotional exhaustion. The second process is the motivational process. This process occurs when there is an excess of job resources. Crucially, the motivational process does not require low job demands. With enough job resources, a person will be energised and engaged, reframing job demands as a challenge to overcome. The JD-R model has had extensive research to confirm its validity. A meta-analysis by Lesener et al. (2019) looked at 74 longitudinal studies of the JD-R with a strong average sample size (n=450) against the assumptions as described by Schaufeli (2017). Although most of the studies appear to come from [[wikipedia:Psychology#WEIRD_bias|WEIRD]] countries (k=60) there were a good number from Asia and Africa (k=13) and a mix of genders. The study confirmed that across the literature, job resources predict burnout negatively (''β'' = −.17; ''p'' < .001) and job demands predict burnout over time (''β'' = .10; ''p'' < .001). Although job resources seem like a cure-all for demands, it's important to note that they do not retroactively treat emotional exhaustion. Under the JD-R model, burnout must be reduced by lowering job demands (Gynning et al., 2025). However, once someone has recovered, it is possible to increase job resources alongside job demands to prevent future burnout and support [[wikipedia:Work_engagement|work engagement]]. Work engagement (engagement) has been discussed as the opposite of burnout (Bakker et al., 2014). Engagement is how people express themselves when performing a role. A role being a pattern of behaviour that a person goes through based on the social expectations of that role, similar to how an actor takes on a role for a movie. For example, when attending a typical restaurant, it is socially expected that a person will engage with the waiter and provide them with their meal order to take to the kitchen. This person is performing the script of a restaurant patron. In the context of a workplace, an employee with high engagement may appear to be energised, enthusiastic, and become absorbed in their work losing track of time (Bakkar et al, 2014). {{RoundBoxTop|theme=3}} '''Why do we feel emotionally exhausted''' Under the JD-R and CORT models, emotional exhaustion occurs when the demand cost outweighs the resources available for an extended period of time, eventually depleting someone's energy reserves. Aelin is experiencing burnout because they increased their job demands by taking on more clients, and has experienced a loss in resources by reducing their free time and social support network. {{RoundBoxBottom}} == Empathy fatigue == You may be familiar with the term [[empathy]]. It is often described in terms of supporting others. For example, when someone close to you experiences a loss and is grieving, you may empathise with them and experience a negative emotion akin to grief too. However, empathy is not exclusive to so called negative emotions. For example, compersion is a positive empathic response when a loved one experiences joy. Parents may experience compersion when something joyful happens to their child. At its core, empathy is the individual ability to imagine or understand another person's feelings and/or emotions (Riess, 2017; Shao et al., 2024). Much like typical physical or cognitive processes, such as running or problem solving, empathising will drain energy and resources. Further, and similar to the difference in energy expenditure between running and jogging, there are distinct types of empathy that each have different energy costs. === Types of empathy === Research makes a clear distinction between two types of empathy: affective and cognitive empathy. Affective empathy occurs when a person has an emotional reaction to someone else's mental state (Shao et al., 2024), whereas cognitive empathy occurs when a person understands someone else's feelings or emotions (Cui et al., 2022). This occurs because (unsure about this next part go through and update accordingly) ''unconscious processes interpret the social stimuli from another person leading to a physiological and cognitive response. These processes may interpret stimuli from any of the five senses.'' Cognitive empathy is when a person understands someone else's feelings or emotions (Cui et al., 2022). * Integrating the two - "affective component is the essence of empathy, and the cognitive component is a process through which that essence comes into being" (Bošnjaković & Radionov, 2018) ** (Author's note: change quote into paraphrasing) * === Empathy or compassion fatigue === * Distinction between empathy and compassion fatigue * Compassion fatigue is closely related to Secondary traumatic stress usually found in healthcare professionals (Figley, 2002) ** Compassion fatigue is to Empathy fatigue as emotional exhaustion is to burnout * Make it clear that Aelin is experiencing Compassion Fatigue vs Empathy fatigue (they are the same but different) (Author's note: include an update to the scenario here!) '''Empathy fatigue questions'''. Choose your answers and click "Submit": <quiz display=simple> {Surface acting is when the observer actually feels the emotion: |type="()"} - True + False {Compassion fatigue and empathy fatigue are synonymous: |type="()"} - True + False </quiz> {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' Under the JD-R and CORT models, a sustained empathic response is a long-term job demand that uses energy to protect existing resources. As Aelin has had their resources is experiencing a * Under the CORT model, a sustained empathic response uses energy to protect existing resources. As resources become low and burnout sets in, cognitive focus narrows leading to empathy fatigue. {{RoundBoxBottom}} == Emotional labor as a mediating factor between empathy fatigue and emotional exhaustion == As a society there are expected [https://en.wikipedia.org/wiki/Social_norm social norms] in behaviour when expressing emotion, regardless if someone is conscious of the process of matching these norms or not. For example, laughing during an emotional speech at a funeral would be considered a rude or offensive expression from the other attendees. As social creatures, humans need a way to regulate socially appropriate emotional expression. This is the essence of Emotional labor: "the regulation of felt and expressed emotions (whether effortful or not) to match these emotional display rules with the objective of fulfilling organizational goals" (Vial & Cowgill, 2022) What is emotional labor (Hochschild, 2012). Theories behind emotional labor involve acting, and an extension of acting known as emotional contagion. === Types of acting === * Surface acting ** This is where emotive dissonance comes into play ** Leads to emotional exhaustion (Kenworthy et al., 2014) ** "Should" language - in this situation, this is what I ''should'' feel * Deep acting ** People who 'deep act', as in genuinely change the way they feel (match the feelings they have with the feelings that they need to display) often report positive relationships with emotional labor ** Actual language - in this situation, this is what I ''actually'' feel Discuss the differences in gender norms and expectations in relationships {| class="wikitable" style="margin: auto;" |- ! !! Deep acting !! Surface acting |- | '''Emotional dissonance''' || || X |- | '''Not known to others''' || Hidden area || Unknown |} === Emotional contagion === Emotional contagion is the idea that emotions are contagious. as the theory to moderate surface/deep acting * Emotional contagion (Panksepp and Lahvis, 2011) ** emotions can spread (expand, combines empathy and acting) ** “the tendency to automatically mimic and synchronize expressions, vocalizations, postures, and movements with those of another person’s and, consequently, to converge emotionally.” (Elaine Hatfield, 1993) * Affective process theory looks at unifying the mechanisms of emotional contagion into three types. Convergent linkage, Divergent linkage, and Complementary linkage (Elfenbein, 2014). Storytelling is a way to promote position emotions (Wei et al., 2024). === The mediating effect of emotional labor === Surface acting does not mediate the effect (high surface acting leads to emotional exhaustion) whereas the inverse is true for deep acting (Wrobel, 2013) Emotional Expression and Emotional Processing (Hoyt et al., 2024) {{RoundBoxTop|theme=3}} '''What role does emotional labor play in emotional exhaustion''' * In relation to the CORT and JD-R theories, emotional labor acts as a job demand and expenditure of resources to protect existing resources {{RoundBoxBottom}} == Strategies for combating empathy fatigue and emotional fatigue == * Talk about emotion regulation (Gross, 1998) and mood regulation strategies as a way to deal with coping (Wrobel, 2013) (Author's note: need more research/references for this section) * Talk about breaks ** Rest and downtime from not performing emotional labor regenerates * How do we reduce the amount of emotional labor we're performing (Jeung et al., 2018) ** Communication/boundaries ** Reducing surface acting (allowing ourselves to feel how we actually feel and reduce emotive dissonance) '''Table 1. Treatment phases of clinical burnout (based on Table 2 from van Dam, A. 2021)''' {| class="wikitable" style="margin: auto; |- ! Phase !! Treatment goal !! Therapeutic interventions |- | '''Phase 1: Crisis''' || Recognition of the patient that the problems are serious, and that serious action is required || Psychoeducation Sick leave Dropping domestic tasks and social obligations Inform social network |- | '''Phase 2: Recovery''' || Recovery of the (physiological) stress system to normal allostatic stress levels || Registration of stress and activities Relaxtion exercises Mindfulness healthy lifestyle advices Graded exercise Gradually resume activities Psychoeducation social network and employer Gradual return to work |- | '''Phase 3: Prevention, learning from the past''' || Acquiring insight and skills to prevent relapse in clinical burnout || Analysing stressful situations Analysing dysfunctional thought patterns Learning new coping skills Learning social skills Job crafting Time management Making choices regarding career and personal life |- | '''Phase 4: Post burnout growth''' || Improving sustainable quality of life. || Setting priorities regarding quality of life and interpersonal relationships. |} === Strategies for the workplace === * Discuss types of strategies that are available in the workplace (organisation models) * SCARF model (Aplin-Houtz, 2025) ** Link back to COR and JDR models e.g. praise could be a job resource vs criticism could be a job demand *Reduce workload to reduce emotional exhaustion, and by extension reduce emotional fatigue. {{RoundBoxTop|theme=3}} '''How does understanding emotional labor and empathy fatigue work to reduce the effects of emotional exhaustion''' *Psychoeducation has been shown to reduce the effects of emotional labor and empathy fatigue on emotional exhaustion {{RoundBoxBottom}} ==Conclusion== * Summarise what empathy fatigue is, and why people become emotionally exhausted (+ the role of emotional labor) * Summarise the symptoms and how to recognise them in yourself and someone else * Emphasise and summarise psychological strategies to support someone going through emotional exhaustion * Summarise ways to prevent emotional exhaustion from happening * Overall message is: *# What is emotional exhaustion *# how do you recognise it *# how do you treat it *# how do you prevent it * Link back to the focus questions ==See also== * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Metacognition_and_emotional_regulation&oldid=2756264 Metacognition and emotional regulation] (Book chapter, 2026) {{ic|Use an internal link style as shown in Tutorial 2}} * [[wikipedia:Conservation_of_resources_theory|Conservation of resources theory]] (Wikipedia) * [[wikipedia:Job_demands-resources_model|Job demands-resources model]] (Wikipedia) ==References== {{Hanging indent|1= Aplin-Houtz, M. (2025). Unraveling the SCARF: How dimensions of the SCARF model influence the relationship between workplace exclusion and turnover intentions. Journal of Management & Organization, 31(6), 2699–2725. https://doi.org/10.1017/jmo.2025.10026 Bošnjaković, J., & Radionov, T. (2018). Empathy: Concepts, Theories and Neuroscientific Basis. Alcoholism (Zagreb), 54(2), 123–150. https://doi.org/10.20471/de..2018.54.02.04 Chan, F. M. (2026). The State of Workplace Burnout 2026: The Workplace Divide. Infinite Potential, Sydney, Australia. https://www.ahri.com.au/wp-content/uploads/The-State-of-Workplace-Burnout-2026-Final.pdf Cui, F., Huang, X., Jing, Y., Luo, Y., Liu, J., & Gu, R. (2022). How resource sharing resists scarcity: the role of cognitive empathy and its neurobiological mechanisms. Cerebral Cortex (New York, N.Y. 1991), 32(23), 5330–5342. https://doi.org/10.1093/cercor/bhac017 De Beer, L. T., van der Vaart, L., Escaffi-Schwarz, M., De Witte, H., & Schaufeli, W. B. (2024). Maslach Burnout Inventory - General Survey: A Systematic Review and Meta-Analysis of Measurement Properties. European journal of psychological assessment : official organ of the European Association of Psychological Assessment, 40(5), 360-375. https://doi.org/10.1027/1015-5759/a000797 Demerouti, E., Bakker, A. B., Nachreiner, F., & Schaufeli, W. B. (2001). The Job Demands-Resources Model of Burnout. Journal of Applied Psychology, 86(3), 499–512. https://doi.org/10.1037/0021-9010.86.3.499 Figley, C. R. (2002). Compassion fatigue: Psychotherapists’ chronic lack of self care. Journal of Clinical Psychology, 58(11), 1433–1441. https://doi.org/10.1002/jclp.10090 Grandey, A. A. (2003). When ‘the show must go on’: surface acting and deep acting as determinants of emotional exhaustion and peer-rated service delivery. Academy of Management Journal, 46(1), 86–96. https://doi.org/10.5465/30040678 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 Halbesleben, J. R. B., Neveu, J.-P., Paustian-Underdahl, S. C., & Westman, M. (2014). Getting to the “COR”: Understanding the Role of Resources in Conservation of Resources Theory. Journal of Management, 40(5), 1334–1364. https://doi.org/10.1177/0149206314527130 Hatfield, E., Cacioppo, J. T., & Rapson, R. L. (1993). Emotional Contagion. Current Directions in Psychological Science : A Journal of the American Psychological Society, 2(3), 96–99. https://doi.org/10.1111/1467-8721.ep10770953 Hollebeek, L. D., Hammedi, W., & Sprott, D. E. (2023). Consumer engagement, stress, and conservation of resources theory: A review, conceptual development, and future research agenda. Psychology & marketing, 40(5), 926-937. https://doi.org/10.1002/mar.21807 Hobfoll, S. E. (1989). Conservation of Resources: A New Attempt at Conceptualizing Stress. The American Psychologist, 44(3), 513–524. https://doi.org/10.1037/0003-066X.44.3.513 Hochschild, A. R. (2012). The managed heart : commercialization of human feeling (Updated, with a new preface.). University of California Press. Jeung, D.-Y., Kim, C., & Chang, S.-J. (2018). Emotional Labor and Burnout: A Review of the Literature. Yonsei Medical Journal, 59(2), 187–193. https://doi.org/10.3349/ymj.2018.59.2.187 Lavelle, J. J., Rupp, D. E., Herda, D. N., Pandey, A., & Lauck, J. R. (2021). Customer Injustice and Employee Performance: Roles of Emotional Exhaustion, Surface Acting, and Emotional Demands–Abilities Fit. Journal of Management, 47(3), Article 0149206319869426. https://doi.org/10.1177/0149206319869426 Lloyd, C., & King, R. (2004). A survey of burnout among Australian mental health occupational therapists and social workers. Social Psychiatry and Psychiatric Epidemiology, 39(9), 752-757. https://doi.org/https://doi.org/10.1007/s00127-004-0808-7 Kenworthy, J., Fay, C., Frame, M., & Petree, R. (2014). A meta-analytic review of the relationship between emotional dissonance and emotional exhaustion. Journal of Applied Social Psychology, 44(2), 94-105. https://doi.org/https://doi.org/10.1111/jasp.12211 Martingano, A. J., Stosic, M. D., & Konrath, S. (2026). Different empathy types show opposing associations with burnout: systematic review and meta-analyses. Psychology, Health & Medicine, 31(3), 798-829. https://doi.org/10.1080/13548506.2025.2591859 Michielsen, H. J., Willemsen, T. M., Croon, M. A., de Vries, J., & Van Heck, G. L. (2004). Determinants of general fatigue and emotional exhaustion: a prospective study. Psychology & health, 19(2), 223-235. https://doi.org/10.1080/08870440310001627135 Panksepp, J. B., & Lahvis, G. P. (2011). Rodent empathy and affective neuroscience. Neuroscience and Biobehavioral Reviews, 35(9), 1864–1875. https://doi.org/10.1016/j.neubiorev.2011.05.013 Pooladvand, S., & Hasanzadeh, S. (2023). Impacts of Stress on Workers’ Risk-Taking Behaviors: Cognitive Tunneling and Impaired Selective Attention. Journal of Construction Engineering and Management, 149(8), 04023060. https://doi.org/doi:10.1061/JCEMD4.COENG-13339 Riess, H. (2017). The Science of Empathy. Journal of Patient Experience, 4(2), 74–77. https://doi.org/10.1177/2374373517699267 Schaufeli, W. B. (2017). Applying the job demands-resources model: a “how to” guide to measuring and tackling work engagement and burnout. Organizational Dynamics, 46(2), 120–132. https://doi.org/10.1016/j.orgdyn.2017.04.008 Schneiderman, N., Ironson, G., & Siegel, S. D. (2005). Stress and Health: Psychological, Behavioral, and Biological Determinants. Annual Review of Clinical Psychology, 1(Volume 1, 2005), 607-628. https://doi.org/https://doi.org/10.1146/annurev.clinpsy.1.102803.144141 Seidler, A., Thinschmidt, M., Deckert, S., Then, F., Hegewald, J., Nieuwenhuijsen, K., & Riedel-Heller, S. G. (2014). The role of psychosocial working conditions on burnout and its core component emotional exhaustion – a systematic review. Journal of Occupational Medicine and Toxicology, 9(1), 10. https://doi.org/10.1186/1745-6673-9-10 Shao, M., Li, L., Li, X., Wei, Z., Wang, J., Hong, M., Liu, X., & Meng, J. (2024). The effect of top-down attention on empathy fatigue. Cerebral Cortex (New York, N.Y. 1991), 34(1), Article bhad441. https://doi.org/10.1093/cercor/bhad441 van Dam, A. (2021). A clinical perspective on burnout: diagnosis, classification, and treatment of clinical burnout. European Journal of Work and Organizational Psychology, 30(5), 732-741. https://doi.org/10.1080/1359432X.2021.1948400 Vial, A. C., & Cowgill, C. M. (2022). Heavier Lies Her Crown: Gendered Patterns of Leader Emotional Labor and Their Downstream Effects. Frontiers in Psychology, 13, 849566. https://doi.org/https://doi.org/10.3389/fpsyg.2022.849566 Wharton, A. S. (2009). The Sociology of Emotional Labor. Annual Review of Sociology, 35(1), 147–165. https://doi.org/10.1146/annurev-soc-070308-115944 Wrobel, M. (2013). Can empathy lead to emotional exhaustion in teachers? The mediating role of emotional labor. International Journal of Occupational Medicine and Environmental Health, 26(4), 581–592. https://doi.org/10.2478/s13382-013-0123-1 }} ==External links== Provide [[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: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Burnout]] [[Category:Motivation and emotion/Book/Empathy]] fe79vkw9voefuzl6akqnhgqedmtfo5v 2844537 2844536 2026-11-04T23:51:49Z U3143751 3106785 /* Types of empathy */ Updated and removed types of empathy heading. Moved emotional contagion here to link through to the next section. 2844537 wikitext text/x-wiki {{title|Empathy fatigue and emotional exhaustion:<br>How does sustained empathic engagement contribute to emotional exhaustion?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}} [[File:Exhausted smartphone user.JPG|right|thumb|150px|'''Figure 1'''. An emotionally exhausted Aelin at the end of the work day]] ; Scenario Aelin, a 23-year-old social worker employed at headspace {{ic|Link to the Wikipedia article to help explain what this term means}}, works with approximately 20 {{ic|Explain what sort of}}Imagine you are only 23 years old and are working as a social worker at headspace. You see around 20 clients each week, many of whom have experienced rough living conditions. During your first six months in this role, you felt highly motivated and excited to help people in need. You are an excellent active listener and can easily remember details about your clients Live’s. You are also good at recognising signs of distress in your client’s body language and adjust yours in order to show warmth and understanding. As you progress in your role, you take on more cases that involve complex past traumas. The workload becomes too much for you to get done at work, but you don’t want your clients to miss out, so you take your work home to complete it in the evenings. Your workload increases again and you now start doing work on your weekends as well. After several months of working this way, you start to notice you feel exhausted every morning before you have even started work. You then start to notice you’re responding to people at work differently. Instead of your usual energised active listening response you feel frustrated, empathetic and detached. After work you are too drained to socialise with friends. This begins to worry you. What if you won’t be able to continue working as a social worker? {{RoundBoxBottom}} Aelin is experiencing '''emotional exhaustion''' from a sustained empathic response. Due to the nature of Aelin's work, they are constantly engaged in an empathic response. Empathic responses require cognitive and physical resources to sustain. Because Aelin is taking their work home and not resting on weekends, they do not have a chance to rest. It is important for Aelin to recognise what is happening and why. Not actioning the problem has led to empathy fatigue and emotional exhaustion. Emotional exhaustion is one of the key aspects of [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FEmpathy_fatigue_and_emotional_exhaustion&diff=2840411&oldid=2832859 burnout], a long-term condition of existing in a constant state of high stress. Psychological science can support Aelin in recognising and mapping the symptoms they are experiencing, understand coping strategies to minimise the current and future effects to improve their wellbeing. Understanding the symptoms will also support their role as a social worker, providing more tools for the people they support on a day-to-day basis. {{RoundBoxTop|theme=3}} '''Focus questions''' * Why do people become emotionally exhausted? * What is the relationship between empathy fatigue and emotional exhaustion? * What role does emotional labor play in emotional exhaustion? * How does understanding emotional labor and empathy fatigue work to reduce the effects of emotional exhaustion? {{RoundBoxBottom}} == Emotional exhaustion == To understand emotional exhaustion, it will be important to first understand burnout. Burnout is a long-term condition of existing in a constant state of high stress and affects millions of people annually. In the early 2000s, a survey from Australia found that more than 50% of social workers and occupational workers reported high levels of burnout symptoms (Llyod and King, 2004) and a recent report from 2026 found that more than one third of all workers in Australia were experiencing burnout (Chan, 2026). Being in a state of high stress for short periods of time is a useful adaptation for humans, so much so that it has been given a name, [[wikipedia:Eustress|Eustress]]. However, as a long-term condition it leads to individual health issues and can be linked to cardiovascular disease, increased vulnerability to upper respiratory diseases, and chronic inflammation (Schneiderman et al., 2005), and cognitive issues such as [[wikipedia:Inattentional_blindness|cognitive tunneling]] and impaired selective attention (Pooladvand and Hasanzadeh, 2023). Burnout can be defined across three dimensions: emotional exhaustion, [[wikipedia:Depersonalization|depersonalisation]], and reduced feelings of personal accomplishment (Maslach, 1993, as cited in Martingano et al., 2026). Emotional exhaustion may be considered the 'core' component of burnout and refers to the depletion of emotional resources, leading to feelings of overwhelm, overextension, and exhaustion (Martingano et al., 2026; De Beer et al., 2024; Seidler et al., 2014). As a major factor, its symptoms are similar to burnout and stress. Although antecedents of burnout are typically agreed upon (Bakker et al., 2014) and divided into individual (e.g. self-efficacy) and situational categories (e.g. work load), there are many theories that attempt to explain why emotional exhaustion occurs. Two of these theories with good evidence are the [[wikipedia:Conservation_of_resources_theory|Conservation of resources stress model]] (Hobfoll, 1989) and the [[wikipedia:Job_demands-resources_model|Job Demands-Resources model]] (Demerouti et al., 2001). === Conservation of resources stress model === The Conservation of resources stress model (CORT; Hobfall, 1989) is a theory that works to understand and explain why stress occurs. Under the model, humans are motivated to maintain, pursue, or attain resources. Stress, defined as "an individual's perceived pressure or strain" (Hollebeek et al., 2023 p.928), occurs when these resources are threatened or lost, or gained at a net loss. As noted above, resources aren't strictly tied to being physical objects. They can also be emotional states, conditions, or even ideas. For example, someone may place value on enjoying a calm, quiet Sunday morning at home. If their neighbour starts using power tools loudly next door, it would disrupt this state and induce stress in the individual. The CORT model focuses primarily on two principles. The first principle is that there is more harm in losing resources than there is in gaining them back (Halbesleben et al., 2014). This is known as loss salience, and some research suggests that the pain of loss is felt twice as strong as the pleasure of gain (Thaler, 2000). Looking at our previous example, imagine that the person who enjoys a quiet Sunday morning at home equally enjoys the sound of a loud car engine. Even though the person enjoys the sound of car engines, if the neighbour started revving their car loudly instead of using power tools, it would still disrupt their desired state of quiet and lead to an increase of stress. The second principle is that people must invest existing resources in order to gain or protect resources (Halbesleben et al., 2014). This resource investment includes recovering from resource loss. Coping is how people manage or overcome a difficult situation, such as protecting or recovering from resource loss. Coping can be broken down into different strategies that people use, but not all strategies are equal (Ito and Brotheridge, 2003). They are divided into two categories: control or escape strategies. Control strategies are considered better at reducing the strain from stress and consist of positive thinking, direct action, and help seeking (Havlovic & Keenan, 1991, as cited in Ito and Brotheridge, 2003). The most useful control strategies incorporate a person's [[locus of control]], emphasise seeking support from co-workers, and highlight the importance of autonomy in decision making. Alarcon (2011) conducted a meta-analysis of 231 studies in a work setting that used the [[wikipedia:Maslach_Burnout_Inventory|Maslach Burnout Inventory]] (MBI) to review the relationship between these principles and burnout to confirm their validity. Although the study had no specific mention of demographics, it was noted that multiple translations of the MBI were used including Dutch (k=51), German (k=7), Italian (k=2), and French (k=2). The findings indicate that workload is positively related to emotional exhaustion (''ρ'' = .49, ''k'' = 86, ''N'' = 51,529) and that control (''ρ'' = − .26, ''k'' = 56, ''N'' = 33,297) and autonomy (''ρ'' = − .24, ''k'' = 34, ''N'' = 17,165) were each negatively associated with emotional exhaustion. The difference of ''ρ'' values is consistent with Thaler (2000) and supports that the benefit of gaining is approximately half that of losing resources. === Job demands-resources model === Another model that attempts to understand why people become emotionally exhausted is the Job Demands-Resources model (JD-R; Demerouti et al., 2001). The JD-R model discusses two concepts: job demands and job resources. Demands and resources are both related to the physical, social, or psychological aspects of a job, but differ in their impact on a person. Job demands require physical or mental effort and exhaust a person's energy supply. Job resources typically support achieving goals, reduce the effort cost of job demands, and stimulate personal growth (Richter and Hacker, 1998, as cited in Demerouti et al., 2001). For example, an unsustainable workload would be considered a job demand, and receiving help and support from others would be considered a resource. The JD-R has two primary processes for how demands and resources interact with one another (Schaufeli, 2017). The stress process occurs when job demands are high and resources are low. As stress increases strain and pressure on an individual, it leads to the exhaustion of their energy eventually, eventually leading to emotional exhaustion. The second process is the motivational process. This process occurs when there is an excess of job resources. Crucially, the motivational process does not require low job demands. With enough job resources, a person will be energised and engaged, reframing job demands as a challenge to overcome. The JD-R model has had extensive research to confirm its validity. A meta-analysis by Lesener et al. (2019) looked at 74 longitudinal studies of the JD-R with a strong average sample size (n=450) against the assumptions as described by Schaufeli (2017). Although most of the studies appear to come from [[wikipedia:Psychology#WEIRD_bias|WEIRD]] countries (k=60) there were a good number from Asia and Africa (k=13) and a mix of genders. The study confirmed that across the literature, job resources predict burnout negatively (''β'' = −.17; ''p'' < .001) and job demands predict burnout over time (''β'' = .10; ''p'' < .001). Although job resources seem like a cure-all for demands, it's important to note that they do not retroactively treat emotional exhaustion. Under the JD-R model, burnout must be reduced by lowering job demands (Gynning et al., 2025). However, once someone has recovered, it is possible to increase job resources alongside job demands to prevent future burnout and support [[wikipedia:Work_engagement|work engagement]]. Work engagement (engagement) has been discussed as the opposite of burnout (Bakker et al., 2014). Engagement is how people express themselves when performing a role. A role being a pattern of behaviour that a person goes through based on the social expectations of that role, similar to how an actor takes on a role for a movie. For example, when attending a typical restaurant, it is socially expected that a person will engage with the waiter and provide them with their meal order to take to the kitchen. This person is performing the script of a restaurant patron. In the context of a workplace, an employee with high engagement may appear to be energised, enthusiastic, and become absorbed in their work losing track of time (Bakkar et al, 2014). {{RoundBoxTop|theme=3}} '''Why do we feel emotionally exhausted''' Under the JD-R and CORT models, emotional exhaustion occurs when the demand cost outweighs the resources available for an extended period of time, eventually depleting someone's energy reserves. Aelin is experiencing burnout because they increased their job demands by taking on more clients, and has experienced a loss in resources by reducing their free time and social support network. {{RoundBoxBottom}} == Empathy fatigue == You may be familiar with the term [[empathy]]. It is often described in terms of supporting others. For example, when someone close to you experiences a loss and is grieving, you may empathise with them and experience a negative emotion akin to grief too. However, empathy is not exclusive to so called negative emotions. For example, compersion is a positive empathic response when a loved one experiences joy. Parents may experience compersion when something joyful happens to their child. At its core, empathy is the individual ability to imagine or understand another person's feelings and/or emotions (Riess, 2017; Shao et al., 2024). Much like typical physical or cognitive processes, such as running or problem solving, empathising will drain energy and resources. Further, and similar to the difference in energy expenditure between running and jogging, there are distinct types of empathy that each have different energy costs. Research makes a clear distinction between two types of empathy: affective and cognitive empathy. Affective empathy occurs when a person has an emotional reaction to someone else's mental state (Shao et al., 2024), whereas cognitive empathy occurs when a person understands someone else's feelings or emotions (Cui et al., 2022). A way to make a distinction between these two types may be considering them as the "what" and "how". Cognitive empathy is the "what" and focuses on understanding the difference between two people. A person who is using cognitive empathy will focus on understanding what the other person is feeling as a concept and use the information processing part of their brain, the pre-frontal cortex (Shao et al., 2024). This allows the empathiser to ask this question from a cognitive, problem solving perspective. As cognitive empathy is processed by the pre-frontal cortex, there is less association with feeling the emotion across the body hence reducing the physiological response and is linked to lower energy expenditure. Affective empathy is the "how" and focuses on the feeling aspect of empathy. This is a more common interpretation of empathy, and utilises * Integrating the two - "affective component is the essence of empathy, and the cognitive component is a process through which that essence comes into being" (Bošnjaković & Radionov, 2018) ** (Author's note: change quote into paraphrasing) === Emotional contagion === Emotional contagion is the idea that emotions are contagious. as the theory to moderate surface/deep acting * Emotional contagion (Panksepp and Lahvis, 2011) ** emotions can spread (expand, combines empathy and acting) ** “the tendency to automatically mimic and synchronize expressions, vocalizations, postures, and movements with those of another person’s and, consequently, to converge emotionally.” (Elaine Hatfield, 1993) * Affective process theory looks at unifying the mechanisms of emotional contagion into three types. Convergent linkage, Divergent linkage, and Complementary linkage (Elfenbein, 2014). Storytelling is a way to promote position emotions (Wei et al., 2024). === Empathy or compassion fatigue === * Distinction between empathy and compassion fatigue * Compassion fatigue is closely related to Secondary traumatic stress usually found in healthcare professionals (Figley, 2002) ** Compassion fatigue is to Empathy fatigue as emotional exhaustion is to burnout * Make it clear that Aelin is experiencing Compassion Fatigue vs Empathy fatigue (they are the same but different) (Author's note: include an update to the scenario here!) '''Empathy fatigue questions'''. Choose your answers and click "Submit": <quiz display=simple> {Surface acting is when the observer actually feels the emotion: |type="()"} - True + False {Compassion fatigue and empathy fatigue are synonymous: |type="()"} - True + False </quiz> {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' Under the JD-R and CORT models, a sustained empathic response is a long-term job demand that uses energy to protect existing resources. As Aelin has had their resources is experiencing a * Under the CORT model, a sustained empathic response uses energy to protect existing resources. As resources become low and burnout sets in, cognitive focus narrows leading to empathy fatigue. {{RoundBoxBottom}} == Emotional labor as a mediating factor between empathy fatigue and emotional exhaustion == As a society there are expected [https://en.wikipedia.org/wiki/Social_norm social norms] in behaviour when expressing emotion, regardless if someone is conscious of the process of matching these norms or not. For example, laughing during an emotional speech at a funeral would be considered a rude or offensive expression from the other attendees. As social creatures, humans need a way to regulate socially appropriate emotional expression. This is the essence of Emotional labor: "the regulation of felt and expressed emotions (whether effortful or not) to match these emotional display rules with the objective of fulfilling organizational goals" (Vial & Cowgill, 2022) What is emotional labor (Hochschild, 2012). Theories behind emotional labor involve acting, and an extension of acting known as emotional contagion. === Types of acting === * Surface acting ** This is where emotive dissonance comes into play ** Leads to emotional exhaustion (Kenworthy et al., 2014) ** "Should" language - in this situation, this is what I ''should'' feel * Deep acting ** People who 'deep act', as in genuinely change the way they feel (match the feelings they have with the feelings that they need to display) often report positive relationships with emotional labor ** Actual language - in this situation, this is what I ''actually'' feel Discuss the differences in gender norms and expectations in relationships {| class="wikitable" style="margin: auto;" |- ! !! Deep acting !! Surface acting |- | '''Emotional dissonance''' || || X |- | '''Not known to others''' || Hidden area || Unknown |} === Emotional contagion === Emotional contagion is the idea that emotions are contagious. as the theory to moderate surface/deep acting * Emotional contagion (Panksepp and Lahvis, 2011) ** emotions can spread (expand, combines empathy and acting) ** “the tendency to automatically mimic and synchronize expressions, vocalizations, postures, and movements with those of another person’s and, consequently, to converge emotionally.” (Elaine Hatfield, 1993) * Affective process theory looks at unifying the mechanisms of emotional contagion into three types. Convergent linkage, Divergent linkage, and Complementary linkage (Elfenbein, 2014). Storytelling is a way to promote position emotions (Wei et al., 2024). === The mediating effect of emotional labor === Surface acting does not mediate the effect (high surface acting leads to emotional exhaustion) whereas the inverse is true for deep acting (Wrobel, 2013) Emotional Expression and Emotional Processing (Hoyt et al., 2024) {{RoundBoxTop|theme=3}} '''What role does emotional labor play in emotional exhaustion''' * In relation to the CORT and JD-R theories, emotional labor acts as a job demand and expenditure of resources to protect existing resources {{RoundBoxBottom}} == Strategies for combating empathy fatigue and emotional fatigue == * Talk about emotion regulation (Gross, 1998) and mood regulation strategies as a way to deal with coping (Wrobel, 2013) (Author's note: need more research/references for this section) * Talk about breaks ** Rest and downtime from not performing emotional labor regenerates * How do we reduce the amount of emotional labor we're performing (Jeung et al., 2018) ** Communication/boundaries ** Reducing surface acting (allowing ourselves to feel how we actually feel and reduce emotive dissonance) '''Table 1. Treatment phases of clinical burnout (based on Table 2 from van Dam, A. 2021)''' {| class="wikitable" style="margin: auto; |- ! Phase !! Treatment goal !! Therapeutic interventions |- | '''Phase 1: Crisis''' || Recognition of the patient that the problems are serious, and that serious action is required || Psychoeducation Sick leave Dropping domestic tasks and social obligations Inform social network |- | '''Phase 2: Recovery''' || Recovery of the (physiological) stress system to normal allostatic stress levels || Registration of stress and activities Relaxtion exercises Mindfulness healthy lifestyle advices Graded exercise Gradually resume activities Psychoeducation social network and employer Gradual return to work |- | '''Phase 3: Prevention, learning from the past''' || Acquiring insight and skills to prevent relapse in clinical burnout || Analysing stressful situations Analysing dysfunctional thought patterns Learning new coping skills Learning social skills Job crafting Time management Making choices regarding career and personal life |- | '''Phase 4: Post burnout growth''' || Improving sustainable quality of life. || Setting priorities regarding quality of life and interpersonal relationships. |} === Strategies for the workplace === * Discuss types of strategies that are available in the workplace (organisation models) * SCARF model (Aplin-Houtz, 2025) ** Link back to COR and JDR models e.g. praise could be a job resource vs criticism could be a job demand *Reduce workload to reduce emotional exhaustion, and by extension reduce emotional fatigue. {{RoundBoxTop|theme=3}} '''How does understanding emotional labor and empathy fatigue work to reduce the effects of emotional exhaustion''' *Psychoeducation has been shown to reduce the effects of emotional labor and empathy fatigue on emotional exhaustion {{RoundBoxBottom}} ==Conclusion== * Summarise what empathy fatigue is, and why people become emotionally exhausted (+ the role of emotional labor) * Summarise the symptoms and how to recognise them in yourself and someone else * Emphasise and summarise psychological strategies to support someone going through emotional exhaustion * Summarise ways to prevent emotional exhaustion from happening * Overall message is: *# What is emotional exhaustion *# how do you recognise it *# how do you treat it *# how do you prevent it * Link back to the focus questions ==See also== * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Metacognition_and_emotional_regulation&oldid=2756264 Metacognition and emotional regulation] (Book chapter, 2026) {{ic|Use an internal link style as shown in Tutorial 2}} * [[wikipedia:Conservation_of_resources_theory|Conservation of resources theory]] (Wikipedia) * [[wikipedia:Job_demands-resources_model|Job demands-resources model]] (Wikipedia) ==References== {{Hanging indent|1= Aplin-Houtz, M. (2025). Unraveling the SCARF: How dimensions of the SCARF model influence the relationship between workplace exclusion and turnover intentions. Journal of Management & Organization, 31(6), 2699–2725. https://doi.org/10.1017/jmo.2025.10026 Bošnjaković, J., & Radionov, T. (2018). Empathy: Concepts, Theories and Neuroscientific Basis. Alcoholism (Zagreb), 54(2), 123–150. https://doi.org/10.20471/de..2018.54.02.04 Chan, F. M. (2026). The State of Workplace Burnout 2026: The Workplace Divide. Infinite Potential, Sydney, Australia. https://www.ahri.com.au/wp-content/uploads/The-State-of-Workplace-Burnout-2026-Final.pdf Cui, F., Huang, X., Jing, Y., Luo, Y., Liu, J., & Gu, R. (2022). How resource sharing resists scarcity: the role of cognitive empathy and its neurobiological mechanisms. Cerebral Cortex (New York, N.Y. 1991), 32(23), 5330–5342. https://doi.org/10.1093/cercor/bhac017 De Beer, L. T., van der Vaart, L., Escaffi-Schwarz, M., De Witte, H., & Schaufeli, W. B. (2024). Maslach Burnout Inventory - General Survey: A Systematic Review and Meta-Analysis of Measurement Properties. European journal of psychological assessment : official organ of the European Association of Psychological Assessment, 40(5), 360-375. https://doi.org/10.1027/1015-5759/a000797 Demerouti, E., Bakker, A. B., Nachreiner, F., & Schaufeli, W. B. (2001). The Job Demands-Resources Model of Burnout. Journal of Applied Psychology, 86(3), 499–512. https://doi.org/10.1037/0021-9010.86.3.499 Figley, C. R. (2002). Compassion fatigue: Psychotherapists’ chronic lack of self care. Journal of Clinical Psychology, 58(11), 1433–1441. https://doi.org/10.1002/jclp.10090 Grandey, A. A. (2003). When ‘the show must go on’: surface acting and deep acting as determinants of emotional exhaustion and peer-rated service delivery. Academy of Management Journal, 46(1), 86–96. https://doi.org/10.5465/30040678 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 Halbesleben, J. R. B., Neveu, J.-P., Paustian-Underdahl, S. C., & Westman, M. (2014). Getting to the “COR”: Understanding the Role of Resources in Conservation of Resources Theory. Journal of Management, 40(5), 1334–1364. https://doi.org/10.1177/0149206314527130 Hatfield, E., Cacioppo, J. T., & Rapson, R. L. (1993). Emotional Contagion. Current Directions in Psychological Science : A Journal of the American Psychological Society, 2(3), 96–99. https://doi.org/10.1111/1467-8721.ep10770953 Hollebeek, L. D., Hammedi, W., & Sprott, D. E. (2023). Consumer engagement, stress, and conservation of resources theory: A review, conceptual development, and future research agenda. Psychology & marketing, 40(5), 926-937. https://doi.org/10.1002/mar.21807 Hobfoll, S. E. (1989). Conservation of Resources: A New Attempt at Conceptualizing Stress. The American Psychologist, 44(3), 513–524. https://doi.org/10.1037/0003-066X.44.3.513 Hochschild, A. R. (2012). The managed heart : commercialization of human feeling (Updated, with a new preface.). University of California Press. Jeung, D.-Y., Kim, C., & Chang, S.-J. (2018). Emotional Labor and Burnout: A Review of the Literature. Yonsei Medical Journal, 59(2), 187–193. https://doi.org/10.3349/ymj.2018.59.2.187 Lavelle, J. J., Rupp, D. E., Herda, D. N., Pandey, A., & Lauck, J. R. (2021). Customer Injustice and Employee Performance: Roles of Emotional Exhaustion, Surface Acting, and Emotional Demands–Abilities Fit. Journal of Management, 47(3), Article 0149206319869426. https://doi.org/10.1177/0149206319869426 Lloyd, C., & King, R. (2004). A survey of burnout among Australian mental health occupational therapists and social workers. Social Psychiatry and Psychiatric Epidemiology, 39(9), 752-757. https://doi.org/https://doi.org/10.1007/s00127-004-0808-7 Kenworthy, J., Fay, C., Frame, M., & Petree, R. (2014). A meta-analytic review of the relationship between emotional dissonance and emotional exhaustion. Journal of Applied Social Psychology, 44(2), 94-105. https://doi.org/https://doi.org/10.1111/jasp.12211 Martingano, A. J., Stosic, M. D., & Konrath, S. (2026). Different empathy types show opposing associations with burnout: systematic review and meta-analyses. Psychology, Health & Medicine, 31(3), 798-829. https://doi.org/10.1080/13548506.2025.2591859 Michielsen, H. J., Willemsen, T. M., Croon, M. A., de Vries, J., & Van Heck, G. L. (2004). Determinants of general fatigue and emotional exhaustion: a prospective study. Psychology & health, 19(2), 223-235. https://doi.org/10.1080/08870440310001627135 Panksepp, J. B., & Lahvis, G. P. (2011). Rodent empathy and affective neuroscience. Neuroscience and Biobehavioral Reviews, 35(9), 1864–1875. https://doi.org/10.1016/j.neubiorev.2011.05.013 Pooladvand, S., & Hasanzadeh, S. (2023). Impacts of Stress on Workers’ Risk-Taking Behaviors: Cognitive Tunneling and Impaired Selective Attention. Journal of Construction Engineering and Management, 149(8), 04023060. https://doi.org/doi:10.1061/JCEMD4.COENG-13339 Riess, H. (2017). The Science of Empathy. Journal of Patient Experience, 4(2), 74–77. https://doi.org/10.1177/2374373517699267 Schaufeli, W. B. (2017). Applying the job demands-resources model: a “how to” guide to measuring and tackling work engagement and burnout. Organizational Dynamics, 46(2), 120–132. https://doi.org/10.1016/j.orgdyn.2017.04.008 Schneiderman, N., Ironson, G., & Siegel, S. D. (2005). Stress and Health: Psychological, Behavioral, and Biological Determinants. Annual Review of Clinical Psychology, 1(Volume 1, 2005), 607-628. https://doi.org/https://doi.org/10.1146/annurev.clinpsy.1.102803.144141 Seidler, A., Thinschmidt, M., Deckert, S., Then, F., Hegewald, J., Nieuwenhuijsen, K., & Riedel-Heller, S. G. (2014). The role of psychosocial working conditions on burnout and its core component emotional exhaustion – a systematic review. Journal of Occupational Medicine and Toxicology, 9(1), 10. https://doi.org/10.1186/1745-6673-9-10 Shao, M., Li, L., Li, X., Wei, Z., Wang, J., Hong, M., Liu, X., & Meng, J. (2024). The effect of top-down attention on empathy fatigue. Cerebral Cortex (New York, N.Y. 1991), 34(1), Article bhad441. https://doi.org/10.1093/cercor/bhad441 van Dam, A. (2021). A clinical perspective on burnout: diagnosis, classification, and treatment of clinical burnout. European Journal of Work and Organizational Psychology, 30(5), 732-741. https://doi.org/10.1080/1359432X.2021.1948400 Vial, A. C., & Cowgill, C. M. (2022). Heavier Lies Her Crown: Gendered Patterns of Leader Emotional Labor and Their Downstream Effects. Frontiers in Psychology, 13, 849566. https://doi.org/https://doi.org/10.3389/fpsyg.2022.849566 Wharton, A. S. (2009). The Sociology of Emotional Labor. Annual Review of Sociology, 35(1), 147–165. https://doi.org/10.1146/annurev-soc-070308-115944 Wrobel, M. (2013). Can empathy lead to emotional exhaustion in teachers? The mediating role of emotional labor. International Journal of Occupational Medicine and Environmental Health, 26(4), 581–592. https://doi.org/10.2478/s13382-013-0123-1 }} ==External links== Provide [[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: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Burnout]] [[Category:Motivation and emotion/Book/Empathy]] qk4yxnq3im9fbpnuhc6pa8lie4htioq 2844538 2844537 2026-11-04T23:52:25Z U3143751 3106785 /* Types of acting */ 2844538 wikitext text/x-wiki {{title|Empathy fatigue and emotional exhaustion:<br>How does sustained empathic engagement contribute to emotional exhaustion?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}} [[File:Exhausted smartphone user.JPG|right|thumb|150px|'''Figure 1'''. An emotionally exhausted Aelin at the end of the work day]] ; Scenario Aelin, a 23-year-old social worker employed at headspace {{ic|Link to the Wikipedia article to help explain what this term means}}, works with approximately 20 {{ic|Explain what sort of}}Imagine you are only 23 years old and are working as a social worker at headspace. You see around 20 clients each week, many of whom have experienced rough living conditions. During your first six months in this role, you felt highly motivated and excited to help people in need. You are an excellent active listener and can easily remember details about your clients Live’s. You are also good at recognising signs of distress in your client’s body language and adjust yours in order to show warmth and understanding. As you progress in your role, you take on more cases that involve complex past traumas. The workload becomes too much for you to get done at work, but you don’t want your clients to miss out, so you take your work home to complete it in the evenings. Your workload increases again and you now start doing work on your weekends as well. After several months of working this way, you start to notice you feel exhausted every morning before you have even started work. You then start to notice you’re responding to people at work differently. Instead of your usual energised active listening response you feel frustrated, empathetic and detached. After work you are too drained to socialise with friends. This begins to worry you. What if you won’t be able to continue working as a social worker? {{RoundBoxBottom}} Aelin is experiencing '''emotional exhaustion''' from a sustained empathic response. Due to the nature of Aelin's work, they are constantly engaged in an empathic response. Empathic responses require cognitive and physical resources to sustain. Because Aelin is taking their work home and not resting on weekends, they do not have a chance to rest. It is important for Aelin to recognise what is happening and why. Not actioning the problem has led to empathy fatigue and emotional exhaustion. Emotional exhaustion is one of the key aspects of [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FEmpathy_fatigue_and_emotional_exhaustion&diff=2840411&oldid=2832859 burnout], a long-term condition of existing in a constant state of high stress. Psychological science can support Aelin in recognising and mapping the symptoms they are experiencing, understand coping strategies to minimise the current and future effects to improve their wellbeing. Understanding the symptoms will also support their role as a social worker, providing more tools for the people they support on a day-to-day basis. {{RoundBoxTop|theme=3}} '''Focus questions''' * Why do people become emotionally exhausted? * What is the relationship between empathy fatigue and emotional exhaustion? * What role does emotional labor play in emotional exhaustion? * How does understanding emotional labor and empathy fatigue work to reduce the effects of emotional exhaustion? {{RoundBoxBottom}} == Emotional exhaustion == To understand emotional exhaustion, it will be important to first understand burnout. Burnout is a long-term condition of existing in a constant state of high stress and affects millions of people annually. In the early 2000s, a survey from Australia found that more than 50% of social workers and occupational workers reported high levels of burnout symptoms (Llyod and King, 2004) and a recent report from 2026 found that more than one third of all workers in Australia were experiencing burnout (Chan, 2026). Being in a state of high stress for short periods of time is a useful adaptation for humans, so much so that it has been given a name, [[wikipedia:Eustress|Eustress]]. However, as a long-term condition it leads to individual health issues and can be linked to cardiovascular disease, increased vulnerability to upper respiratory diseases, and chronic inflammation (Schneiderman et al., 2005), and cognitive issues such as [[wikipedia:Inattentional_blindness|cognitive tunneling]] and impaired selective attention (Pooladvand and Hasanzadeh, 2023). Burnout can be defined across three dimensions: emotional exhaustion, [[wikipedia:Depersonalization|depersonalisation]], and reduced feelings of personal accomplishment (Maslach, 1993, as cited in Martingano et al., 2026). Emotional exhaustion may be considered the 'core' component of burnout and refers to the depletion of emotional resources, leading to feelings of overwhelm, overextension, and exhaustion (Martingano et al., 2026; De Beer et al., 2024; Seidler et al., 2014). As a major factor, its symptoms are similar to burnout and stress. Although antecedents of burnout are typically agreed upon (Bakker et al., 2014) and divided into individual (e.g. self-efficacy) and situational categories (e.g. work load), there are many theories that attempt to explain why emotional exhaustion occurs. Two of these theories with good evidence are the [[wikipedia:Conservation_of_resources_theory|Conservation of resources stress model]] (Hobfoll, 1989) and the [[wikipedia:Job_demands-resources_model|Job Demands-Resources model]] (Demerouti et al., 2001). === Conservation of resources stress model === The Conservation of resources stress model (CORT; Hobfall, 1989) is a theory that works to understand and explain why stress occurs. Under the model, humans are motivated to maintain, pursue, or attain resources. Stress, defined as "an individual's perceived pressure or strain" (Hollebeek et al., 2023 p.928), occurs when these resources are threatened or lost, or gained at a net loss. As noted above, resources aren't strictly tied to being physical objects. They can also be emotional states, conditions, or even ideas. For example, someone may place value on enjoying a calm, quiet Sunday morning at home. If their neighbour starts using power tools loudly next door, it would disrupt this state and induce stress in the individual. The CORT model focuses primarily on two principles. The first principle is that there is more harm in losing resources than there is in gaining them back (Halbesleben et al., 2014). This is known as loss salience, and some research suggests that the pain of loss is felt twice as strong as the pleasure of gain (Thaler, 2000). Looking at our previous example, imagine that the person who enjoys a quiet Sunday morning at home equally enjoys the sound of a loud car engine. Even though the person enjoys the sound of car engines, if the neighbour started revving their car loudly instead of using power tools, it would still disrupt their desired state of quiet and lead to an increase of stress. The second principle is that people must invest existing resources in order to gain or protect resources (Halbesleben et al., 2014). This resource investment includes recovering from resource loss. Coping is how people manage or overcome a difficult situation, such as protecting or recovering from resource loss. Coping can be broken down into different strategies that people use, but not all strategies are equal (Ito and Brotheridge, 2003). They are divided into two categories: control or escape strategies. Control strategies are considered better at reducing the strain from stress and consist of positive thinking, direct action, and help seeking (Havlovic & Keenan, 1991, as cited in Ito and Brotheridge, 2003). The most useful control strategies incorporate a person's [[locus of control]], emphasise seeking support from co-workers, and highlight the importance of autonomy in decision making. Alarcon (2011) conducted a meta-analysis of 231 studies in a work setting that used the [[wikipedia:Maslach_Burnout_Inventory|Maslach Burnout Inventory]] (MBI) to review the relationship between these principles and burnout to confirm their validity. Although the study had no specific mention of demographics, it was noted that multiple translations of the MBI were used including Dutch (k=51), German (k=7), Italian (k=2), and French (k=2). The findings indicate that workload is positively related to emotional exhaustion (''ρ'' = .49, ''k'' = 86, ''N'' = 51,529) and that control (''ρ'' = − .26, ''k'' = 56, ''N'' = 33,297) and autonomy (''ρ'' = − .24, ''k'' = 34, ''N'' = 17,165) were each negatively associated with emotional exhaustion. The difference of ''ρ'' values is consistent with Thaler (2000) and supports that the benefit of gaining is approximately half that of losing resources. === Job demands-resources model === Another model that attempts to understand why people become emotionally exhausted is the Job Demands-Resources model (JD-R; Demerouti et al., 2001). The JD-R model discusses two concepts: job demands and job resources. Demands and resources are both related to the physical, social, or psychological aspects of a job, but differ in their impact on a person. Job demands require physical or mental effort and exhaust a person's energy supply. Job resources typically support achieving goals, reduce the effort cost of job demands, and stimulate personal growth (Richter and Hacker, 1998, as cited in Demerouti et al., 2001). For example, an unsustainable workload would be considered a job demand, and receiving help and support from others would be considered a resource. The JD-R has two primary processes for how demands and resources interact with one another (Schaufeli, 2017). The stress process occurs when job demands are high and resources are low. As stress increases strain and pressure on an individual, it leads to the exhaustion of their energy eventually, eventually leading to emotional exhaustion. The second process is the motivational process. This process occurs when there is an excess of job resources. Crucially, the motivational process does not require low job demands. With enough job resources, a person will be energised and engaged, reframing job demands as a challenge to overcome. The JD-R model has had extensive research to confirm its validity. A meta-analysis by Lesener et al. (2019) looked at 74 longitudinal studies of the JD-R with a strong average sample size (n=450) against the assumptions as described by Schaufeli (2017). Although most of the studies appear to come from [[wikipedia:Psychology#WEIRD_bias|WEIRD]] countries (k=60) there were a good number from Asia and Africa (k=13) and a mix of genders. The study confirmed that across the literature, job resources predict burnout negatively (''β'' = −.17; ''p'' < .001) and job demands predict burnout over time (''β'' = .10; ''p'' < .001). Although job resources seem like a cure-all for demands, it's important to note that they do not retroactively treat emotional exhaustion. Under the JD-R model, burnout must be reduced by lowering job demands (Gynning et al., 2025). However, once someone has recovered, it is possible to increase job resources alongside job demands to prevent future burnout and support [[wikipedia:Work_engagement|work engagement]]. Work engagement (engagement) has been discussed as the opposite of burnout (Bakker et al., 2014). Engagement is how people express themselves when performing a role. A role being a pattern of behaviour that a person goes through based on the social expectations of that role, similar to how an actor takes on a role for a movie. For example, when attending a typical restaurant, it is socially expected that a person will engage with the waiter and provide them with their meal order to take to the kitchen. This person is performing the script of a restaurant patron. In the context of a workplace, an employee with high engagement may appear to be energised, enthusiastic, and become absorbed in their work losing track of time (Bakkar et al, 2014). {{RoundBoxTop|theme=3}} '''Why do we feel emotionally exhausted''' Under the JD-R and CORT models, emotional exhaustion occurs when the demand cost outweighs the resources available for an extended period of time, eventually depleting someone's energy reserves. Aelin is experiencing burnout because they increased their job demands by taking on more clients, and has experienced a loss in resources by reducing their free time and social support network. {{RoundBoxBottom}} == Empathy fatigue == You may be familiar with the term [[empathy]]. It is often described in terms of supporting others. For example, when someone close to you experiences a loss and is grieving, you may empathise with them and experience a negative emotion akin to grief too. However, empathy is not exclusive to so called negative emotions. For example, compersion is a positive empathic response when a loved one experiences joy. Parents may experience compersion when something joyful happens to their child. At its core, empathy is the individual ability to imagine or understand another person's feelings and/or emotions (Riess, 2017; Shao et al., 2024). Much like typical physical or cognitive processes, such as running or problem solving, empathising will drain energy and resources. Further, and similar to the difference in energy expenditure between running and jogging, there are distinct types of empathy that each have different energy costs. Research makes a clear distinction between two types of empathy: affective and cognitive empathy. Affective empathy occurs when a person has an emotional reaction to someone else's mental state (Shao et al., 2024), whereas cognitive empathy occurs when a person understands someone else's feelings or emotions (Cui et al., 2022). A way to make a distinction between these two types may be considering them as the "what" and "how". Cognitive empathy is the "what" and focuses on understanding the difference between two people. A person who is using cognitive empathy will focus on understanding what the other person is feeling as a concept and use the information processing part of their brain, the pre-frontal cortex (Shao et al., 2024). This allows the empathiser to ask this question from a cognitive, problem solving perspective. As cognitive empathy is processed by the pre-frontal cortex, there is less association with feeling the emotion across the body hence reducing the physiological response and is linked to lower energy expenditure. Affective empathy is the "how" and focuses on the feeling aspect of empathy. This is a more common interpretation of empathy, and utilises * Integrating the two - "affective component is the essence of empathy, and the cognitive component is a process through which that essence comes into being" (Bošnjaković & Radionov, 2018) ** (Author's note: change quote into paraphrasing) === Emotional contagion === Emotional contagion is the idea that emotions are contagious. as the theory to moderate surface/deep acting * Emotional contagion (Panksepp and Lahvis, 2011) ** emotions can spread (expand, combines empathy and acting) ** “the tendency to automatically mimic and synchronize expressions, vocalizations, postures, and movements with those of another person’s and, consequently, to converge emotionally.” (Elaine Hatfield, 1993) * Affective process theory looks at unifying the mechanisms of emotional contagion into three types. Convergent linkage, Divergent linkage, and Complementary linkage (Elfenbein, 2014). Storytelling is a way to promote position emotions (Wei et al., 2024). === Empathy or compassion fatigue === * Distinction between empathy and compassion fatigue * Compassion fatigue is closely related to Secondary traumatic stress usually found in healthcare professionals (Figley, 2002) ** Compassion fatigue is to Empathy fatigue as emotional exhaustion is to burnout * Make it clear that Aelin is experiencing Compassion Fatigue vs Empathy fatigue (they are the same but different) (Author's note: include an update to the scenario here!) '''Empathy fatigue questions'''. Choose your answers and click "Submit": <quiz display=simple> {Surface acting is when the observer actually feels the emotion: |type="()"} - True + False {Compassion fatigue and empathy fatigue are synonymous: |type="()"} - True + False </quiz> {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' Under the JD-R and CORT models, a sustained empathic response is a long-term job demand that uses energy to protect existing resources. As Aelin has had their resources is experiencing a * Under the CORT model, a sustained empathic response uses energy to protect existing resources. As resources become low and burnout sets in, cognitive focus narrows leading to empathy fatigue. {{RoundBoxBottom}} == Emotional labor as a mediating factor between empathy fatigue and emotional exhaustion == As a society there are expected [https://en.wikipedia.org/wiki/Social_norm social norms] in behaviour when expressing emotion, regardless if someone is conscious of the process of matching these norms or not. For example, laughing during an emotional speech at a funeral would be considered a rude or offensive expression from the other attendees. As social creatures, humans need a way to regulate socially appropriate emotional expression. This is the essence of Emotional labor: "the regulation of felt and expressed emotions (whether effortful or not) to match these emotional display rules with the objective of fulfilling organizational goals" (Vial & Cowgill, 2022) What is emotional labor (Hochschild, 2012). Theories behind emotional labor involve acting, and an extension of acting known as emotional contagion. === Types of acting === * Surface acting ** This is where emotive dissonance comes into play ** Leads to emotional exhaustion (Kenworthy et al., 2014) ** "Should" language - in this situation, this is what I ''should'' feel * Deep acting ** People who 'deep act', as in genuinely change the way they feel (match the feelings they have with the feelings that they need to display) often report positive relationships with emotional labor ** Actual language - in this situation, this is what I ''actually'' feel Discuss the differences in gender norms and expectations in relationships {| class="wikitable" style="margin: auto;" |- ! !! Deep acting !! Surface acting |- | '''Emotional dissonance''' || || X |- | '''Not known to others''' || Hidden area || Unknown |} * === The mediating effect of emotional labor === Surface acting does not mediate the effect (high surface acting leads to emotional exhaustion) whereas the inverse is true for deep acting (Wrobel, 2013) Emotional Expression and Emotional Processing (Hoyt et al., 2024) {{RoundBoxTop|theme=3}} '''What role does emotional labor play in emotional exhaustion''' * In relation to the CORT and JD-R theories, emotional labor acts as a job demand and expenditure of resources to protect existing resources {{RoundBoxBottom}} == Strategies for combating empathy fatigue and emotional fatigue == * Talk about emotion regulation (Gross, 1998) and mood regulation strategies as a way to deal with coping (Wrobel, 2013) (Author's note: need more research/references for this section) * Talk about breaks ** Rest and downtime from not performing emotional labor regenerates * How do we reduce the amount of emotional labor we're performing (Jeung et al., 2018) ** Communication/boundaries ** Reducing surface acting (allowing ourselves to feel how we actually feel and reduce emotive dissonance) '''Table 1. Treatment phases of clinical burnout (based on Table 2 from van Dam, A. 2021)''' {| class="wikitable" style="margin: auto; |- ! Phase !! Treatment goal !! Therapeutic interventions |- | '''Phase 1: Crisis''' || Recognition of the patient that the problems are serious, and that serious action is required || Psychoeducation Sick leave Dropping domestic tasks and social obligations Inform social network |- | '''Phase 2: Recovery''' || Recovery of the (physiological) stress system to normal allostatic stress levels || Registration of stress and activities Relaxtion exercises Mindfulness healthy lifestyle advices Graded exercise Gradually resume activities Psychoeducation social network and employer Gradual return to work |- | '''Phase 3: Prevention, learning from the past''' || Acquiring insight and skills to prevent relapse in clinical burnout || Analysing stressful situations Analysing dysfunctional thought patterns Learning new coping skills Learning social skills Job crafting Time management Making choices regarding career and personal life |- | '''Phase 4: Post burnout growth''' || Improving sustainable quality of life. || Setting priorities regarding quality of life and interpersonal relationships. |} === Strategies for the workplace === * Discuss types of strategies that are available in the workplace (organisation models) * SCARF model (Aplin-Houtz, 2025) ** Link back to COR and JDR models e.g. praise could be a job resource vs criticism could be a job demand *Reduce workload to reduce emotional exhaustion, and by extension reduce emotional fatigue. {{RoundBoxTop|theme=3}} '''How does understanding emotional labor and empathy fatigue work to reduce the effects of emotional exhaustion''' *Psychoeducation has been shown to reduce the effects of emotional labor and empathy fatigue on emotional exhaustion {{RoundBoxBottom}} ==Conclusion== * Summarise what empathy fatigue is, and why people become emotionally exhausted (+ the role of emotional labor) * Summarise the symptoms and how to recognise them in yourself and someone else * Emphasise and summarise psychological strategies to support someone going through emotional exhaustion * Summarise ways to prevent emotional exhaustion from happening * Overall message is: *# What is emotional exhaustion *# how do you recognise it *# how do you treat it *# how do you prevent it * Link back to the focus questions ==See also== * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Metacognition_and_emotional_regulation&oldid=2756264 Metacognition and emotional regulation] (Book chapter, 2026) {{ic|Use an internal link style as shown in Tutorial 2}} * [[wikipedia:Conservation_of_resources_theory|Conservation of resources theory]] (Wikipedia) * [[wikipedia:Job_demands-resources_model|Job demands-resources model]] (Wikipedia) ==References== {{Hanging indent|1= Aplin-Houtz, M. (2025). Unraveling the SCARF: How dimensions of the SCARF model influence the relationship between workplace exclusion and turnover intentions. Journal of Management & Organization, 31(6), 2699–2725. https://doi.org/10.1017/jmo.2025.10026 Bošnjaković, J., & Radionov, T. (2018). Empathy: Concepts, Theories and Neuroscientific Basis. Alcoholism (Zagreb), 54(2), 123–150. https://doi.org/10.20471/de..2018.54.02.04 Chan, F. M. (2026). The State of Workplace Burnout 2026: The Workplace Divide. Infinite Potential, Sydney, Australia. https://www.ahri.com.au/wp-content/uploads/The-State-of-Workplace-Burnout-2026-Final.pdf Cui, F., Huang, X., Jing, Y., Luo, Y., Liu, J., & Gu, R. (2022). How resource sharing resists scarcity: the role of cognitive empathy and its neurobiological mechanisms. Cerebral Cortex (New York, N.Y. 1991), 32(23), 5330–5342. https://doi.org/10.1093/cercor/bhac017 De Beer, L. T., van der Vaart, L., Escaffi-Schwarz, M., De Witte, H., & Schaufeli, W. B. (2024). Maslach Burnout Inventory - General Survey: A Systematic Review and Meta-Analysis of Measurement Properties. European journal of psychological assessment : official organ of the European Association of Psychological Assessment, 40(5), 360-375. https://doi.org/10.1027/1015-5759/a000797 Demerouti, E., Bakker, A. B., Nachreiner, F., & Schaufeli, W. B. (2001). The Job Demands-Resources Model of Burnout. Journal of Applied Psychology, 86(3), 499–512. https://doi.org/10.1037/0021-9010.86.3.499 Figley, C. R. (2002). Compassion fatigue: Psychotherapists’ chronic lack of self care. Journal of Clinical Psychology, 58(11), 1433–1441. https://doi.org/10.1002/jclp.10090 Grandey, A. A. (2003). When ‘the show must go on’: surface acting and deep acting as determinants of emotional exhaustion and peer-rated service delivery. Academy of Management Journal, 46(1), 86–96. https://doi.org/10.5465/30040678 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 Halbesleben, J. R. B., Neveu, J.-P., Paustian-Underdahl, S. C., & Westman, M. (2014). Getting to the “COR”: Understanding the Role of Resources in Conservation of Resources Theory. Journal of Management, 40(5), 1334–1364. https://doi.org/10.1177/0149206314527130 Hatfield, E., Cacioppo, J. T., & Rapson, R. L. (1993). Emotional Contagion. Current Directions in Psychological Science : A Journal of the American Psychological Society, 2(3), 96–99. https://doi.org/10.1111/1467-8721.ep10770953 Hollebeek, L. D., Hammedi, W., & Sprott, D. E. (2023). Consumer engagement, stress, and conservation of resources theory: A review, conceptual development, and future research agenda. Psychology & marketing, 40(5), 926-937. https://doi.org/10.1002/mar.21807 Hobfoll, S. E. (1989). Conservation of Resources: A New Attempt at Conceptualizing Stress. The American Psychologist, 44(3), 513–524. https://doi.org/10.1037/0003-066X.44.3.513 Hochschild, A. R. (2012). The managed heart : commercialization of human feeling (Updated, with a new preface.). University of California Press. Jeung, D.-Y., Kim, C., & Chang, S.-J. (2018). Emotional Labor and Burnout: A Review of the Literature. Yonsei Medical Journal, 59(2), 187–193. https://doi.org/10.3349/ymj.2018.59.2.187 Lavelle, J. J., Rupp, D. E., Herda, D. N., Pandey, A., & Lauck, J. R. (2021). Customer Injustice and Employee Performance: Roles of Emotional Exhaustion, Surface Acting, and Emotional Demands–Abilities Fit. Journal of Management, 47(3), Article 0149206319869426. https://doi.org/10.1177/0149206319869426 Lloyd, C., & King, R. (2004). A survey of burnout among Australian mental health occupational therapists and social workers. Social Psychiatry and Psychiatric Epidemiology, 39(9), 752-757. https://doi.org/https://doi.org/10.1007/s00127-004-0808-7 Kenworthy, J., Fay, C., Frame, M., & Petree, R. (2014). A meta-analytic review of the relationship between emotional dissonance and emotional exhaustion. Journal of Applied Social Psychology, 44(2), 94-105. https://doi.org/https://doi.org/10.1111/jasp.12211 Martingano, A. J., Stosic, M. D., & Konrath, S. (2026). Different empathy types show opposing associations with burnout: systematic review and meta-analyses. Psychology, Health & Medicine, 31(3), 798-829. https://doi.org/10.1080/13548506.2025.2591859 Michielsen, H. J., Willemsen, T. M., Croon, M. A., de Vries, J., & Van Heck, G. L. (2004). Determinants of general fatigue and emotional exhaustion: a prospective study. Psychology & health, 19(2), 223-235. https://doi.org/10.1080/08870440310001627135 Panksepp, J. B., & Lahvis, G. P. (2011). Rodent empathy and affective neuroscience. Neuroscience and Biobehavioral Reviews, 35(9), 1864–1875. https://doi.org/10.1016/j.neubiorev.2011.05.013 Pooladvand, S., & Hasanzadeh, S. (2023). Impacts of Stress on Workers’ Risk-Taking Behaviors: Cognitive Tunneling and Impaired Selective Attention. Journal of Construction Engineering and Management, 149(8), 04023060. https://doi.org/doi:10.1061/JCEMD4.COENG-13339 Riess, H. (2017). The Science of Empathy. Journal of Patient Experience, 4(2), 74–77. https://doi.org/10.1177/2374373517699267 Schaufeli, W. B. (2017). Applying the job demands-resources model: a “how to” guide to measuring and tackling work engagement and burnout. Organizational Dynamics, 46(2), 120–132. https://doi.org/10.1016/j.orgdyn.2017.04.008 Schneiderman, N., Ironson, G., & Siegel, S. D. (2005). Stress and Health: Psychological, Behavioral, and Biological Determinants. Annual Review of Clinical Psychology, 1(Volume 1, 2005), 607-628. https://doi.org/https://doi.org/10.1146/annurev.clinpsy.1.102803.144141 Seidler, A., Thinschmidt, M., Deckert, S., Then, F., Hegewald, J., Nieuwenhuijsen, K., & Riedel-Heller, S. G. (2014). The role of psychosocial working conditions on burnout and its core component emotional exhaustion – a systematic review. Journal of Occupational Medicine and Toxicology, 9(1), 10. https://doi.org/10.1186/1745-6673-9-10 Shao, M., Li, L., Li, X., Wei, Z., Wang, J., Hong, M., Liu, X., & Meng, J. (2024). The effect of top-down attention on empathy fatigue. Cerebral Cortex (New York, N.Y. 1991), 34(1), Article bhad441. https://doi.org/10.1093/cercor/bhad441 van Dam, A. (2021). A clinical perspective on burnout: diagnosis, classification, and treatment of clinical burnout. European Journal of Work and Organizational Psychology, 30(5), 732-741. https://doi.org/10.1080/1359432X.2021.1948400 Vial, A. C., & Cowgill, C. M. (2022). Heavier Lies Her Crown: Gendered Patterns of Leader Emotional Labor and Their Downstream Effects. Frontiers in Psychology, 13, 849566. https://doi.org/https://doi.org/10.3389/fpsyg.2022.849566 Wharton, A. S. (2009). The Sociology of Emotional Labor. Annual Review of Sociology, 35(1), 147–165. https://doi.org/10.1146/annurev-soc-070308-115944 Wrobel, M. (2013). Can empathy lead to emotional exhaustion in teachers? The mediating role of emotional labor. International Journal of Occupational Medicine and Environmental Health, 26(4), 581–592. https://doi.org/10.2478/s13382-013-0123-1 }} ==External links== Provide [[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: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Burnout]] [[Category:Motivation and emotion/Book/Empathy]] culvvjng0qmnn3n7s1hswbx1cxlmnkc 2844539 2844538 2026-11-04T23:56:48Z U3143751 3106785 /* Overview */ updated introduction scenario 2844539 wikitext text/x-wiki {{title|Empathy fatigue and emotional exhaustion:<br>How does sustained empathic engagement contribute to emotional exhaustion?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}} [[File:Exhausted smartphone user.JPG|right|thumb|150px|'''Figure 1'''. An emotionally exhausted Aelin at the end of the work day]] ; Scenario Aelin is a 23-year-old social worker employed at a non-profit organisation providing mental health services to teenagers and young adults. Aelin's clients are aged between 18 and 25, many of whom having experienced rough living conditions. They work with approximately 20 clients each week. During Aelin's first six months in the role, they feel highly motivated and are excited about helping people. They are an excellent active listener, and can easily remember details about their clients' lives. During their sessions, Aelin recognises when a client is distressed and adjusts their body language to warm and understanding in an effort to provide a safe and welcoming environment to hold space for their emotions. As Aelin gains more experience working in this role, their caseload increases, and they are exposed to clients with increasing complexity and past traumas. Aelin wants to provide the best possible outcome for all of their clients, and begins to take their work home with them in the evenings, but finds it starts creeping into their weekends as well. After several months of working this way, Aelin begins to notice they feel exhausted every morning - even before the workday begins. Fast-forward a few more months, and Aelin notices a change in how they start responding to people, including clients. Instead of feeling engaged and actively listening, they feel detached and find their thoughts feeling more frustrated than empathetic. After finishing the work day, Aelin is drained and has no energy for friends or family. With increasing irritability, Aelin begins to worry that they won't be able to continue working as a social-worker. {{RoundBoxBottom}} Aelin is experiencing '''emotional exhaustion''' from a sustained empathic response. Due to the nature of Aelin's work, they are constantly engaged in an empathic response. Empathic responses require cognitive and physical resources to sustain. Because Aelin is taking their work home and not resting on weekends, they do not have a chance to rest. It is important for Aelin to recognise what is happening and why. Not actioning the problem has led to empathy fatigue and emotional exhaustion. Emotional exhaustion is one of the key aspects of [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FEmpathy_fatigue_and_emotional_exhaustion&diff=2840411&oldid=2832859 burnout], a long-term condition of existing in a constant state of high stress. Psychological science can support Aelin in recognising and mapping the symptoms they are experiencing, understand coping strategies to minimise the current and future effects to improve their wellbeing. Understanding the symptoms will also support their role as a social worker, providing more tools for the people they support on a day-to-day basis. {{RoundBoxTop|theme=3}} '''Focus questions''' * Why do people become emotionally exhausted? * What is the relationship between empathy fatigue and emotional exhaustion? * What role does emotional labor play in emotional exhaustion? * How does understanding emotional labor and empathy fatigue work to reduce the effects of emotional exhaustion? {{RoundBoxBottom}} == Emotional exhaustion == To understand emotional exhaustion, it will be important to first understand burnout. Burnout is a long-term condition of existing in a constant state of high stress and affects millions of people annually. In the early 2000s, a survey from Australia found that more than 50% of social workers and occupational workers reported high levels of burnout symptoms (Llyod and King, 2004) and a recent report from 2026 found that more than one third of all workers in Australia were experiencing burnout (Chan, 2026). Being in a state of high stress for short periods of time is a useful adaptation for humans, so much so that it has been given a name, [[wikipedia:Eustress|Eustress]]. However, as a long-term condition it leads to individual health issues and can be linked to cardiovascular disease, increased vulnerability to upper respiratory diseases, and chronic inflammation (Schneiderman et al., 2005), and cognitive issues such as [[wikipedia:Inattentional_blindness|cognitive tunneling]] and impaired selective attention (Pooladvand and Hasanzadeh, 2023). Burnout can be defined across three dimensions: emotional exhaustion, [[wikipedia:Depersonalization|depersonalisation]], and reduced feelings of personal accomplishment (Maslach, 1993, as cited in Martingano et al., 2026). Emotional exhaustion may be considered the 'core' component of burnout and refers to the depletion of emotional resources, leading to feelings of overwhelm, overextension, and exhaustion (Martingano et al., 2026; De Beer et al., 2024; Seidler et al., 2014). As a major factor, its symptoms are similar to burnout and stress. Although antecedents of burnout are typically agreed upon (Bakker et al., 2014) and divided into individual (e.g. self-efficacy) and situational categories (e.g. work load), there are many theories that attempt to explain why emotional exhaustion occurs. Two of these theories with good evidence are the [[wikipedia:Conservation_of_resources_theory|Conservation of resources stress model]] (Hobfoll, 1989) and the [[wikipedia:Job_demands-resources_model|Job Demands-Resources model]] (Demerouti et al., 2001). === Conservation of resources stress model === The Conservation of resources stress model (CORT; Hobfall, 1989) is a theory that works to understand and explain why stress occurs. Under the model, humans are motivated to maintain, pursue, or attain resources. Stress, defined as "an individual's perceived pressure or strain" (Hollebeek et al., 2023 p.928), occurs when these resources are threatened or lost, or gained at a net loss. As noted above, resources aren't strictly tied to being physical objects. They can also be emotional states, conditions, or even ideas. For example, someone may place value on enjoying a calm, quiet Sunday morning at home. If their neighbour starts using power tools loudly next door, it would disrupt this state and induce stress in the individual. The CORT model focuses primarily on two principles. The first principle is that there is more harm in losing resources than there is in gaining them back (Halbesleben et al., 2014). This is known as loss salience, and some research suggests that the pain of loss is felt twice as strong as the pleasure of gain (Thaler, 2000). Looking at our previous example, imagine that the person who enjoys a quiet Sunday morning at home equally enjoys the sound of a loud car engine. Even though the person enjoys the sound of car engines, if the neighbour started revving their car loudly instead of using power tools, it would still disrupt their desired state of quiet and lead to an increase of stress. The second principle is that people must invest existing resources in order to gain or protect resources (Halbesleben et al., 2014). This resource investment includes recovering from resource loss. Coping is how people manage or overcome a difficult situation, such as protecting or recovering from resource loss. Coping can be broken down into different strategies that people use, but not all strategies are equal (Ito and Brotheridge, 2003). They are divided into two categories: control or escape strategies. Control strategies are considered better at reducing the strain from stress and consist of positive thinking, direct action, and help seeking (Havlovic & Keenan, 1991, as cited in Ito and Brotheridge, 2003). The most useful control strategies incorporate a person's [[locus of control]], emphasise seeking support from co-workers, and highlight the importance of autonomy in decision making. Alarcon (2011) conducted a meta-analysis of 231 studies in a work setting that used the [[wikipedia:Maslach_Burnout_Inventory|Maslach Burnout Inventory]] (MBI) to review the relationship between these principles and burnout to confirm their validity. Although the study had no specific mention of demographics, it was noted that multiple translations of the MBI were used including Dutch (k=51), German (k=7), Italian (k=2), and French (k=2). The findings indicate that workload is positively related to emotional exhaustion (''ρ'' = .49, ''k'' = 86, ''N'' = 51,529) and that control (''ρ'' = − .26, ''k'' = 56, ''N'' = 33,297) and autonomy (''ρ'' = − .24, ''k'' = 34, ''N'' = 17,165) were each negatively associated with emotional exhaustion. The difference of ''ρ'' values is consistent with Thaler (2000) and supports that the benefit of gaining is approximately half that of losing resources. === Job demands-resources model === Another model that attempts to understand why people become emotionally exhausted is the Job Demands-Resources model (JD-R; Demerouti et al., 2001). The JD-R model discusses two concepts: job demands and job resources. Demands and resources are both related to the physical, social, or psychological aspects of a job, but differ in their impact on a person. Job demands require physical or mental effort and exhaust a person's energy supply. Job resources typically support achieving goals, reduce the effort cost of job demands, and stimulate personal growth (Richter and Hacker, 1998, as cited in Demerouti et al., 2001). For example, an unsustainable workload would be considered a job demand, and receiving help and support from others would be considered a resource. The JD-R has two primary processes for how demands and resources interact with one another (Schaufeli, 2017). The stress process occurs when job demands are high and resources are low. As stress increases strain and pressure on an individual, it leads to the exhaustion of their energy eventually, eventually leading to emotional exhaustion. The second process is the motivational process. This process occurs when there is an excess of job resources. Crucially, the motivational process does not require low job demands. With enough job resources, a person will be energised and engaged, reframing job demands as a challenge to overcome. The JD-R model has had extensive research to confirm its validity. A meta-analysis by Lesener et al. (2019) looked at 74 longitudinal studies of the JD-R with a strong average sample size (n=450) against the assumptions as described by Schaufeli (2017). Although most of the studies appear to come from [[wikipedia:Psychology#WEIRD_bias|WEIRD]] countries (k=60) there were a good number from Asia and Africa (k=13) and a mix of genders. The study confirmed that across the literature, job resources predict burnout negatively (''β'' = −.17; ''p'' < .001) and job demands predict burnout over time (''β'' = .10; ''p'' < .001). Although job resources seem like a cure-all for demands, it's important to note that they do not retroactively treat emotional exhaustion. Under the JD-R model, burnout must be reduced by lowering job demands (Gynning et al., 2025). However, once someone has recovered, it is possible to increase job resources alongside job demands to prevent future burnout and support [[wikipedia:Work_engagement|work engagement]]. Work engagement (engagement) has been discussed as the opposite of burnout (Bakker et al., 2014). Engagement is how people express themselves when performing a role. A role being a pattern of behaviour that a person goes through based on the social expectations of that role, similar to how an actor takes on a role for a movie. For example, when attending a typical restaurant, it is socially expected that a person will engage with the waiter and provide them with their meal order to take to the kitchen. This person is performing the script of a restaurant patron. In the context of a workplace, an employee with high engagement may appear to be energised, enthusiastic, and become absorbed in their work losing track of time (Bakkar et al, 2014). {{RoundBoxTop|theme=3}} '''Why do we feel emotionally exhausted''' Under the JD-R and CORT models, emotional exhaustion occurs when the demand cost outweighs the resources available for an extended period of time, eventually depleting someone's energy reserves. Aelin is experiencing burnout because they increased their job demands by taking on more clients, and has experienced a loss in resources by reducing their free time and social support network. {{RoundBoxBottom}} == Empathy fatigue == You may be familiar with the term [[empathy]]. It is often described in terms of supporting others. For example, when someone close to you experiences a loss and is grieving, you may empathise with them and experience a negative emotion akin to grief too. However, empathy is not exclusive to so called negative emotions. For example, compersion is a positive empathic response when a loved one experiences joy. Parents may experience compersion when something joyful happens to their child. At its core, empathy is the individual ability to imagine or understand another person's feelings and/or emotions (Riess, 2017; Shao et al., 2024). Much like typical physical or cognitive processes, such as running or problem solving, empathising will drain energy and resources. Further, and similar to the difference in energy expenditure between running and jogging, there are distinct types of empathy that each have different energy costs. Research makes a clear distinction between two types of empathy: affective and cognitive empathy. Affective empathy occurs when a person has an emotional reaction to someone else's mental state (Shao et al., 2024), whereas cognitive empathy occurs when a person understands someone else's feelings or emotions (Cui et al., 2022). A way to make a distinction between these two types may be considering them as the "what" and "how". Cognitive empathy is the "what" and focuses on understanding the difference between two people. A person who is using cognitive empathy will focus on understanding what the other person is feeling as a concept and use the information processing part of their brain, the pre-frontal cortex (Shao et al., 2024). This allows the empathiser to ask this question from a cognitive, problem solving perspective. As cognitive empathy is processed by the pre-frontal cortex, there is less association with feeling the emotion across the body hence reducing the physiological response and is linked to lower energy expenditure. Affective empathy is the "how" and focuses on the feeling aspect of empathy. This is a more common interpretation of empathy, and utilises * Integrating the two - "affective component is the essence of empathy, and the cognitive component is a process through which that essence comes into being" (Bošnjaković & Radionov, 2018) ** (Author's note: change quote into paraphrasing) === Emotional contagion === Emotional contagion is the idea that emotions are contagious. as the theory to moderate surface/deep acting * Emotional contagion (Panksepp and Lahvis, 2011) ** emotions can spread (expand, combines empathy and acting) ** “the tendency to automatically mimic and synchronize expressions, vocalizations, postures, and movements with those of another person’s and, consequently, to converge emotionally.” (Elaine Hatfield, 1993) * Affective process theory looks at unifying the mechanisms of emotional contagion into three types. Convergent linkage, Divergent linkage, and Complementary linkage (Elfenbein, 2014). Storytelling is a way to promote position emotions (Wei et al., 2024). === Empathy or compassion fatigue === * Distinction between empathy and compassion fatigue * Compassion fatigue is closely related to Secondary traumatic stress usually found in healthcare professionals (Figley, 2002) ** Compassion fatigue is to Empathy fatigue as emotional exhaustion is to burnout * Make it clear that Aelin is experiencing Compassion Fatigue vs Empathy fatigue (they are the same but different) (Author's note: include an update to the scenario here!) '''Empathy fatigue questions'''. Choose your answers and click "Submit": <quiz display=simple> {Surface acting is when the observer actually feels the emotion: |type="()"} - True + False {Compassion fatigue and empathy fatigue are synonymous: |type="()"} - True + False </quiz> {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' Under the JD-R and CORT models, a sustained empathic response is a long-term job demand that uses energy to protect existing resources. As Aelin has had their resources is experiencing a * Under the CORT model, a sustained empathic response uses energy to protect existing resources. As resources become low and burnout sets in, cognitive focus narrows leading to empathy fatigue. {{RoundBoxBottom}} == Emotional labor as a mediating factor between empathy fatigue and emotional exhaustion == As a society there are expected [https://en.wikipedia.org/wiki/Social_norm social norms] in behaviour when expressing emotion, regardless if someone is conscious of the process of matching these norms or not. For example, laughing during an emotional speech at a funeral would be considered a rude or offensive expression from the other attendees. As social creatures, humans need a way to regulate socially appropriate emotional expression. This is the essence of Emotional labor: "the regulation of felt and expressed emotions (whether effortful or not) to match these emotional display rules with the objective of fulfilling organizational goals" (Vial & Cowgill, 2022) What is emotional labor (Hochschild, 2012). Theories behind emotional labor involve acting, and an extension of acting known as emotional contagion. === Types of acting === * Surface acting ** This is where emotive dissonance comes into play ** Leads to emotional exhaustion (Kenworthy et al., 2014) ** "Should" language - in this situation, this is what I ''should'' feel * Deep acting ** People who 'deep act', as in genuinely change the way they feel (match the feelings they have with the feelings that they need to display) often report positive relationships with emotional labor ** Actual language - in this situation, this is what I ''actually'' feel Discuss the differences in gender norms and expectations in relationships {| class="wikitable" style="margin: auto;" |- ! !! Deep acting !! Surface acting |- | '''Emotional dissonance''' || || X |- | '''Not known to others''' || Hidden area || Unknown |} * === The mediating effect of emotional labor === Surface acting does not mediate the effect (high surface acting leads to emotional exhaustion) whereas the inverse is true for deep acting (Wrobel, 2013) Emotional Expression and Emotional Processing (Hoyt et al., 2024) {{RoundBoxTop|theme=3}} '''What role does emotional labor play in emotional exhaustion''' * In relation to the CORT and JD-R theories, emotional labor acts as a job demand and expenditure of resources to protect existing resources {{RoundBoxBottom}} == Strategies for combating empathy fatigue and emotional fatigue == * Talk about emotion regulation (Gross, 1998) and mood regulation strategies as a way to deal with coping (Wrobel, 2013) (Author's note: need more research/references for this section) * Talk about breaks ** Rest and downtime from not performing emotional labor regenerates * How do we reduce the amount of emotional labor we're performing (Jeung et al., 2018) ** Communication/boundaries ** Reducing surface acting (allowing ourselves to feel how we actually feel and reduce emotive dissonance) '''Table 1. Treatment phases of clinical burnout (based on Table 2 from van Dam, A. 2021)''' {| class="wikitable" style="margin: auto; |- ! Phase !! Treatment goal !! Therapeutic interventions |- | '''Phase 1: Crisis''' || Recognition of the patient that the problems are serious, and that serious action is required || Psychoeducation Sick leave Dropping domestic tasks and social obligations Inform social network |- | '''Phase 2: Recovery''' || Recovery of the (physiological) stress system to normal allostatic stress levels || Registration of stress and activities Relaxtion exercises Mindfulness healthy lifestyle advices Graded exercise Gradually resume activities Psychoeducation social network and employer Gradual return to work |- | '''Phase 3: Prevention, learning from the past''' || Acquiring insight and skills to prevent relapse in clinical burnout || Analysing stressful situations Analysing dysfunctional thought patterns Learning new coping skills Learning social skills Job crafting Time management Making choices regarding career and personal life |- | '''Phase 4: Post burnout growth''' || Improving sustainable quality of life. || Setting priorities regarding quality of life and interpersonal relationships. |} === Strategies for the workplace === * Discuss types of strategies that are available in the workplace (organisation models) * SCARF model (Aplin-Houtz, 2025) ** Link back to COR and JDR models e.g. praise could be a job resource vs criticism could be a job demand *Reduce workload to reduce emotional exhaustion, and by extension reduce emotional fatigue. {{RoundBoxTop|theme=3}} '''How does understanding emotional labor and empathy fatigue work to reduce the effects of emotional exhaustion''' *Psychoeducation has been shown to reduce the effects of emotional labor and empathy fatigue on emotional exhaustion {{RoundBoxBottom}} ==Conclusion== * Summarise what empathy fatigue is, and why people become emotionally exhausted (+ the role of emotional labor) * Summarise the symptoms and how to recognise them in yourself and someone else * Emphasise and summarise psychological strategies to support someone going through emotional exhaustion * Summarise ways to prevent emotional exhaustion from happening * Overall message is: *# What is emotional exhaustion *# how do you recognise it *# how do you treat it *# how do you prevent it * Link back to the focus questions ==See also== * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Metacognition_and_emotional_regulation&oldid=2756264 Metacognition and emotional regulation] (Book chapter, 2026) {{ic|Use an internal link style as shown in Tutorial 2}} * [[wikipedia:Conservation_of_resources_theory|Conservation of resources theory]] (Wikipedia) * [[wikipedia:Job_demands-resources_model|Job demands-resources model]] (Wikipedia) ==References== {{Hanging indent|1= Aplin-Houtz, M. (2025). Unraveling the SCARF: How dimensions of the SCARF model influence the relationship between workplace exclusion and turnover intentions. Journal of Management & Organization, 31(6), 2699–2725. https://doi.org/10.1017/jmo.2025.10026 Bošnjaković, J., & Radionov, T. (2018). Empathy: Concepts, Theories and Neuroscientific Basis. Alcoholism (Zagreb), 54(2), 123–150. https://doi.org/10.20471/de..2018.54.02.04 Chan, F. M. (2026). The State of Workplace Burnout 2026: The Workplace Divide. Infinite Potential, Sydney, Australia. https://www.ahri.com.au/wp-content/uploads/The-State-of-Workplace-Burnout-2026-Final.pdf Cui, F., Huang, X., Jing, Y., Luo, Y., Liu, J., & Gu, R. (2022). How resource sharing resists scarcity: the role of cognitive empathy and its neurobiological mechanisms. Cerebral Cortex (New York, N.Y. 1991), 32(23), 5330–5342. https://doi.org/10.1093/cercor/bhac017 De Beer, L. T., van der Vaart, L., Escaffi-Schwarz, M., De Witte, H., & Schaufeli, W. B. (2024). Maslach Burnout Inventory - General Survey: A Systematic Review and Meta-Analysis of Measurement Properties. European journal of psychological assessment : official organ of the European Association of Psychological Assessment, 40(5), 360-375. https://doi.org/10.1027/1015-5759/a000797 Demerouti, E., Bakker, A. B., Nachreiner, F., & Schaufeli, W. B. (2001). The Job Demands-Resources Model of Burnout. Journal of Applied Psychology, 86(3), 499–512. https://doi.org/10.1037/0021-9010.86.3.499 Figley, C. R. (2002). Compassion fatigue: Psychotherapists’ chronic lack of self care. Journal of Clinical Psychology, 58(11), 1433–1441. https://doi.org/10.1002/jclp.10090 Grandey, A. A. (2003). When ‘the show must go on’: surface acting and deep acting as determinants of emotional exhaustion and peer-rated service delivery. Academy of Management Journal, 46(1), 86–96. https://doi.org/10.5465/30040678 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 Halbesleben, J. R. B., Neveu, J.-P., Paustian-Underdahl, S. C., & Westman, M. (2014). Getting to the “COR”: Understanding the Role of Resources in Conservation of Resources Theory. Journal of Management, 40(5), 1334–1364. https://doi.org/10.1177/0149206314527130 Hatfield, E., Cacioppo, J. T., & Rapson, R. L. (1993). Emotional Contagion. Current Directions in Psychological Science : A Journal of the American Psychological Society, 2(3), 96–99. https://doi.org/10.1111/1467-8721.ep10770953 Hollebeek, L. D., Hammedi, W., & Sprott, D. E. (2023). Consumer engagement, stress, and conservation of resources theory: A review, conceptual development, and future research agenda. Psychology & marketing, 40(5), 926-937. https://doi.org/10.1002/mar.21807 Hobfoll, S. E. (1989). Conservation of Resources: A New Attempt at Conceptualizing Stress. The American Psychologist, 44(3), 513–524. https://doi.org/10.1037/0003-066X.44.3.513 Hochschild, A. R. (2012). The managed heart : commercialization of human feeling (Updated, with a new preface.). University of California Press. Jeung, D.-Y., Kim, C., & Chang, S.-J. (2018). Emotional Labor and Burnout: A Review of the Literature. Yonsei Medical Journal, 59(2), 187–193. https://doi.org/10.3349/ymj.2018.59.2.187 Lavelle, J. J., Rupp, D. E., Herda, D. N., Pandey, A., & Lauck, J. R. (2021). Customer Injustice and Employee Performance: Roles of Emotional Exhaustion, Surface Acting, and Emotional Demands–Abilities Fit. Journal of Management, 47(3), Article 0149206319869426. https://doi.org/10.1177/0149206319869426 Lloyd, C., & King, R. (2004). A survey of burnout among Australian mental health occupational therapists and social workers. Social Psychiatry and Psychiatric Epidemiology, 39(9), 752-757. https://doi.org/https://doi.org/10.1007/s00127-004-0808-7 Kenworthy, J., Fay, C., Frame, M., & Petree, R. (2014). A meta-analytic review of the relationship between emotional dissonance and emotional exhaustion. Journal of Applied Social Psychology, 44(2), 94-105. https://doi.org/https://doi.org/10.1111/jasp.12211 Martingano, A. J., Stosic, M. D., & Konrath, S. (2026). Different empathy types show opposing associations with burnout: systematic review and meta-analyses. Psychology, Health & Medicine, 31(3), 798-829. https://doi.org/10.1080/13548506.2025.2591859 Michielsen, H. J., Willemsen, T. M., Croon, M. A., de Vries, J., & Van Heck, G. L. (2004). Determinants of general fatigue and emotional exhaustion: a prospective study. Psychology & health, 19(2), 223-235. https://doi.org/10.1080/08870440310001627135 Panksepp, J. B., & Lahvis, G. P. (2011). Rodent empathy and affective neuroscience. Neuroscience and Biobehavioral Reviews, 35(9), 1864–1875. https://doi.org/10.1016/j.neubiorev.2011.05.013 Pooladvand, S., & Hasanzadeh, S. (2023). Impacts of Stress on Workers’ Risk-Taking Behaviors: Cognitive Tunneling and Impaired Selective Attention. Journal of Construction Engineering and Management, 149(8), 04023060. https://doi.org/doi:10.1061/JCEMD4.COENG-13339 Riess, H. (2017). The Science of Empathy. Journal of Patient Experience, 4(2), 74–77. https://doi.org/10.1177/2374373517699267 Schaufeli, W. B. (2017). Applying the job demands-resources model: a “how to” guide to measuring and tackling work engagement and burnout. Organizational Dynamics, 46(2), 120–132. https://doi.org/10.1016/j.orgdyn.2017.04.008 Schneiderman, N., Ironson, G., & Siegel, S. D. (2005). Stress and Health: Psychological, Behavioral, and Biological Determinants. Annual Review of Clinical Psychology, 1(Volume 1, 2005), 607-628. https://doi.org/https://doi.org/10.1146/annurev.clinpsy.1.102803.144141 Seidler, A., Thinschmidt, M., Deckert, S., Then, F., Hegewald, J., Nieuwenhuijsen, K., & Riedel-Heller, S. G. (2014). The role of psychosocial working conditions on burnout and its core component emotional exhaustion – a systematic review. Journal of Occupational Medicine and Toxicology, 9(1), 10. https://doi.org/10.1186/1745-6673-9-10 Shao, M., Li, L., Li, X., Wei, Z., Wang, J., Hong, M., Liu, X., & Meng, J. (2024). The effect of top-down attention on empathy fatigue. Cerebral Cortex (New York, N.Y. 1991), 34(1), Article bhad441. https://doi.org/10.1093/cercor/bhad441 van Dam, A. (2021). A clinical perspective on burnout: diagnosis, classification, and treatment of clinical burnout. European Journal of Work and Organizational Psychology, 30(5), 732-741. https://doi.org/10.1080/1359432X.2021.1948400 Vial, A. C., & Cowgill, C. M. (2022). Heavier Lies Her Crown: Gendered Patterns of Leader Emotional Labor and Their Downstream Effects. Frontiers in Psychology, 13, 849566. https://doi.org/https://doi.org/10.3389/fpsyg.2022.849566 Wharton, A. S. (2009). The Sociology of Emotional Labor. Annual Review of Sociology, 35(1), 147–165. https://doi.org/10.1146/annurev-soc-070308-115944 Wrobel, M. (2013). Can empathy lead to emotional exhaustion in teachers? The mediating role of emotional labor. International Journal of Occupational Medicine and Environmental Health, 26(4), 581–592. https://doi.org/10.2478/s13382-013-0123-1 }} ==External links== Provide [[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: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Burnout]] [[Category:Motivation and emotion/Book/Empathy]] groyd9tq9mnxhhvskbz96bmbs6nz4lh 2844543 2844539 2026-11-05T01:27:17Z U3143751 3106785 /* Empathy fatigue */ 2844543 wikitext text/x-wiki {{title|Empathy fatigue and emotional exhaustion:<br>How does sustained empathic engagement contribute to emotional exhaustion?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}} [[File:Exhausted smartphone user.JPG|right|thumb|150px|'''Figure 1'''. An emotionally exhausted Aelin at the end of the work day]] ; Scenario Aelin is a 23-year-old social worker employed at a non-profit organisation providing mental health services to teenagers and young adults. Aelin's clients are aged between 18 and 25, many of whom having experienced rough living conditions. They work with approximately 20 clients each week. During Aelin's first six months in the role, they feel highly motivated and are excited about helping people. They are an excellent active listener, and can easily remember details about their clients' lives. During their sessions, Aelin recognises when a client is distressed and adjusts their body language to warm and understanding in an effort to provide a safe and welcoming environment to hold space for their emotions. As Aelin gains more experience working in this role, their caseload increases, and they are exposed to clients with increasing complexity and past traumas. Aelin wants to provide the best possible outcome for all of their clients, and begins to take their work home with them in the evenings, but finds it starts creeping into their weekends as well. After several months of working this way, Aelin begins to notice they feel exhausted every morning - even before the workday begins. Fast-forward a few more months, and Aelin notices a change in how they start responding to people, including clients. Instead of feeling engaged and actively listening, they feel detached and find their thoughts feeling more frustrated than empathetic. After finishing the work day, Aelin is drained and has no energy for friends or family. With increasing irritability, Aelin begins to worry that they won't be able to continue working as a social-worker. {{RoundBoxBottom}} Aelin is experiencing '''emotional exhaustion''' from a sustained empathic response. Due to the nature of Aelin's work, they are constantly engaged in an empathic response. Empathic responses require cognitive and physical resources to sustain. Because Aelin is taking their work home and not resting on weekends, they do not have a chance to rest. It is important for Aelin to recognise what is happening and why. Not actioning the problem has led to empathy fatigue and emotional exhaustion. Emotional exhaustion is one of the key aspects of [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FEmpathy_fatigue_and_emotional_exhaustion&diff=2840411&oldid=2832859 burnout], a long-term condition of existing in a constant state of high stress. Psychological science can support Aelin in recognising and mapping the symptoms they are experiencing, understand coping strategies to minimise the current and future effects to improve their wellbeing. Understanding the symptoms will also support their role as a social worker, providing more tools for the people they support on a day-to-day basis. {{RoundBoxTop|theme=3}} '''Focus questions''' * Why do people become emotionally exhausted? * What is the relationship between empathy fatigue and emotional exhaustion? * What role does emotional labor play in emotional exhaustion? * How does understanding emotional labor and empathy fatigue work to reduce the effects of emotional exhaustion? {{RoundBoxBottom}} == Emotional exhaustion == To understand emotional exhaustion, it will be important to first understand burnout. Burnout is a long-term condition of existing in a constant state of high stress and affects millions of people annually. In the early 2000s, a survey from Australia found that more than 50% of social workers and occupational workers reported high levels of burnout symptoms (Llyod and King, 2004) and a recent report from 2026 found that more than one third of all workers in Australia were experiencing burnout (Chan, 2026). Being in a state of high stress for short periods of time is a useful adaptation for humans, so much so that it has been given a name, [[wikipedia:Eustress|Eustress]]. However, as a long-term condition it leads to individual health issues and can be linked to cardiovascular disease, increased vulnerability to upper respiratory diseases, and chronic inflammation (Schneiderman et al., 2005), and cognitive issues such as [[wikipedia:Inattentional_blindness|cognitive tunneling]] and impaired selective attention (Pooladvand and Hasanzadeh, 2023). Burnout can be defined across three dimensions: emotional exhaustion, [[wikipedia:Depersonalization|depersonalisation]], and reduced feelings of personal accomplishment (Maslach, 1993, as cited in Martingano et al., 2026). Emotional exhaustion may be considered the 'core' component of burnout and refers to the depletion of emotional resources, leading to feelings of overwhelm, overextension, and exhaustion (Martingano et al., 2026; De Beer et al., 2024; Seidler et al., 2014). As a major factor, its symptoms are similar to burnout and stress. Although antecedents of burnout are typically agreed upon (Bakker et al., 2014) and divided into individual (e.g. self-efficacy) and situational categories (e.g. work load), there are many theories that attempt to explain why emotional exhaustion occurs. Two of these theories with good evidence are the [[wikipedia:Conservation_of_resources_theory|Conservation of resources stress model]] (Hobfoll, 1989) and the [[wikipedia:Job_demands-resources_model|Job Demands-Resources model]] (Demerouti et al., 2001). === Conservation of resources stress model === The Conservation of resources stress model (CORT; Hobfall, 1989) is a theory that works to understand and explain why stress occurs. Under the model, humans are motivated to maintain, pursue, or attain resources. Stress, defined as "an individual's perceived pressure or strain" (Hollebeek et al., 2023 p.928), occurs when these resources are threatened or lost, or gained at a net loss. As noted above, resources aren't strictly tied to being physical objects. They can also be emotional states, conditions, or even ideas. For example, someone may place value on enjoying a calm, quiet Sunday morning at home. If their neighbour starts using power tools loudly next door, it would disrupt this state and induce stress in the individual. The CORT model focuses primarily on two principles. The first principle is that there is more harm in losing resources than there is in gaining them back (Halbesleben et al., 2014). This is known as loss salience, and some research suggests that the pain of loss is felt twice as strong as the pleasure of gain (Thaler, 2000). Looking at our previous example, imagine that the person who enjoys a quiet Sunday morning at home equally enjoys the sound of a loud car engine. Even though the person enjoys the sound of car engines, if the neighbour started revving their car loudly instead of using power tools, it would still disrupt their desired state of quiet and lead to an increase of stress. The second principle is that people must invest existing resources in order to gain or protect resources (Halbesleben et al., 2014). This resource investment includes recovering from resource loss. Coping is how people manage or overcome a difficult situation, such as protecting or recovering from resource loss. Coping can be broken down into different strategies that people use, but not all strategies are equal (Ito and Brotheridge, 2003). They are divided into two categories: control or escape strategies. Control strategies are considered better at reducing the strain from stress and consist of positive thinking, direct action, and help seeking (Havlovic & Keenan, 1991, as cited in Ito and Brotheridge, 2003). The most useful control strategies incorporate a person's [[locus of control]], emphasise seeking support from co-workers, and highlight the importance of autonomy in decision making. Alarcon (2011) conducted a meta-analysis of 231 studies in a work setting that used the [[wikipedia:Maslach_Burnout_Inventory|Maslach Burnout Inventory]] (MBI) to review the relationship between these principles and burnout to confirm their validity. Although the study had no specific mention of demographics, it was noted that multiple translations of the MBI were used including Dutch (k=51), German (k=7), Italian (k=2), and French (k=2). The findings indicate that workload is positively related to emotional exhaustion (''ρ'' = .49, ''k'' = 86, ''N'' = 51,529) and that control (''ρ'' = − .26, ''k'' = 56, ''N'' = 33,297) and autonomy (''ρ'' = − .24, ''k'' = 34, ''N'' = 17,165) were each negatively associated with emotional exhaustion. The difference of ''ρ'' values is consistent with Thaler (2000) and supports that the benefit of gaining is approximately half that of losing resources. === Job demands-resources model === Another model that attempts to understand why people become emotionally exhausted is the Job Demands-Resources model (JD-R; Demerouti et al., 2001). The JD-R model discusses two concepts: job demands and job resources. Demands and resources are both related to the physical, social, or psychological aspects of a job, but differ in their impact on a person. Job demands require physical or mental effort and exhaust a person's energy supply. Job resources typically support achieving goals, reduce the effort cost of job demands, and stimulate personal growth (Richter and Hacker, 1998, as cited in Demerouti et al., 2001). For example, an unsustainable workload would be considered a job demand, and receiving help and support from others would be considered a resource. The JD-R has two primary processes for how demands and resources interact with one another (Schaufeli, 2017). The stress process occurs when job demands are high and resources are low. As stress increases strain and pressure on an individual, it leads to the exhaustion of their energy eventually, eventually leading to emotional exhaustion. The second process is the motivational process. This process occurs when there is an excess of job resources. Crucially, the motivational process does not require low job demands. With enough job resources, a person will be energised and engaged, reframing job demands as a challenge to overcome. The JD-R model has had extensive research to confirm its validity. A meta-analysis by Lesener et al. (2019) looked at 74 longitudinal studies of the JD-R with a strong average sample size (n=450) against the assumptions as described by Schaufeli (2017). Although most of the studies appear to come from [[wikipedia:Psychology#WEIRD_bias|WEIRD]] countries (k=60) there were a good number from Asia and Africa (k=13) and a mix of genders. The study confirmed that across the literature, job resources predict burnout negatively (''β'' = −.17; ''p'' < .001) and job demands predict burnout over time (''β'' = .10; ''p'' < .001). Although job resources seem like a cure-all for demands, it's important to note that they do not retroactively treat emotional exhaustion. Under the JD-R model, burnout must be reduced by lowering job demands (Gynning et al., 2025). However, once someone has recovered, it is possible to increase job resources alongside job demands to prevent future burnout and support [[wikipedia:Work_engagement|work engagement]]. Work engagement (engagement) has been discussed as the opposite of burnout (Bakker et al., 2014). Engagement is how people express themselves when performing a role. A role being a pattern of behaviour that a person goes through based on the social expectations of that role, similar to how an actor takes on a role for a movie. For example, when attending a typical restaurant, it is socially expected that a person will engage with the waiter and provide them with their meal order to take to the kitchen. This person is performing the script of a restaurant patron. In the context of a workplace, an employee with high engagement may appear to be energised, enthusiastic, and become absorbed in their work losing track of time (Bakkar et al, 2014). {{RoundBoxTop|theme=3}} '''Why do we feel emotionally exhausted''' Under the JD-R and CORT models, emotional exhaustion occurs when the demand cost outweighs the resources available for an extended period of time, eventually depleting someone's energy reserves. Aelin is experiencing burnout because they increased their job demands by taking on more clients, and has experienced a loss in resources by reducing their free time and social support network. {{RoundBoxBottom}} == Empathy fatigue == You may be familiar with the term [[empathy]]. It is often described in terms of supporting others. For example, when someone close to you experiences a loss and is grieving, you may empathise with them and experience a negative emotion akin to grief too. However, empathy is not exclusive to so called negative emotions. For example, compersion is a positive empathic response when a loved one experiences joy. Parents may experience compersion when something joyful happens to their child. At its core, empathy is the individual ability to imagine or understand another person's feelings and/or emotions (Riess, 2017; Shao et al., 2024). Much like typical physical or cognitive processes, such as running or problem solving, empathising will drain energy and resources. Further, and similar to the difference in energy expenditure between running and jogging, there are distinct types of empathy that each have different energy costs. Research makes a clear distinction between two types of empathy: affective and cognitive empathy. Affective empathy occurs when a person has an emotional reaction to someone else's mental state (Shao et al., 2024), whereas cognitive empathy occurs when a person understands someone else's feelings or emotions (Cui et al., 2022). A way to make a distinction between these two types may be considering them as the "what" and "how". Cognitive empathy is the "what" and focuses on understanding the difference between two people. A person who is using cognitive empathy will focus on understanding what the other person is feeling as a concept and use the information processing part of their brain, the pre-frontal cortex (Shao et al., 2024). This allows the empathiser to ask this question from a cognitive, problem solving perspective. As cognitive empathy is processed by the pre-frontal cortex, there is less association with feeling the emotion across the body hence reducing the physiological response and is linked to lower energy expenditure. Affective empathy is the "how" and focuses on the feeling aspect of empathy. This is a more common understanding of empathy, and utilises * Integrating the two - "affective component is the essence of empathy, and the cognitive component is a process through which that essence comes into being" (Bošnjaković & Radionov, 2018) ** (Author's note: change quote into paraphrasing) === Emotional contagion === Emotional contagion is the idea that emotions are contagious. as the theory to moderate surface/deep acting * Emotional contagion (Panksepp and Lahvis, 2011) ** emotions can spread (expand, combines empathy and acting) ** “the tendency to automatically mimic and synchronize expressions, vocalizations, postures, and movements with those of another person’s and, consequently, to converge emotionally.” (Elaine Hatfield, 1993) * Affective process theory looks at unifying the mechanisms of emotional contagion into three types. Convergent linkage, Divergent linkage, and Complementary linkage (Elfenbein, 2014). Storytelling is a way to promote position emotions (Wei et al., 2024). === Empathy or compassion fatigue === * Distinction between empathy and compassion fatigue * Compassion fatigue is closely related to Secondary traumatic stress usually found in healthcare professionals (Figley, 2002) ** Compassion fatigue is to Empathy fatigue as emotional exhaustion is to burnout * Make it clear that Aelin is experiencing Compassion Fatigue vs Empathy fatigue (they are the same but different) (Author's note: include an update to the scenario here!) '''Empathy fatigue questions'''. Choose your answers and click "Submit": <quiz display=simple> {Surface acting is when the observer actually feels the emotion: |type="()"} - True + False {Compassion fatigue and empathy fatigue are synonymous: |type="()"} - True + False </quiz> {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' Under the JD-R and CORT models, a sustained empathic response is a long-term job demand that uses energy to protect existing resources. As Aelin has had their resources is experiencing a * Under the CORT model, a sustained empathic response uses energy to protect existing resources. As resources become low and burnout sets in, cognitive focus narrows leading to empathy fatigue. {{RoundBoxBottom}} == Emotional labor as a mediating factor between empathy fatigue and emotional exhaustion == As a society there are expected [https://en.wikipedia.org/wiki/Social_norm social norms] in behaviour when expressing emotion, regardless if someone is conscious of the process of matching these norms or not. For example, laughing during an emotional speech at a funeral would be considered a rude or offensive expression from the other attendees. As social creatures, humans need a way to regulate socially appropriate emotional expression. This is the essence of Emotional labor: "the regulation of felt and expressed emotions (whether effortful or not) to match these emotional display rules with the objective of fulfilling organizational goals" (Vial & Cowgill, 2022) What is emotional labor (Hochschild, 2012). Theories behind emotional labor involve acting, and an extension of acting known as emotional contagion. === Types of acting === * Surface acting ** This is where emotive dissonance comes into play ** Leads to emotional exhaustion (Kenworthy et al., 2014) ** "Should" language - in this situation, this is what I ''should'' feel * Deep acting ** People who 'deep act', as in genuinely change the way they feel (match the feelings they have with the feelings that they need to display) often report positive relationships with emotional labor ** Actual language - in this situation, this is what I ''actually'' feel Discuss the differences in gender norms and expectations in relationships {| class="wikitable" style="margin: auto;" |- ! !! Deep acting !! Surface acting |- | '''Emotional dissonance''' || || X |- | '''Not known to others''' || Hidden area || Unknown |} * === The mediating effect of emotional labor === Surface acting does not mediate the effect (high surface acting leads to emotional exhaustion) whereas the inverse is true for deep acting (Wrobel, 2013) Emotional Expression and Emotional Processing (Hoyt et al., 2024) {{RoundBoxTop|theme=3}} '''What role does emotional labor play in emotional exhaustion''' * In relation to the CORT and JD-R theories, emotional labor acts as a job demand and expenditure of resources to protect existing resources {{RoundBoxBottom}} == Strategies for combating empathy fatigue and emotional fatigue == * Talk about emotion regulation (Gross, 1998) and mood regulation strategies as a way to deal with coping (Wrobel, 2013) (Author's note: need more research/references for this section) * Talk about breaks ** Rest and downtime from not performing emotional labor regenerates * How do we reduce the amount of emotional labor we're performing (Jeung et al., 2018) ** Communication/boundaries ** Reducing surface acting (allowing ourselves to feel how we actually feel and reduce emotive dissonance) '''Table 1. Treatment phases of clinical burnout (based on Table 2 from van Dam, A. 2021)''' {| class="wikitable" style="margin: auto; |- ! Phase !! Treatment goal !! Therapeutic interventions |- | '''Phase 1: Crisis''' || Recognition of the patient that the problems are serious, and that serious action is required || Psychoeducation Sick leave Dropping domestic tasks and social obligations Inform social network |- | '''Phase 2: Recovery''' || Recovery of the (physiological) stress system to normal allostatic stress levels || Registration of stress and activities Relaxtion exercises Mindfulness healthy lifestyle advices Graded exercise Gradually resume activities Psychoeducation social network and employer Gradual return to work |- | '''Phase 3: Prevention, learning from the past''' || Acquiring insight and skills to prevent relapse in clinical burnout || Analysing stressful situations Analysing dysfunctional thought patterns Learning new coping skills Learning social skills Job crafting Time management Making choices regarding career and personal life |- | '''Phase 4: Post burnout growth''' || Improving sustainable quality of life. || Setting priorities regarding quality of life and interpersonal relationships. |} === Strategies for the workplace === * Discuss types of strategies that are available in the workplace (organisation models) * SCARF model (Aplin-Houtz, 2025) ** Link back to COR and JDR models e.g. praise could be a job resource vs criticism could be a job demand *Reduce workload to reduce emotional exhaustion, and by extension reduce emotional fatigue. {{RoundBoxTop|theme=3}} '''How does understanding emotional labor and empathy fatigue work to reduce the effects of emotional exhaustion''' *Psychoeducation has been shown to reduce the effects of emotional labor and empathy fatigue on emotional exhaustion {{RoundBoxBottom}} ==Conclusion== * Summarise what empathy fatigue is, and why people become emotionally exhausted (+ the role of emotional labor) * Summarise the symptoms and how to recognise them in yourself and someone else * Emphasise and summarise psychological strategies to support someone going through emotional exhaustion * Summarise ways to prevent emotional exhaustion from happening * Overall message is: *# What is emotional exhaustion *# how do you recognise it *# how do you treat it *# how do you prevent it * Link back to the focus questions ==See also== * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Metacognition_and_emotional_regulation&oldid=2756264 Metacognition and emotional regulation] (Book chapter, 2026) {{ic|Use an internal link style as shown in Tutorial 2}} * [[wikipedia:Conservation_of_resources_theory|Conservation of resources theory]] (Wikipedia) * [[wikipedia:Job_demands-resources_model|Job demands-resources model]] (Wikipedia) ==References== {{Hanging indent|1= Aplin-Houtz, M. (2025). Unraveling the SCARF: How dimensions of the SCARF model influence the relationship between workplace exclusion and turnover intentions. Journal of Management & Organization, 31(6), 2699–2725. https://doi.org/10.1017/jmo.2025.10026 Bošnjaković, J., & Radionov, T. (2018). Empathy: Concepts, Theories and Neuroscientific Basis. Alcoholism (Zagreb), 54(2), 123–150. https://doi.org/10.20471/de..2018.54.02.04 Chan, F. M. (2026). The State of Workplace Burnout 2026: The Workplace Divide. Infinite Potential, Sydney, Australia. https://www.ahri.com.au/wp-content/uploads/The-State-of-Workplace-Burnout-2026-Final.pdf Cui, F., Huang, X., Jing, Y., Luo, Y., Liu, J., & Gu, R. (2022). How resource sharing resists scarcity: the role of cognitive empathy and its neurobiological mechanisms. Cerebral Cortex (New York, N.Y. 1991), 32(23), 5330–5342. https://doi.org/10.1093/cercor/bhac017 De Beer, L. T., van der Vaart, L., Escaffi-Schwarz, M., De Witte, H., & Schaufeli, W. B. (2024). Maslach Burnout Inventory - General Survey: A Systematic Review and Meta-Analysis of Measurement Properties. European journal of psychological assessment : official organ of the European Association of Psychological Assessment, 40(5), 360-375. https://doi.org/10.1027/1015-5759/a000797 Demerouti, E., Bakker, A. B., Nachreiner, F., & Schaufeli, W. B. (2001). The Job Demands-Resources Model of Burnout. Journal of Applied Psychology, 86(3), 499–512. https://doi.org/10.1037/0021-9010.86.3.499 Figley, C. R. (2002). Compassion fatigue: Psychotherapists’ chronic lack of self care. Journal of Clinical Psychology, 58(11), 1433–1441. https://doi.org/10.1002/jclp.10090 Grandey, A. A. (2003). When ‘the show must go on’: surface acting and deep acting as determinants of emotional exhaustion and peer-rated service delivery. Academy of Management Journal, 46(1), 86–96. https://doi.org/10.5465/30040678 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 Halbesleben, J. R. B., Neveu, J.-P., Paustian-Underdahl, S. C., & Westman, M. (2014). Getting to the “COR”: Understanding the Role of Resources in Conservation of Resources Theory. Journal of Management, 40(5), 1334–1364. https://doi.org/10.1177/0149206314527130 Hatfield, E., Cacioppo, J. T., & Rapson, R. L. (1993). Emotional Contagion. Current Directions in Psychological Science : A Journal of the American Psychological Society, 2(3), 96–99. https://doi.org/10.1111/1467-8721.ep10770953 Hollebeek, L. D., Hammedi, W., & Sprott, D. E. (2023). Consumer engagement, stress, and conservation of resources theory: A review, conceptual development, and future research agenda. Psychology & marketing, 40(5), 926-937. https://doi.org/10.1002/mar.21807 Hobfoll, S. E. (1989). Conservation of Resources: A New Attempt at Conceptualizing Stress. The American Psychologist, 44(3), 513–524. https://doi.org/10.1037/0003-066X.44.3.513 Hochschild, A. R. (2012). The managed heart : commercialization of human feeling (Updated, with a new preface.). University of California Press. Jeung, D.-Y., Kim, C., & Chang, S.-J. (2018). Emotional Labor and Burnout: A Review of the Literature. Yonsei Medical Journal, 59(2), 187–193. https://doi.org/10.3349/ymj.2018.59.2.187 Lavelle, J. J., Rupp, D. E., Herda, D. N., Pandey, A., & Lauck, J. R. (2021). Customer Injustice and Employee Performance: Roles of Emotional Exhaustion, Surface Acting, and Emotional Demands–Abilities Fit. Journal of Management, 47(3), Article 0149206319869426. https://doi.org/10.1177/0149206319869426 Lloyd, C., & King, R. (2004). A survey of burnout among Australian mental health occupational therapists and social workers. Social Psychiatry and Psychiatric Epidemiology, 39(9), 752-757. https://doi.org/https://doi.org/10.1007/s00127-004-0808-7 Kenworthy, J., Fay, C., Frame, M., & Petree, R. (2014). A meta-analytic review of the relationship between emotional dissonance and emotional exhaustion. Journal of Applied Social Psychology, 44(2), 94-105. https://doi.org/https://doi.org/10.1111/jasp.12211 Martingano, A. J., Stosic, M. D., & Konrath, S. (2026). Different empathy types show opposing associations with burnout: systematic review and meta-analyses. Psychology, Health & Medicine, 31(3), 798-829. https://doi.org/10.1080/13548506.2025.2591859 Michielsen, H. J., Willemsen, T. M., Croon, M. A., de Vries, J., & Van Heck, G. L. (2004). Determinants of general fatigue and emotional exhaustion: a prospective study. Psychology & health, 19(2), 223-235. https://doi.org/10.1080/08870440310001627135 Panksepp, J. B., & Lahvis, G. P. (2011). Rodent empathy and affective neuroscience. Neuroscience and Biobehavioral Reviews, 35(9), 1864–1875. https://doi.org/10.1016/j.neubiorev.2011.05.013 Pooladvand, S., & Hasanzadeh, S. (2023). Impacts of Stress on Workers’ Risk-Taking Behaviors: Cognitive Tunneling and Impaired Selective Attention. Journal of Construction Engineering and Management, 149(8), 04023060. https://doi.org/doi:10.1061/JCEMD4.COENG-13339 Riess, H. (2017). The Science of Empathy. Journal of Patient Experience, 4(2), 74–77. https://doi.org/10.1177/2374373517699267 Schaufeli, W. B. (2017). Applying the job demands-resources model: a “how to” guide to measuring and tackling work engagement and burnout. Organizational Dynamics, 46(2), 120–132. https://doi.org/10.1016/j.orgdyn.2017.04.008 Schneiderman, N., Ironson, G., & Siegel, S. D. (2005). Stress and Health: Psychological, Behavioral, and Biological Determinants. Annual Review of Clinical Psychology, 1(Volume 1, 2005), 607-628. https://doi.org/https://doi.org/10.1146/annurev.clinpsy.1.102803.144141 Seidler, A., Thinschmidt, M., Deckert, S., Then, F., Hegewald, J., Nieuwenhuijsen, K., & Riedel-Heller, S. G. (2014). The role of psychosocial working conditions on burnout and its core component emotional exhaustion – a systematic review. Journal of Occupational Medicine and Toxicology, 9(1), 10. https://doi.org/10.1186/1745-6673-9-10 Shao, M., Li, L., Li, X., Wei, Z., Wang, J., Hong, M., Liu, X., & Meng, J. (2024). The effect of top-down attention on empathy fatigue. Cerebral Cortex (New York, N.Y. 1991), 34(1), Article bhad441. https://doi.org/10.1093/cercor/bhad441 van Dam, A. (2021). A clinical perspective on burnout: diagnosis, classification, and treatment of clinical burnout. European Journal of Work and Organizational Psychology, 30(5), 732-741. https://doi.org/10.1080/1359432X.2021.1948400 Vial, A. C., & Cowgill, C. M. (2022). Heavier Lies Her Crown: Gendered Patterns of Leader Emotional Labor and Their Downstream Effects. Frontiers in Psychology, 13, 849566. https://doi.org/https://doi.org/10.3389/fpsyg.2022.849566 Wharton, A. S. (2009). The Sociology of Emotional Labor. Annual Review of Sociology, 35(1), 147–165. https://doi.org/10.1146/annurev-soc-070308-115944 Wrobel, M. (2013). Can empathy lead to emotional exhaustion in teachers? The mediating role of emotional labor. International Journal of Occupational Medicine and Environmental Health, 26(4), 581–592. https://doi.org/10.2478/s13382-013-0123-1 }} ==External links== Provide [[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: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Burnout]] [[Category:Motivation and emotion/Book/Empathy]] cy74el6d3tvsys0hxwpif45m3943qx1 2844547 2844543 2026-11-05T01:41:29Z U3143751 3106785 /* Empathy fatigue */ 2844547 wikitext text/x-wiki {{title|Empathy fatigue and emotional exhaustion:<br>How does sustained empathic engagement contribute to emotional exhaustion?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}} [[File:Exhausted smartphone user.JPG|right|thumb|150px|'''Figure 1'''. An emotionally exhausted Aelin at the end of the work day]] ; Scenario Aelin is a 23-year-old social worker employed at a non-profit organisation providing mental health services to teenagers and young adults. Aelin's clients are aged between 18 and 25, many of whom having experienced rough living conditions. They work with approximately 20 clients each week. During Aelin's first six months in the role, they feel highly motivated and are excited about helping people. They are an excellent active listener, and can easily remember details about their clients' lives. During their sessions, Aelin recognises when a client is distressed and adjusts their body language to warm and understanding in an effort to provide a safe and welcoming environment to hold space for their emotions. As Aelin gains more experience working in this role, their caseload increases, and they are exposed to clients with increasing complexity and past traumas. Aelin wants to provide the best possible outcome for all of their clients, and begins to take their work home with them in the evenings, but finds it starts creeping into their weekends as well. After several months of working this way, Aelin begins to notice they feel exhausted every morning - even before the workday begins. Fast-forward a few more months, and Aelin notices a change in how they start responding to people, including clients. Instead of feeling engaged and actively listening, they feel detached and find their thoughts feeling more frustrated than empathetic. After finishing the work day, Aelin is drained and has no energy for friends or family. With increasing irritability, Aelin begins to worry that they won't be able to continue working as a social-worker. {{RoundBoxBottom}} Aelin is experiencing '''emotional exhaustion''' from a sustained empathic response. Due to the nature of Aelin's work, they are constantly engaged in an empathic response. Empathic responses require cognitive and physical resources to sustain. Because Aelin is taking their work home and not resting on weekends, they do not have a chance to rest. It is important for Aelin to recognise what is happening and why. Not actioning the problem has led to empathy fatigue and emotional exhaustion. Emotional exhaustion is one of the key aspects of [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FEmpathy_fatigue_and_emotional_exhaustion&diff=2840411&oldid=2832859 burnout], a long-term condition of existing in a constant state of high stress. Psychological science can support Aelin in recognising and mapping the symptoms they are experiencing, understand coping strategies to minimise the current and future effects to improve their wellbeing. Understanding the symptoms will also support their role as a social worker, providing more tools for the people they support on a day-to-day basis. {{RoundBoxTop|theme=3}} '''Focus questions''' * Why do people become emotionally exhausted? * What is the relationship between empathy fatigue and emotional exhaustion? * What role does emotional labor play in emotional exhaustion? * How does understanding emotional labor and empathy fatigue work to reduce the effects of emotional exhaustion? {{RoundBoxBottom}} == Emotional exhaustion == To understand emotional exhaustion, it will be important to first understand burnout. Burnout is a long-term condition of existing in a constant state of high stress and affects millions of people annually. In the early 2000s, a survey from Australia found that more than 50% of social workers and occupational workers reported high levels of burnout symptoms (Llyod and King, 2004) and a recent report from 2026 found that more than one third of all workers in Australia were experiencing burnout (Chan, 2026). Being in a state of high stress for short periods of time is a useful adaptation for humans, so much so that it has been given a name, [[wikipedia:Eustress|Eustress]]. However, as a long-term condition it leads to individual health issues and can be linked to cardiovascular disease, increased vulnerability to upper respiratory diseases, and chronic inflammation (Schneiderman et al., 2005), and cognitive issues such as [[wikipedia:Inattentional_blindness|cognitive tunneling]] and impaired selective attention (Pooladvand and Hasanzadeh, 2023). Burnout can be defined across three dimensions: emotional exhaustion, [[wikipedia:Depersonalization|depersonalisation]], and reduced feelings of personal accomplishment (Maslach, 1993, as cited in Martingano et al., 2026). Emotional exhaustion may be considered the 'core' component of burnout and refers to the depletion of emotional resources, leading to feelings of overwhelm, overextension, and exhaustion (Martingano et al., 2026; De Beer et al., 2024; Seidler et al., 2014). As a major factor, its symptoms are similar to burnout and stress. Although antecedents of burnout are typically agreed upon (Bakker et al., 2014) and divided into individual (e.g. self-efficacy) and situational categories (e.g. work load), there are many theories that attempt to explain why emotional exhaustion occurs. Two of these theories with good evidence are the [[wikipedia:Conservation_of_resources_theory|Conservation of resources stress model]] (Hobfoll, 1989) and the [[wikipedia:Job_demands-resources_model|Job Demands-Resources model]] (Demerouti et al., 2001). === Conservation of resources stress model === The Conservation of resources stress model (CORT; Hobfall, 1989) is a theory that works to understand and explain why stress occurs. Under the model, humans are motivated to maintain, pursue, or attain resources. Stress, defined as "an individual's perceived pressure or strain" (Hollebeek et al., 2023 p.928), occurs when these resources are threatened or lost, or gained at a net loss. As noted above, resources aren't strictly tied to being physical objects. They can also be emotional states, conditions, or even ideas. For example, someone may place value on enjoying a calm, quiet Sunday morning at home. If their neighbour starts using power tools loudly next door, it would disrupt this state and induce stress in the individual. The CORT model focuses primarily on two principles. The first principle is that there is more harm in losing resources than there is in gaining them back (Halbesleben et al., 2014). This is known as loss salience, and some research suggests that the pain of loss is felt twice as strong as the pleasure of gain (Thaler, 2000). Looking at our previous example, imagine that the person who enjoys a quiet Sunday morning at home equally enjoys the sound of a loud car engine. Even though the person enjoys the sound of car engines, if the neighbour started revving their car loudly instead of using power tools, it would still disrupt their desired state of quiet and lead to an increase of stress. The second principle is that people must invest existing resources in order to gain or protect resources (Halbesleben et al., 2014). This resource investment includes recovering from resource loss. Coping is how people manage or overcome a difficult situation, such as protecting or recovering from resource loss. Coping can be broken down into different strategies that people use, but not all strategies are equal (Ito and Brotheridge, 2003). They are divided into two categories: control or escape strategies. Control strategies are considered better at reducing the strain from stress and consist of positive thinking, direct action, and help seeking (Havlovic & Keenan, 1991, as cited in Ito and Brotheridge, 2003). The most useful control strategies incorporate a person's [[locus of control]], emphasise seeking support from co-workers, and highlight the importance of autonomy in decision making. Alarcon (2011) conducted a meta-analysis of 231 studies in a work setting that used the [[wikipedia:Maslach_Burnout_Inventory|Maslach Burnout Inventory]] (MBI) to review the relationship between these principles and burnout to confirm their validity. Although the study had no specific mention of demographics, it was noted that multiple translations of the MBI were used including Dutch (k=51), German (k=7), Italian (k=2), and French (k=2). The findings indicate that workload is positively related to emotional exhaustion (''ρ'' = .49, ''k'' = 86, ''N'' = 51,529) and that control (''ρ'' = − .26, ''k'' = 56, ''N'' = 33,297) and autonomy (''ρ'' = − .24, ''k'' = 34, ''N'' = 17,165) were each negatively associated with emotional exhaustion. The difference of ''ρ'' values is consistent with Thaler (2000) and supports that the benefit of gaining is approximately half that of losing resources. === Job demands-resources model === Another model that attempts to understand why people become emotionally exhausted is the Job Demands-Resources model (JD-R; Demerouti et al., 2001). The JD-R model discusses two concepts: job demands and job resources. Demands and resources are both related to the physical, social, or psychological aspects of a job, but differ in their impact on a person. Job demands require physical or mental effort and exhaust a person's energy supply. Job resources typically support achieving goals, reduce the effort cost of job demands, and stimulate personal growth (Richter and Hacker, 1998, as cited in Demerouti et al., 2001). For example, an unsustainable workload would be considered a job demand, and receiving help and support from others would be considered a resource. The JD-R has two primary processes for how demands and resources interact with one another (Schaufeli, 2017). The stress process occurs when job demands are high and resources are low. As stress increases strain and pressure on an individual, it leads to the exhaustion of their energy eventually, eventually leading to emotional exhaustion. The second process is the motivational process. This process occurs when there is an excess of job resources. Crucially, the motivational process does not require low job demands. With enough job resources, a person will be energised and engaged, reframing job demands as a challenge to overcome. The JD-R model has had extensive research to confirm its validity. A meta-analysis by Lesener et al. (2019) looked at 74 longitudinal studies of the JD-R with a strong average sample size (n=450) against the assumptions as described by Schaufeli (2017). Although most of the studies appear to come from [[wikipedia:Psychology#WEIRD_bias|WEIRD]] countries (k=60) there were a good number from Asia and Africa (k=13) and a mix of genders. The study confirmed that across the literature, job resources predict burnout negatively (''β'' = −.17; ''p'' < .001) and job demands predict burnout over time (''β'' = .10; ''p'' < .001). Although job resources seem like a cure-all for demands, it's important to note that they do not retroactively treat emotional exhaustion. Under the JD-R model, burnout must be reduced by lowering job demands (Gynning et al., 2025). However, once someone has recovered, it is possible to increase job resources alongside job demands to prevent future burnout and support [[wikipedia:Work_engagement|work engagement]]. Work engagement (engagement) has been discussed as the opposite of burnout (Bakker et al., 2014). Engagement is how people express themselves when performing a role. A role being a pattern of behaviour that a person goes through based on the social expectations of that role, similar to how an actor takes on a role for a movie. For example, when attending a typical restaurant, it is socially expected that a person will engage with the waiter and provide them with their meal order to take to the kitchen. This person is performing the script of a restaurant patron. In the context of a workplace, an employee with high engagement may appear to be energised, enthusiastic, and become absorbed in their work losing track of time (Bakkar et al, 2014). {{RoundBoxTop|theme=3}} '''Why do we feel emotionally exhausted''' Under the JD-R and CORT models, emotional exhaustion occurs when the demand cost outweighs the resources available for an extended period of time, eventually depleting someone's energy reserves. Aelin is experiencing burnout because they increased their job demands by taking on more clients, and has experienced a loss in resources by reducing their free time and social support network. {{RoundBoxBottom}} == Empathy fatigue == You may be familiar with the term [[empathy]]. It is often described in terms of supporting others. For example, when someone close to you experiences a loss and is grieving, you may empathise with them and experience a negative emotion akin to grief too. However, empathy is not exclusive to so called negative emotions. For example, compersion is a positive empathic response when a loved one experiences joy. Parents may experience compersion when something joyful happens to their child. At its core, empathy is the individual ability to imagine or understand another person's feelings and/or emotions (Riess, 2017; Shao et al., 2024). Much like typical physical or cognitive processes, such as running or problem solving, empathising will drain energy and resources. Further, and similar to the difference in energy expenditure between running and jogging, there are distinct types of empathy that each have different energy costs. A way to make a distinction between these two types may be considering them as the "what" and "how". Research makes a clear distinction between two types of empathy: affective and cognitive empathy. Affective empathy occurs when a person has an emotional reaction to someone else's mental state (Shao et al., 2024), whereas cognitive empathy occurs when a person understands someone else's feelings or emotions (Cui et al., 2022). === Affective empathy === Affective empathy is the "how" and focuses on the feeling aspect of empathy. This is a more common understanding of empathy, and utilises * Integrating the two - "affective component is the essence of empathy, and the cognitive component is a process through which that essence comes into being" (Bošnjaković & Radionov, 2018) ** (Author's note: change quote into paraphrasing) Emotional contagion is the idea that emotions are contagious. as the theory to moderate surface/deep acting * Emotional contagion (Panksepp and Lahvis, 2011) ** emotions can spread (expand, combines empathy and acting) ** “the tendency to automatically mimic and synchronize expressions, vocalizations, postures, and movements with those of another person’s and, consequently, to converge emotionally.” (Elaine Hatfield, 1993) * Affective process theory looks at unifying the mechanisms of emotional contagion into three types. Convergent linkage, Divergent linkage, and Complementary linkage (Elfenbein, 2014). Storytelling is a way to promote position emotions (Wei et al., 2024). === Cognitive empathy === Cognitive empathy is the "what" and focuses on a person trying to tangibly understand what another person is feeling. This information is processed through a top-down approach (), and primarily utilises the pre-frontal cortex (Shao et al., 2024). This allows the empathiser to feel the emotion from a cognitive, problem solving perspective. As cognitive empathy is processed by the pre-frontal cortex, there is less association with feeling the emotion across the body hence reducing the physiological response and is linked to lower energy expenditure. Paragraph on compassion (use that study in your one-note and look for an additional meta-analysis) === Empathy fatigue and compassion fatigue === * Distinction between empathy and compassion fatigue * Compassion fatigue is closely related to Secondary traumatic stress usually found in healthcare professionals (Figley, 2002) ** Compassion fatigue is to Empathy fatigue as emotional exhaustion is to burnout * Make it clear that Aelin is experiencing Compassion Fatigue vs Empathy fatigue (they are the same but different) (Author's note: include an update to the scenario here!) '''Empathy fatigue questions'''. Choose your answers and click "Submit": <quiz display=simple> {Surface acting is when the observer actually feels the emotion: |type="()"} - True + False {Compassion fatigue and empathy fatigue are synonymous: |type="()"} - True + False </quiz> {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' Under the JD-R and CORT models, a sustained empathic response is a long-term job demand that uses energy to protect existing resources. As Aelin has had their resources is experiencing a * Under the CORT model, a sustained empathic response uses energy to protect existing resources. As resources become low and burnout sets in, cognitive focus narrows leading to empathy fatigue. {{RoundBoxBottom}} == Emotional labor as a mediating factor between empathy fatigue and emotional exhaustion == As a society there are expected [https://en.wikipedia.org/wiki/Social_norm social norms] in behaviour when expressing emotion, regardless if someone is conscious of the process of matching these norms or not. For example, laughing during an emotional speech at a funeral would be considered a rude or offensive expression from the other attendees. As social creatures, humans need a way to regulate socially appropriate emotional expression. This is the essence of Emotional labor: "the regulation of felt and expressed emotions (whether effortful or not) to match these emotional display rules with the objective of fulfilling organizational goals" (Vial & Cowgill, 2022) What is emotional labor (Hochschild, 2012). This section is really where all the above research comes together, and should highlight how the models of stress (JD-R and CORT), relate to the types of empathy (affective/contagion vs cognitive/compassion), and how types of acting (expectations of what you should feel (Emotional labor!!!!) vs accepting what you actually feel) contribute and '''''mediate''''' the relationship between empathy fatigue and emotional exhaustion. This is the penultimate section and should reflect that accordingly: "This is what is happening, and this is why its happening" === Types of acting === * Surface acting ** This is where emotive dissonance comes into play ** Leads to emotional exhaustion (Kenworthy et al., 2014) ** "Should" language - in this situation, this is what I ''should'' feel * Deep acting ** People who 'deep act', as in genuinely, intrinsically adjust the way they feel (match the feelings they have with the feelings that are appropriate to display in a given situation) often report positive relationships with emotional labor ** Actual language - in this situation, this is what I ''actually'' feel Discuss the differences in gender norms and expectations in relationships {| class="wikitable" style="margin: auto;" |- ! !! Deep acting !! Surface acting |- | '''Emotional dissonance''' || || X |- | '''Not known to others''' || Hidden area || Unknown |} * === The mediating effect of emotional labor === Surface acting does not mediate the effect (high surface acting leads to emotional exhaustion) whereas the inverse is true for deep acting (Wrobel, 2013) Emotional Expression and Emotional Processing (Hoyt et al., 2024) {{RoundBoxTop|theme=3}} '''What role does emotional labor play in emotional exhaustion''' * In relation to the CORT and JD-R theories, emotional labor acts as a job demand and expenditure of resources to protect existing resources {{RoundBoxBottom}} == Strategies for combating empathy fatigue and emotional fatigue == * Talk about emotion regulation (Gross, 1998) and mood regulation strategies as a way to deal with coping (Wrobel, 2013) (Author's note: need more research/references for this section) * Talk about breaks ** Rest and downtime from not performing emotional labor regenerates * How do we reduce the amount of emotional labor we're performing (Jeung et al., 2018) ** Communication/boundaries ** Reducing surface acting (allowing ourselves to feel how we actually feel and reduce emotive dissonance) '''Table 1. Treatment phases of clinical burnout (based on Table 2 from van Dam, A. 2021)''' {| class="wikitable" style="margin: auto; |- ! Phase !! Treatment goal !! Therapeutic interventions |- | '''Phase 1: Crisis''' || Recognition of the patient that the problems are serious, and that serious action is required || Psychoeducation Sick leave Dropping domestic tasks and social obligations Inform social network |- | '''Phase 2: Recovery''' || Recovery of the (physiological) stress system to normal allostatic stress levels || Registration of stress and activities Relaxtion exercises Mindfulness healthy lifestyle advices Graded exercise Gradually resume activities Psychoeducation social network and employer Gradual return to work |- | '''Phase 3: Prevention, learning from the past''' || Acquiring insight and skills to prevent relapse in clinical burnout || Analysing stressful situations Analysing dysfunctional thought patterns Learning new coping skills Learning social skills Job crafting Time management Making choices regarding career and personal life |- | '''Phase 4: Post burnout growth''' || Improving sustainable quality of life. || Setting priorities regarding quality of life and interpersonal relationships. |} === Strategies for the workplace === * Discuss types of strategies that are available in the workplace (organisation models) * SCARF model (Aplin-Houtz, 2025) ** Link back to COR and JDR models e.g. praise could be a job resource vs criticism could be a job demand *Reduce workload to reduce emotional exhaustion, and by extension reduce emotional fatigue. {{RoundBoxTop|theme=3}} '''How does understanding emotional labor and empathy fatigue work to reduce the effects of emotional exhaustion''' *Psychoeducation has been shown to reduce the effects of emotional labor and empathy fatigue on emotional exhaustion {{RoundBoxBottom}} ==Conclusion== * Summarise what empathy fatigue is, and why people become emotionally exhausted (+ the role of emotional labor) * Summarise the symptoms and how to recognise them in yourself and someone else * Emphasise and summarise psychological strategies to support someone going through emotional exhaustion * Summarise ways to prevent emotional exhaustion from happening * Overall message is: *# What is emotional exhaustion *# how do you recognise it *# how do you treat it *# how do you prevent it * Link back to the focus questions ==See also== * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Metacognition_and_emotional_regulation&oldid=2756264 Metacognition and emotional regulation] (Book chapter, 2026) {{ic|Use an internal link style as shown in Tutorial 2}} * [[wikipedia:Conservation_of_resources_theory|Conservation of resources theory]] (Wikipedia) * [[wikipedia:Job_demands-resources_model|Job demands-resources model]] (Wikipedia) ==References== {{Hanging indent|1= Aplin-Houtz, M. (2025). Unraveling the SCARF: How dimensions of the SCARF model influence the relationship between workplace exclusion and turnover intentions. Journal of Management & Organization, 31(6), 2699–2725. https://doi.org/10.1017/jmo.2025.10026 Bošnjaković, J., & Radionov, T. (2018). Empathy: Concepts, Theories and Neuroscientific Basis. Alcoholism (Zagreb), 54(2), 123–150. https://doi.org/10.20471/de..2018.54.02.04 Chan, F. M. (2026). The State of Workplace Burnout 2026: The Workplace Divide. Infinite Potential, Sydney, Australia. https://www.ahri.com.au/wp-content/uploads/The-State-of-Workplace-Burnout-2026-Final.pdf Cui, F., Huang, X., Jing, Y., Luo, Y., Liu, J., & Gu, R. (2022). How resource sharing resists scarcity: the role of cognitive empathy and its neurobiological mechanisms. Cerebral Cortex (New York, N.Y. 1991), 32(23), 5330–5342. https://doi.org/10.1093/cercor/bhac017 De Beer, L. T., van der Vaart, L., Escaffi-Schwarz, M., De Witte, H., & Schaufeli, W. B. (2024). Maslach Burnout Inventory - General Survey: A Systematic Review and Meta-Analysis of Measurement Properties. European journal of psychological assessment : official organ of the European Association of Psychological Assessment, 40(5), 360-375. https://doi.org/10.1027/1015-5759/a000797 Demerouti, E., Bakker, A. B., Nachreiner, F., & Schaufeli, W. B. (2001). The Job Demands-Resources Model of Burnout. Journal of Applied Psychology, 86(3), 499–512. https://doi.org/10.1037/0021-9010.86.3.499 Figley, C. R. (2002). Compassion fatigue: Psychotherapists’ chronic lack of self care. Journal of Clinical Psychology, 58(11), 1433–1441. https://doi.org/10.1002/jclp.10090 Grandey, A. A. (2003). When ‘the show must go on’: surface acting and deep acting as determinants of emotional exhaustion and peer-rated service delivery. Academy of Management Journal, 46(1), 86–96. https://doi.org/10.5465/30040678 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 Halbesleben, J. R. B., Neveu, J.-P., Paustian-Underdahl, S. C., & Westman, M. (2014). Getting to the “COR”: Understanding the Role of Resources in Conservation of Resources Theory. Journal of Management, 40(5), 1334–1364. https://doi.org/10.1177/0149206314527130 Hatfield, E., Cacioppo, J. T., & Rapson, R. L. (1993). Emotional Contagion. Current Directions in Psychological Science : A Journal of the American Psychological Society, 2(3), 96–99. https://doi.org/10.1111/1467-8721.ep10770953 Hollebeek, L. D., Hammedi, W., & Sprott, D. E. (2023). Consumer engagement, stress, and conservation of resources theory: A review, conceptual development, and future research agenda. Psychology & marketing, 40(5), 926-937. https://doi.org/10.1002/mar.21807 Hobfoll, S. E. (1989). Conservation of Resources: A New Attempt at Conceptualizing Stress. The American Psychologist, 44(3), 513–524. https://doi.org/10.1037/0003-066X.44.3.513 Hochschild, A. R. (2012). The managed heart : commercialization of human feeling (Updated, with a new preface.). University of California Press. Jeung, D.-Y., Kim, C., & Chang, S.-J. (2018). Emotional Labor and Burnout: A Review of the Literature. Yonsei Medical Journal, 59(2), 187–193. https://doi.org/10.3349/ymj.2018.59.2.187 Lavelle, J. J., Rupp, D. E., Herda, D. N., Pandey, A., & Lauck, J. R. (2021). Customer Injustice and Employee Performance: Roles of Emotional Exhaustion, Surface Acting, and Emotional Demands–Abilities Fit. Journal of Management, 47(3), Article 0149206319869426. https://doi.org/10.1177/0149206319869426 Lloyd, C., & King, R. (2004). A survey of burnout among Australian mental health occupational therapists and social workers. Social Psychiatry and Psychiatric Epidemiology, 39(9), 752-757. https://doi.org/https://doi.org/10.1007/s00127-004-0808-7 Kenworthy, J., Fay, C., Frame, M., & Petree, R. (2014). A meta-analytic review of the relationship between emotional dissonance and emotional exhaustion. Journal of Applied Social Psychology, 44(2), 94-105. https://doi.org/https://doi.org/10.1111/jasp.12211 Martingano, A. J., Stosic, M. D., & Konrath, S. (2026). Different empathy types show opposing associations with burnout: systematic review and meta-analyses. Psychology, Health & Medicine, 31(3), 798-829. https://doi.org/10.1080/13548506.2025.2591859 Michielsen, H. J., Willemsen, T. M., Croon, M. A., de Vries, J., & Van Heck, G. L. (2004). Determinants of general fatigue and emotional exhaustion: a prospective study. Psychology & health, 19(2), 223-235. https://doi.org/10.1080/08870440310001627135 Panksepp, J. B., & Lahvis, G. P. (2011). Rodent empathy and affective neuroscience. Neuroscience and Biobehavioral Reviews, 35(9), 1864–1875. https://doi.org/10.1016/j.neubiorev.2011.05.013 Pooladvand, S., & Hasanzadeh, S. (2023). Impacts of Stress on Workers’ Risk-Taking Behaviors: Cognitive Tunneling and Impaired Selective Attention. Journal of Construction Engineering and Management, 149(8), 04023060. https://doi.org/doi:10.1061/JCEMD4.COENG-13339 Riess, H. (2017). The Science of Empathy. Journal of Patient Experience, 4(2), 74–77. https://doi.org/10.1177/2374373517699267 Schaufeli, W. B. (2017). Applying the job demands-resources model: a “how to” guide to measuring and tackling work engagement and burnout. Organizational Dynamics, 46(2), 120–132. https://doi.org/10.1016/j.orgdyn.2017.04.008 Schneiderman, N., Ironson, G., & Siegel, S. D. (2005). Stress and Health: Psychological, Behavioral, and Biological Determinants. Annual Review of Clinical Psychology, 1(Volume 1, 2005), 607-628. https://doi.org/https://doi.org/10.1146/annurev.clinpsy.1.102803.144141 Seidler, A., Thinschmidt, M., Deckert, S., Then, F., Hegewald, J., Nieuwenhuijsen, K., & Riedel-Heller, S. G. (2014). The role of psychosocial working conditions on burnout and its core component emotional exhaustion – a systematic review. Journal of Occupational Medicine and Toxicology, 9(1), 10. https://doi.org/10.1186/1745-6673-9-10 Shao, M., Li, L., Li, X., Wei, Z., Wang, J., Hong, M., Liu, X., & Meng, J. (2024). The effect of top-down attention on empathy fatigue. Cerebral Cortex (New York, N.Y. 1991), 34(1), Article bhad441. https://doi.org/10.1093/cercor/bhad441 van Dam, A. (2021). A clinical perspective on burnout: diagnosis, classification, and treatment of clinical burnout. European Journal of Work and Organizational Psychology, 30(5), 732-741. https://doi.org/10.1080/1359432X.2021.1948400 Vial, A. C., & Cowgill, C. M. (2022). Heavier Lies Her Crown: Gendered Patterns of Leader Emotional Labor and Their Downstream Effects. Frontiers in Psychology, 13, 849566. https://doi.org/https://doi.org/10.3389/fpsyg.2022.849566 Wharton, A. S. (2009). The Sociology of Emotional Labor. Annual Review of Sociology, 35(1), 147–165. https://doi.org/10.1146/annurev-soc-070308-115944 Wrobel, M. (2013). Can empathy lead to emotional exhaustion in teachers? The mediating role of emotional labor. International Journal of Occupational Medicine and Environmental Health, 26(4), 581–592. https://doi.org/10.2478/s13382-013-0123-1 }} ==External links== Provide [[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: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Burnout]] [[Category:Motivation and emotion/Book/Empathy]] 7wesrf4pmnnd90xsvadoak4p3921gr2 2844548 2844547 2026-11-05T01:42:08Z U3143751 3106785 /* Emotional labor as a mediating factor between empathy fatigue and emotional exhaustion */ 2844548 wikitext text/x-wiki {{title|Empathy fatigue and emotional exhaustion:<br>How does sustained empathic engagement contribute to emotional exhaustion?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}} [[File:Exhausted smartphone user.JPG|right|thumb|150px|'''Figure 1'''. An emotionally exhausted Aelin at the end of the work day]] ; Scenario Aelin is a 23-year-old social worker employed at a non-profit organisation providing mental health services to teenagers and young adults. Aelin's clients are aged between 18 and 25, many of whom having experienced rough living conditions. They work with approximately 20 clients each week. During Aelin's first six months in the role, they feel highly motivated and are excited about helping people. They are an excellent active listener, and can easily remember details about their clients' lives. During their sessions, Aelin recognises when a client is distressed and adjusts their body language to warm and understanding in an effort to provide a safe and welcoming environment to hold space for their emotions. As Aelin gains more experience working in this role, their caseload increases, and they are exposed to clients with increasing complexity and past traumas. Aelin wants to provide the best possible outcome for all of their clients, and begins to take their work home with them in the evenings, but finds it starts creeping into their weekends as well. After several months of working this way, Aelin begins to notice they feel exhausted every morning - even before the workday begins. Fast-forward a few more months, and Aelin notices a change in how they start responding to people, including clients. Instead of feeling engaged and actively listening, they feel detached and find their thoughts feeling more frustrated than empathetic. After finishing the work day, Aelin is drained and has no energy for friends or family. With increasing irritability, Aelin begins to worry that they won't be able to continue working as a social-worker. {{RoundBoxBottom}} Aelin is experiencing '''emotional exhaustion''' from a sustained empathic response. Due to the nature of Aelin's work, they are constantly engaged in an empathic response. Empathic responses require cognitive and physical resources to sustain. Because Aelin is taking their work home and not resting on weekends, they do not have a chance to rest. It is important for Aelin to recognise what is happening and why. Not actioning the problem has led to empathy fatigue and emotional exhaustion. Emotional exhaustion is one of the key aspects of [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FEmpathy_fatigue_and_emotional_exhaustion&diff=2840411&oldid=2832859 burnout], a long-term condition of existing in a constant state of high stress. Psychological science can support Aelin in recognising and mapping the symptoms they are experiencing, understand coping strategies to minimise the current and future effects to improve their wellbeing. Understanding the symptoms will also support their role as a social worker, providing more tools for the people they support on a day-to-day basis. {{RoundBoxTop|theme=3}} '''Focus questions''' * Why do people become emotionally exhausted? * What is the relationship between empathy fatigue and emotional exhaustion? * What role does emotional labor play in emotional exhaustion? * How does understanding emotional labor and empathy fatigue work to reduce the effects of emotional exhaustion? {{RoundBoxBottom}} == Emotional exhaustion == To understand emotional exhaustion, it will be important to first understand burnout. Burnout is a long-term condition of existing in a constant state of high stress and affects millions of people annually. In the early 2000s, a survey from Australia found that more than 50% of social workers and occupational workers reported high levels of burnout symptoms (Llyod and King, 2004) and a recent report from 2026 found that more than one third of all workers in Australia were experiencing burnout (Chan, 2026). Being in a state of high stress for short periods of time is a useful adaptation for humans, so much so that it has been given a name, [[wikipedia:Eustress|Eustress]]. However, as a long-term condition it leads to individual health issues and can be linked to cardiovascular disease, increased vulnerability to upper respiratory diseases, and chronic inflammation (Schneiderman et al., 2005), and cognitive issues such as [[wikipedia:Inattentional_blindness|cognitive tunneling]] and impaired selective attention (Pooladvand and Hasanzadeh, 2023). Burnout can be defined across three dimensions: emotional exhaustion, [[wikipedia:Depersonalization|depersonalisation]], and reduced feelings of personal accomplishment (Maslach, 1993, as cited in Martingano et al., 2026). Emotional exhaustion may be considered the 'core' component of burnout and refers to the depletion of emotional resources, leading to feelings of overwhelm, overextension, and exhaustion (Martingano et al., 2026; De Beer et al., 2024; Seidler et al., 2014). As a major factor, its symptoms are similar to burnout and stress. Although antecedents of burnout are typically agreed upon (Bakker et al., 2014) and divided into individual (e.g. self-efficacy) and situational categories (e.g. work load), there are many theories that attempt to explain why emotional exhaustion occurs. Two of these theories with good evidence are the [[wikipedia:Conservation_of_resources_theory|Conservation of resources stress model]] (Hobfoll, 1989) and the [[wikipedia:Job_demands-resources_model|Job Demands-Resources model]] (Demerouti et al., 2001). === Conservation of resources stress model === The Conservation of resources stress model (CORT; Hobfall, 1989) is a theory that works to understand and explain why stress occurs. Under the model, humans are motivated to maintain, pursue, or attain resources. Stress, defined as "an individual's perceived pressure or strain" (Hollebeek et al., 2023 p.928), occurs when these resources are threatened or lost, or gained at a net loss. As noted above, resources aren't strictly tied to being physical objects. They can also be emotional states, conditions, or even ideas. For example, someone may place value on enjoying a calm, quiet Sunday morning at home. If their neighbour starts using power tools loudly next door, it would disrupt this state and induce stress in the individual. The CORT model focuses primarily on two principles. The first principle is that there is more harm in losing resources than there is in gaining them back (Halbesleben et al., 2014). This is known as loss salience, and some research suggests that the pain of loss is felt twice as strong as the pleasure of gain (Thaler, 2000). Looking at our previous example, imagine that the person who enjoys a quiet Sunday morning at home equally enjoys the sound of a loud car engine. Even though the person enjoys the sound of car engines, if the neighbour started revving their car loudly instead of using power tools, it would still disrupt their desired state of quiet and lead to an increase of stress. The second principle is that people must invest existing resources in order to gain or protect resources (Halbesleben et al., 2014). This resource investment includes recovering from resource loss. Coping is how people manage or overcome a difficult situation, such as protecting or recovering from resource loss. Coping can be broken down into different strategies that people use, but not all strategies are equal (Ito and Brotheridge, 2003). They are divided into two categories: control or escape strategies. Control strategies are considered better at reducing the strain from stress and consist of positive thinking, direct action, and help seeking (Havlovic & Keenan, 1991, as cited in Ito and Brotheridge, 2003). The most useful control strategies incorporate a person's [[locus of control]], emphasise seeking support from co-workers, and highlight the importance of autonomy in decision making. Alarcon (2011) conducted a meta-analysis of 231 studies in a work setting that used the [[wikipedia:Maslach_Burnout_Inventory|Maslach Burnout Inventory]] (MBI) to review the relationship between these principles and burnout to confirm their validity. Although the study had no specific mention of demographics, it was noted that multiple translations of the MBI were used including Dutch (k=51), German (k=7), Italian (k=2), and French (k=2). The findings indicate that workload is positively related to emotional exhaustion (''ρ'' = .49, ''k'' = 86, ''N'' = 51,529) and that control (''ρ'' = − .26, ''k'' = 56, ''N'' = 33,297) and autonomy (''ρ'' = − .24, ''k'' = 34, ''N'' = 17,165) were each negatively associated with emotional exhaustion. The difference of ''ρ'' values is consistent with Thaler (2000) and supports that the benefit of gaining is approximately half that of losing resources. === Job demands-resources model === Another model that attempts to understand why people become emotionally exhausted is the Job Demands-Resources model (JD-R; Demerouti et al., 2001). The JD-R model discusses two concepts: job demands and job resources. Demands and resources are both related to the physical, social, or psychological aspects of a job, but differ in their impact on a person. Job demands require physical or mental effort and exhaust a person's energy supply. Job resources typically support achieving goals, reduce the effort cost of job demands, and stimulate personal growth (Richter and Hacker, 1998, as cited in Demerouti et al., 2001). For example, an unsustainable workload would be considered a job demand, and receiving help and support from others would be considered a resource. The JD-R has two primary processes for how demands and resources interact with one another (Schaufeli, 2017). The stress process occurs when job demands are high and resources are low. As stress increases strain and pressure on an individual, it leads to the exhaustion of their energy eventually, eventually leading to emotional exhaustion. The second process is the motivational process. This process occurs when there is an excess of job resources. Crucially, the motivational process does not require low job demands. With enough job resources, a person will be energised and engaged, reframing job demands as a challenge to overcome. The JD-R model has had extensive research to confirm its validity. A meta-analysis by Lesener et al. (2019) looked at 74 longitudinal studies of the JD-R with a strong average sample size (n=450) against the assumptions as described by Schaufeli (2017). Although most of the studies appear to come from [[wikipedia:Psychology#WEIRD_bias|WEIRD]] countries (k=60) there were a good number from Asia and Africa (k=13) and a mix of genders. The study confirmed that across the literature, job resources predict burnout negatively (''β'' = −.17; ''p'' < .001) and job demands predict burnout over time (''β'' = .10; ''p'' < .001). Although job resources seem like a cure-all for demands, it's important to note that they do not retroactively treat emotional exhaustion. Under the JD-R model, burnout must be reduced by lowering job demands (Gynning et al., 2025). However, once someone has recovered, it is possible to increase job resources alongside job demands to prevent future burnout and support [[wikipedia:Work_engagement|work engagement]]. Work engagement (engagement) has been discussed as the opposite of burnout (Bakker et al., 2014). Engagement is how people express themselves when performing a role. A role being a pattern of behaviour that a person goes through based on the social expectations of that role, similar to how an actor takes on a role for a movie. For example, when attending a typical restaurant, it is socially expected that a person will engage with the waiter and provide them with their meal order to take to the kitchen. This person is performing the script of a restaurant patron. In the context of a workplace, an employee with high engagement may appear to be energised, enthusiastic, and become absorbed in their work losing track of time (Bakkar et al, 2014). {{RoundBoxTop|theme=3}} '''Why do we feel emotionally exhausted''' Under the JD-R and CORT models, emotional exhaustion occurs when the demand cost outweighs the resources available for an extended period of time, eventually depleting someone's energy reserves. Aelin is experiencing burnout because they increased their job demands by taking on more clients, and has experienced a loss in resources by reducing their free time and social support network. {{RoundBoxBottom}} == Empathy fatigue == You may be familiar with the term [[empathy]]. It is often described in terms of supporting others. For example, when someone close to you experiences a loss and is grieving, you may empathise with them and experience a negative emotion akin to grief too. However, empathy is not exclusive to so called negative emotions. For example, compersion is a positive empathic response when a loved one experiences joy. Parents may experience compersion when something joyful happens to their child. At its core, empathy is the individual ability to imagine or understand another person's feelings and/or emotions (Riess, 2017; Shao et al., 2024). Much like typical physical or cognitive processes, such as running or problem solving, empathising will drain energy and resources. Further, and similar to the difference in energy expenditure between running and jogging, there are distinct types of empathy that each have different energy costs. A way to make a distinction between these two types may be considering them as the "what" and "how". Research makes a clear distinction between two types of empathy: affective and cognitive empathy. Affective empathy occurs when a person has an emotional reaction to someone else's mental state (Shao et al., 2024), whereas cognitive empathy occurs when a person understands someone else's feelings or emotions (Cui et al., 2022). === Affective empathy === Affective empathy is the "how" and focuses on the feeling aspect of empathy. This is a more common understanding of empathy, and utilises * Integrating the two - "affective component is the essence of empathy, and the cognitive component is a process through which that essence comes into being" (Bošnjaković & Radionov, 2018) ** (Author's note: change quote into paraphrasing) Emotional contagion is the idea that emotions are contagious. as the theory to moderate surface/deep acting * Emotional contagion (Panksepp and Lahvis, 2011) ** emotions can spread (expand, combines empathy and acting) ** “the tendency to automatically mimic and synchronize expressions, vocalizations, postures, and movements with those of another person’s and, consequently, to converge emotionally.” (Elaine Hatfield, 1993) * Affective process theory looks at unifying the mechanisms of emotional contagion into three types. Convergent linkage, Divergent linkage, and Complementary linkage (Elfenbein, 2014). Storytelling is a way to promote position emotions (Wei et al., 2024). === Cognitive empathy === Cognitive empathy is the "what" and focuses on a person trying to tangibly understand what another person is feeling. This information is processed through a top-down approach (), and primarily utilises the pre-frontal cortex (Shao et al., 2024). This allows the empathiser to feel the emotion from a cognitive, problem solving perspective. As cognitive empathy is processed by the pre-frontal cortex, there is less association with feeling the emotion across the body hence reducing the physiological response and is linked to lower energy expenditure. Paragraph on compassion (use that study in your one-note and look for an additional meta-analysis) === Empathy fatigue and compassion fatigue === * Distinction between empathy and compassion fatigue * Compassion fatigue is closely related to Secondary traumatic stress usually found in healthcare professionals (Figley, 2002) ** Compassion fatigue is to Empathy fatigue as emotional exhaustion is to burnout * Make it clear that Aelin is experiencing Compassion Fatigue vs Empathy fatigue (they are the same but different) (Author's note: include an update to the scenario here!) '''Empathy fatigue questions'''. Choose your answers and click "Submit": <quiz display=simple> {Surface acting is when the observer actually feels the emotion: |type="()"} - True + False {Compassion fatigue and empathy fatigue are synonymous: |type="()"} - True + False </quiz> {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' Under the JD-R and CORT models, a sustained empathic response is a long-term job demand that uses energy to protect existing resources. As Aelin has had their resources is experiencing a * Under the CORT model, a sustained empathic response uses energy to protect existing resources. As resources become low and burnout sets in, cognitive focus narrows leading to empathy fatigue. {{RoundBoxBottom}} == Emotional labor as a mediating factor between empathy fatigue and emotional exhaustion == As a society there are expected [https://en.wikipedia.org/wiki/Social_norm social norms] in behaviour when expressing emotion, regardless if someone is conscious of the process of matching these norms or not. For example, laughing during an emotional speech at a funeral would be considered a rude or offensive expression from the other attendees. As social creatures, humans need a way to regulate socially appropriate emotional expression. This is the essence of Emotional labor: "the regulation of felt and expressed emotions (whether effortful or not) to match these emotional display rules with the objective of fulfilling organizational goals" (Vial & Cowgill, 2022) What is emotional labor (Hochschild, 2012). This section is really where all the above research comes together, and should highlight how the models of stress (JD-R and CORT), relate to the types of empathy (affective/contagion vs cognitive/compassion), and how types of acting (expectations of what you should feel (Emotional labor!!!!) vs accepting what you actually feel) contribute and '''''mediate''''' the relationship between empathy fatigue and emotional exhaustion. This is the penultimate section and should reflect that accordingly: "This is what is happening, and this is how and why its happening" === Types of acting === * Surface acting ** This is where emotive dissonance comes into play ** Leads to emotional exhaustion (Kenworthy et al., 2014) ** "Should" language - in this situation, this is what I ''should'' feel * Deep acting ** People who 'deep act', as in genuinely, intrinsically adjust the way they feel (match the feelings they have with the feelings that are appropriate to display in a given situation) often report positive relationships with emotional labor ** Actual language - in this situation, this is what I ''actually'' feel Discuss the differences in gender norms and expectations in relationships {| class="wikitable" style="margin: auto;" |- ! !! Deep acting !! Surface acting |- | '''Emotional dissonance''' || || X |- | '''Not known to others''' || Hidden area || Unknown |} * === The mediating effect of emotional labor === Surface acting does not mediate the effect (high surface acting leads to emotional exhaustion) whereas the inverse is true for deep acting (Wrobel, 2013) Emotional Expression and Emotional Processing (Hoyt et al., 2024) {{RoundBoxTop|theme=3}} '''What role does emotional labor play in emotional exhaustion''' * In relation to the CORT and JD-R theories, emotional labor acts as a job demand and expenditure of resources to protect existing resources {{RoundBoxBottom}} == Strategies for combating empathy fatigue and emotional fatigue == * Talk about emotion regulation (Gross, 1998) and mood regulation strategies as a way to deal with coping (Wrobel, 2013) (Author's note: need more research/references for this section) * Talk about breaks ** Rest and downtime from not performing emotional labor regenerates * How do we reduce the amount of emotional labor we're performing (Jeung et al., 2018) ** Communication/boundaries ** Reducing surface acting (allowing ourselves to feel how we actually feel and reduce emotive dissonance) '''Table 1. Treatment phases of clinical burnout (based on Table 2 from van Dam, A. 2021)''' {| class="wikitable" style="margin: auto; |- ! Phase !! Treatment goal !! Therapeutic interventions |- | '''Phase 1: Crisis''' || Recognition of the patient that the problems are serious, and that serious action is required || Psychoeducation Sick leave Dropping domestic tasks and social obligations Inform social network |- | '''Phase 2: Recovery''' || Recovery of the (physiological) stress system to normal allostatic stress levels || Registration of stress and activities Relaxtion exercises Mindfulness healthy lifestyle advices Graded exercise Gradually resume activities Psychoeducation social network and employer Gradual return to work |- | '''Phase 3: Prevention, learning from the past''' || Acquiring insight and skills to prevent relapse in clinical burnout || Analysing stressful situations Analysing dysfunctional thought patterns Learning new coping skills Learning social skills Job crafting Time management Making choices regarding career and personal life |- | '''Phase 4: Post burnout growth''' || Improving sustainable quality of life. || Setting priorities regarding quality of life and interpersonal relationships. |} === Strategies for the workplace === * Discuss types of strategies that are available in the workplace (organisation models) * SCARF model (Aplin-Houtz, 2025) ** Link back to COR and JDR models e.g. praise could be a job resource vs criticism could be a job demand *Reduce workload to reduce emotional exhaustion, and by extension reduce emotional fatigue. {{RoundBoxTop|theme=3}} '''How does understanding emotional labor and empathy fatigue work to reduce the effects of emotional exhaustion''' *Psychoeducation has been shown to reduce the effects of emotional labor and empathy fatigue on emotional exhaustion {{RoundBoxBottom}} ==Conclusion== * Summarise what empathy fatigue is, and why people become emotionally exhausted (+ the role of emotional labor) * Summarise the symptoms and how to recognise them in yourself and someone else * Emphasise and summarise psychological strategies to support someone going through emotional exhaustion * Summarise ways to prevent emotional exhaustion from happening * Overall message is: *# What is emotional exhaustion *# how do you recognise it *# how do you treat it *# how do you prevent it * Link back to the focus questions ==See also== * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Metacognition_and_emotional_regulation&oldid=2756264 Metacognition and emotional regulation] (Book chapter, 2026) {{ic|Use an internal link style as shown in Tutorial 2}} * [[wikipedia:Conservation_of_resources_theory|Conservation of resources theory]] (Wikipedia) * [[wikipedia:Job_demands-resources_model|Job demands-resources model]] (Wikipedia) ==References== {{Hanging indent|1= Aplin-Houtz, M. (2025). Unraveling the SCARF: How dimensions of the SCARF model influence the relationship between workplace exclusion and turnover intentions. Journal of Management & Organization, 31(6), 2699–2725. https://doi.org/10.1017/jmo.2025.10026 Bošnjaković, J., & Radionov, T. (2018). Empathy: Concepts, Theories and Neuroscientific Basis. Alcoholism (Zagreb), 54(2), 123–150. https://doi.org/10.20471/de..2018.54.02.04 Chan, F. M. (2026). The State of Workplace Burnout 2026: The Workplace Divide. Infinite Potential, Sydney, Australia. https://www.ahri.com.au/wp-content/uploads/The-State-of-Workplace-Burnout-2026-Final.pdf Cui, F., Huang, X., Jing, Y., Luo, Y., Liu, J., & Gu, R. (2022). How resource sharing resists scarcity: the role of cognitive empathy and its neurobiological mechanisms. Cerebral Cortex (New York, N.Y. 1991), 32(23), 5330–5342. https://doi.org/10.1093/cercor/bhac017 De Beer, L. T., van der Vaart, L., Escaffi-Schwarz, M., De Witte, H., & Schaufeli, W. B. (2024). Maslach Burnout Inventory - General Survey: A Systematic Review and Meta-Analysis of Measurement Properties. European journal of psychological assessment : official organ of the European Association of Psychological Assessment, 40(5), 360-375. https://doi.org/10.1027/1015-5759/a000797 Demerouti, E., Bakker, A. B., Nachreiner, F., & Schaufeli, W. B. (2001). The Job Demands-Resources Model of Burnout. Journal of Applied Psychology, 86(3), 499–512. https://doi.org/10.1037/0021-9010.86.3.499 Figley, C. R. (2002). Compassion fatigue: Psychotherapists’ chronic lack of self care. Journal of Clinical Psychology, 58(11), 1433–1441. https://doi.org/10.1002/jclp.10090 Grandey, A. A. (2003). When ‘the show must go on’: surface acting and deep acting as determinants of emotional exhaustion and peer-rated service delivery. Academy of Management Journal, 46(1), 86–96. https://doi.org/10.5465/30040678 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 Halbesleben, J. R. B., Neveu, J.-P., Paustian-Underdahl, S. C., & Westman, M. (2014). Getting to the “COR”: Understanding the Role of Resources in Conservation of Resources Theory. Journal of Management, 40(5), 1334–1364. https://doi.org/10.1177/0149206314527130 Hatfield, E., Cacioppo, J. T., & Rapson, R. L. (1993). Emotional Contagion. Current Directions in Psychological Science : A Journal of the American Psychological Society, 2(3), 96–99. https://doi.org/10.1111/1467-8721.ep10770953 Hollebeek, L. D., Hammedi, W., & Sprott, D. E. (2023). Consumer engagement, stress, and conservation of resources theory: A review, conceptual development, and future research agenda. Psychology & marketing, 40(5), 926-937. https://doi.org/10.1002/mar.21807 Hobfoll, S. E. (1989). Conservation of Resources: A New Attempt at Conceptualizing Stress. The American Psychologist, 44(3), 513–524. https://doi.org/10.1037/0003-066X.44.3.513 Hochschild, A. R. (2012). The managed heart : commercialization of human feeling (Updated, with a new preface.). University of California Press. Jeung, D.-Y., Kim, C., & Chang, S.-J. (2018). Emotional Labor and Burnout: A Review of the Literature. Yonsei Medical Journal, 59(2), 187–193. https://doi.org/10.3349/ymj.2018.59.2.187 Lavelle, J. J., Rupp, D. E., Herda, D. N., Pandey, A., & Lauck, J. R. (2021). Customer Injustice and Employee Performance: Roles of Emotional Exhaustion, Surface Acting, and Emotional Demands–Abilities Fit. Journal of Management, 47(3), Article 0149206319869426. https://doi.org/10.1177/0149206319869426 Lloyd, C., & King, R. (2004). A survey of burnout among Australian mental health occupational therapists and social workers. Social Psychiatry and Psychiatric Epidemiology, 39(9), 752-757. https://doi.org/https://doi.org/10.1007/s00127-004-0808-7 Kenworthy, J., Fay, C., Frame, M., & Petree, R. (2014). A meta-analytic review of the relationship between emotional dissonance and emotional exhaustion. Journal of Applied Social Psychology, 44(2), 94-105. https://doi.org/https://doi.org/10.1111/jasp.12211 Martingano, A. J., Stosic, M. D., & Konrath, S. (2026). Different empathy types show opposing associations with burnout: systematic review and meta-analyses. Psychology, Health & Medicine, 31(3), 798-829. https://doi.org/10.1080/13548506.2025.2591859 Michielsen, H. J., Willemsen, T. M., Croon, M. A., de Vries, J., & Van Heck, G. L. (2004). Determinants of general fatigue and emotional exhaustion: a prospective study. Psychology & health, 19(2), 223-235. https://doi.org/10.1080/08870440310001627135 Panksepp, J. B., & Lahvis, G. P. (2011). Rodent empathy and affective neuroscience. Neuroscience and Biobehavioral Reviews, 35(9), 1864–1875. https://doi.org/10.1016/j.neubiorev.2011.05.013 Pooladvand, S., & Hasanzadeh, S. (2023). Impacts of Stress on Workers’ Risk-Taking Behaviors: Cognitive Tunneling and Impaired Selective Attention. Journal of Construction Engineering and Management, 149(8), 04023060. https://doi.org/doi:10.1061/JCEMD4.COENG-13339 Riess, H. (2017). The Science of Empathy. Journal of Patient Experience, 4(2), 74–77. https://doi.org/10.1177/2374373517699267 Schaufeli, W. B. (2017). Applying the job demands-resources model: a “how to” guide to measuring and tackling work engagement and burnout. Organizational Dynamics, 46(2), 120–132. https://doi.org/10.1016/j.orgdyn.2017.04.008 Schneiderman, N., Ironson, G., & Siegel, S. D. (2005). Stress and Health: Psychological, Behavioral, and Biological Determinants. Annual Review of Clinical Psychology, 1(Volume 1, 2005), 607-628. https://doi.org/https://doi.org/10.1146/annurev.clinpsy.1.102803.144141 Seidler, A., Thinschmidt, M., Deckert, S., Then, F., Hegewald, J., Nieuwenhuijsen, K., & Riedel-Heller, S. G. (2014). The role of psychosocial working conditions on burnout and its core component emotional exhaustion – a systematic review. Journal of Occupational Medicine and Toxicology, 9(1), 10. https://doi.org/10.1186/1745-6673-9-10 Shao, M., Li, L., Li, X., Wei, Z., Wang, J., Hong, M., Liu, X., & Meng, J. (2024). The effect of top-down attention on empathy fatigue. Cerebral Cortex (New York, N.Y. 1991), 34(1), Article bhad441. https://doi.org/10.1093/cercor/bhad441 van Dam, A. (2021). A clinical perspective on burnout: diagnosis, classification, and treatment of clinical burnout. European Journal of Work and Organizational Psychology, 30(5), 732-741. https://doi.org/10.1080/1359432X.2021.1948400 Vial, A. C., & Cowgill, C. M. (2022). Heavier Lies Her Crown: Gendered Patterns of Leader Emotional Labor and Their Downstream Effects. Frontiers in Psychology, 13, 849566. https://doi.org/https://doi.org/10.3389/fpsyg.2022.849566 Wharton, A. S. (2009). The Sociology of Emotional Labor. Annual Review of Sociology, 35(1), 147–165. https://doi.org/10.1146/annurev-soc-070308-115944 Wrobel, M. (2013). Can empathy lead to emotional exhaustion in teachers? The mediating role of emotional labor. International Journal of Occupational Medicine and Environmental Health, 26(4), 581–592. https://doi.org/10.2478/s13382-013-0123-1 }} ==External links== Provide [[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: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Burnout]] [[Category:Motivation and emotion/Book/Empathy]] iyljm73xh4r7qum902i9v2aapo4wary 2844549 2844548 2026-11-05T01:49:49Z U3143751 3106785 /* Strategies for combating empathy fatigue and emotional fatigue */ Removed strategies in the workplace section as deemed out of scope 2844549 wikitext text/x-wiki {{title|Empathy fatigue and emotional exhaustion:<br>How does sustained empathic engagement contribute to emotional exhaustion?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}} [[File:Exhausted smartphone user.JPG|right|thumb|150px|'''Figure 1'''. An emotionally exhausted Aelin at the end of the work day]] ; Scenario Aelin is a 23-year-old social worker employed at a non-profit organisation providing mental health services to teenagers and young adults. Aelin's clients are aged between 18 and 25, many of whom having experienced rough living conditions. They work with approximately 20 clients each week. During Aelin's first six months in the role, they feel highly motivated and are excited about helping people. They are an excellent active listener, and can easily remember details about their clients' lives. During their sessions, Aelin recognises when a client is distressed and adjusts their body language to warm and understanding in an effort to provide a safe and welcoming environment to hold space for their emotions. As Aelin gains more experience working in this role, their caseload increases, and they are exposed to clients with increasing complexity and past traumas. Aelin wants to provide the best possible outcome for all of their clients, and begins to take their work home with them in the evenings, but finds it starts creeping into their weekends as well. After several months of working this way, Aelin begins to notice they feel exhausted every morning - even before the workday begins. Fast-forward a few more months, and Aelin notices a change in how they start responding to people, including clients. Instead of feeling engaged and actively listening, they feel detached and find their thoughts feeling more frustrated than empathetic. After finishing the work day, Aelin is drained and has no energy for friends or family. With increasing irritability, Aelin begins to worry that they won't be able to continue working as a social-worker. {{RoundBoxBottom}} Aelin is experiencing '''emotional exhaustion''' from a sustained empathic response. Due to the nature of Aelin's work, they are constantly engaged in an empathic response. Empathic responses require cognitive and physical resources to sustain. Because Aelin is taking their work home and not resting on weekends, they do not have a chance to rest. It is important for Aelin to recognise what is happening and why. Not actioning the problem has led to empathy fatigue and emotional exhaustion. Emotional exhaustion is one of the key aspects of [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FEmpathy_fatigue_and_emotional_exhaustion&diff=2840411&oldid=2832859 burnout], a long-term condition of existing in a constant state of high stress. Psychological science can support Aelin in recognising and mapping the symptoms they are experiencing, understand coping strategies to minimise the current and future effects to improve their wellbeing. Understanding the symptoms will also support their role as a social worker, providing more tools for the people they support on a day-to-day basis. {{RoundBoxTop|theme=3}} '''Focus questions''' * Why do people become emotionally exhausted? * What is the relationship between empathy fatigue and emotional exhaustion? * What role does emotional labor play in emotional exhaustion? * How does understanding emotional labor and empathy fatigue work to reduce the effects of emotional exhaustion? {{RoundBoxBottom}} == Emotional exhaustion == To understand emotional exhaustion, it will be important to first understand burnout. Burnout is a long-term condition of existing in a constant state of high stress and affects millions of people annually. In the early 2000s, a survey from Australia found that more than 50% of social workers and occupational workers reported high levels of burnout symptoms (Llyod and King, 2004) and a recent report from 2026 found that more than one third of all workers in Australia were experiencing burnout (Chan, 2026). Being in a state of high stress for short periods of time is a useful adaptation for humans, so much so that it has been given a name, [[wikipedia:Eustress|Eustress]]. However, as a long-term condition it leads to individual health issues and can be linked to cardiovascular disease, increased vulnerability to upper respiratory diseases, and chronic inflammation (Schneiderman et al., 2005), and cognitive issues such as [[wikipedia:Inattentional_blindness|cognitive tunneling]] and impaired selective attention (Pooladvand and Hasanzadeh, 2023). Burnout can be defined across three dimensions: emotional exhaustion, [[wikipedia:Depersonalization|depersonalisation]], and reduced feelings of personal accomplishment (Maslach, 1993, as cited in Martingano et al., 2026). Emotional exhaustion may be considered the 'core' component of burnout and refers to the depletion of emotional resources, leading to feelings of overwhelm, overextension, and exhaustion (Martingano et al., 2026; De Beer et al., 2024; Seidler et al., 2014). As a major factor, its symptoms are similar to burnout and stress. Although antecedents of burnout are typically agreed upon (Bakker et al., 2014) and divided into individual (e.g. self-efficacy) and situational categories (e.g. work load), there are many theories that attempt to explain why emotional exhaustion occurs. Two of these theories with good evidence are the [[wikipedia:Conservation_of_resources_theory|Conservation of resources stress model]] (Hobfoll, 1989) and the [[wikipedia:Job_demands-resources_model|Job Demands-Resources model]] (Demerouti et al., 2001). === Conservation of resources stress model === The Conservation of resources stress model (CORT; Hobfall, 1989) is a theory that works to understand and explain why stress occurs. Under the model, humans are motivated to maintain, pursue, or attain resources. Stress, defined as "an individual's perceived pressure or strain" (Hollebeek et al., 2023 p.928), occurs when these resources are threatened or lost, or gained at a net loss. As noted above, resources aren't strictly tied to being physical objects. They can also be emotional states, conditions, or even ideas. For example, someone may place value on enjoying a calm, quiet Sunday morning at home. If their neighbour starts using power tools loudly next door, it would disrupt this state and induce stress in the individual. The CORT model focuses primarily on two principles. The first principle is that there is more harm in losing resources than there is in gaining them back (Halbesleben et al., 2014). This is known as loss salience, and some research suggests that the pain of loss is felt twice as strong as the pleasure of gain (Thaler, 2000). Looking at our previous example, imagine that the person who enjoys a quiet Sunday morning at home equally enjoys the sound of a loud car engine. Even though the person enjoys the sound of car engines, if the neighbour started revving their car loudly instead of using power tools, it would still disrupt their desired state of quiet and lead to an increase of stress. The second principle is that people must invest existing resources in order to gain or protect resources (Halbesleben et al., 2014). This resource investment includes recovering from resource loss. Coping is how people manage or overcome a difficult situation, such as protecting or recovering from resource loss. Coping can be broken down into different strategies that people use, but not all strategies are equal (Ito and Brotheridge, 2003). They are divided into two categories: control or escape strategies. Control strategies are considered better at reducing the strain from stress and consist of positive thinking, direct action, and help seeking (Havlovic & Keenan, 1991, as cited in Ito and Brotheridge, 2003). The most useful control strategies incorporate a person's [[locus of control]], emphasise seeking support from co-workers, and highlight the importance of autonomy in decision making. Alarcon (2011) conducted a meta-analysis of 231 studies in a work setting that used the [[wikipedia:Maslach_Burnout_Inventory|Maslach Burnout Inventory]] (MBI) to review the relationship between these principles and burnout to confirm their validity. Although the study had no specific mention of demographics, it was noted that multiple translations of the MBI were used including Dutch (k=51), German (k=7), Italian (k=2), and French (k=2). The findings indicate that workload is positively related to emotional exhaustion (''ρ'' = .49, ''k'' = 86, ''N'' = 51,529) and that control (''ρ'' = − .26, ''k'' = 56, ''N'' = 33,297) and autonomy (''ρ'' = − .24, ''k'' = 34, ''N'' = 17,165) were each negatively associated with emotional exhaustion. The difference of ''ρ'' values is consistent with Thaler (2000) and supports that the benefit of gaining is approximately half that of losing resources. === Job demands-resources model === Another model that attempts to understand why people become emotionally exhausted is the Job Demands-Resources model (JD-R; Demerouti et al., 2001). The JD-R model discusses two concepts: job demands and job resources. Demands and resources are both related to the physical, social, or psychological aspects of a job, but differ in their impact on a person. Job demands require physical or mental effort and exhaust a person's energy supply. Job resources typically support achieving goals, reduce the effort cost of job demands, and stimulate personal growth (Richter and Hacker, 1998, as cited in Demerouti et al., 2001). For example, an unsustainable workload would be considered a job demand, and receiving help and support from others would be considered a resource. The JD-R has two primary processes for how demands and resources interact with one another (Schaufeli, 2017). The stress process occurs when job demands are high and resources are low. As stress increases strain and pressure on an individual, it leads to the exhaustion of their energy eventually, eventually leading to emotional exhaustion. The second process is the motivational process. This process occurs when there is an excess of job resources. Crucially, the motivational process does not require low job demands. With enough job resources, a person will be energised and engaged, reframing job demands as a challenge to overcome. The JD-R model has had extensive research to confirm its validity. A meta-analysis by Lesener et al. (2019) looked at 74 longitudinal studies of the JD-R with a strong average sample size (n=450) against the assumptions as described by Schaufeli (2017). Although most of the studies appear to come from [[wikipedia:Psychology#WEIRD_bias|WEIRD]] countries (k=60) there were a good number from Asia and Africa (k=13) and a mix of genders. The study confirmed that across the literature, job resources predict burnout negatively (''β'' = −.17; ''p'' < .001) and job demands predict burnout over time (''β'' = .10; ''p'' < .001). Although job resources seem like a cure-all for demands, it's important to note that they do not retroactively treat emotional exhaustion. Under the JD-R model, burnout must be reduced by lowering job demands (Gynning et al., 2025). However, once someone has recovered, it is possible to increase job resources alongside job demands to prevent future burnout and support [[wikipedia:Work_engagement|work engagement]]. Work engagement (engagement) has been discussed as the opposite of burnout (Bakker et al., 2014). Engagement is how people express themselves when performing a role. A role being a pattern of behaviour that a person goes through based on the social expectations of that role, similar to how an actor takes on a role for a movie. For example, when attending a typical restaurant, it is socially expected that a person will engage with the waiter and provide them with their meal order to take to the kitchen. This person is performing the script of a restaurant patron. In the context of a workplace, an employee with high engagement may appear to be energised, enthusiastic, and become absorbed in their work losing track of time (Bakkar et al, 2014). {{RoundBoxTop|theme=3}} '''Why do we feel emotionally exhausted''' Under the JD-R and CORT models, emotional exhaustion occurs when the demand cost outweighs the resources available for an extended period of time, eventually depleting someone's energy reserves. Aelin is experiencing burnout because they increased their job demands by taking on more clients, and has experienced a loss in resources by reducing their free time and social support network. {{RoundBoxBottom}} == Empathy fatigue == You may be familiar with the term [[empathy]]. It is often described in terms of supporting others. For example, when someone close to you experiences a loss and is grieving, you may empathise with them and experience a negative emotion akin to grief too. However, empathy is not exclusive to so called negative emotions. For example, compersion is a positive empathic response when a loved one experiences joy. Parents may experience compersion when something joyful happens to their child. At its core, empathy is the individual ability to imagine or understand another person's feelings and/or emotions (Riess, 2017; Shao et al., 2024). Much like typical physical or cognitive processes, such as running or problem solving, empathising will drain energy and resources. Further, and similar to the difference in energy expenditure between running and jogging, there are distinct types of empathy that each have different energy costs. A way to make a distinction between these two types may be considering them as the "what" and "how". Research makes a clear distinction between two types of empathy: affective and cognitive empathy. Affective empathy occurs when a person has an emotional reaction to someone else's mental state (Shao et al., 2024), whereas cognitive empathy occurs when a person understands someone else's feelings or emotions (Cui et al., 2022). === Affective empathy === Affective empathy is the "how" and focuses on the feeling aspect of empathy. This is a more common understanding of empathy, and utilises * Integrating the two - "affective component is the essence of empathy, and the cognitive component is a process through which that essence comes into being" (Bošnjaković & Radionov, 2018) ** (Author's note: change quote into paraphrasing) Emotional contagion is the idea that emotions are contagious. as the theory to moderate surface/deep acting * Emotional contagion (Panksepp and Lahvis, 2011) ** emotions can spread (expand, combines empathy and acting) ** “the tendency to automatically mimic and synchronize expressions, vocalizations, postures, and movements with those of another person’s and, consequently, to converge emotionally.” (Elaine Hatfield, 1993) * Affective process theory looks at unifying the mechanisms of emotional contagion into three types. Convergent linkage, Divergent linkage, and Complementary linkage (Elfenbein, 2014). Storytelling is a way to promote position emotions (Wei et al., 2024). === Cognitive empathy === Cognitive empathy is the "what" and focuses on a person trying to tangibly understand what another person is feeling. This information is processed through a top-down approach (), and primarily utilises the pre-frontal cortex (Shao et al., 2024). This allows the empathiser to feel the emotion from a cognitive, problem solving perspective. As cognitive empathy is processed by the pre-frontal cortex, there is less association with feeling the emotion across the body hence reducing the physiological response and is linked to lower energy expenditure. Paragraph on compassion (use that study in your one-note and look for an additional meta-analysis) === Empathy fatigue and compassion fatigue === * Distinction between empathy and compassion fatigue * Compassion fatigue is closely related to Secondary traumatic stress usually found in healthcare professionals (Figley, 2002) ** Compassion fatigue is to Empathy fatigue as emotional exhaustion is to burnout * Make it clear that Aelin is experiencing Compassion Fatigue vs Empathy fatigue (they are the same but different) (Author's note: include an update to the scenario here!) '''Empathy fatigue questions'''. Choose your answers and click "Submit": <quiz display=simple> {Surface acting is when the observer actually feels the emotion: |type="()"} - True + False {Compassion fatigue and empathy fatigue are synonymous: |type="()"} - True + False </quiz> {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' Under the JD-R and CORT models, a sustained empathic response is a long-term job demand that uses energy to protect existing resources. As Aelin has had their resources is experiencing a * Under the CORT model, a sustained empathic response uses energy to protect existing resources. As resources become low and burnout sets in, cognitive focus narrows leading to empathy fatigue. {{RoundBoxBottom}} == Emotional labor as a mediating factor between empathy fatigue and emotional exhaustion == As a society there are expected [https://en.wikipedia.org/wiki/Social_norm social norms] in behaviour when expressing emotion, regardless if someone is conscious of the process of matching these norms or not. For example, laughing during an emotional speech at a funeral would be considered a rude or offensive expression from the other attendees. As social creatures, humans need a way to regulate socially appropriate emotional expression. This is the essence of Emotional labor: "the regulation of felt and expressed emotions (whether effortful or not) to match these emotional display rules with the objective of fulfilling organizational goals" (Vial & Cowgill, 2022) What is emotional labor (Hochschild, 2012). This section is really where all the above research comes together, and should highlight how the models of stress (JD-R and CORT), relate to the types of empathy (affective/contagion vs cognitive/compassion), and how types of acting (expectations of what you should feel (Emotional labor!!!!) vs accepting what you actually feel) contribute and '''''mediate''''' the relationship between empathy fatigue and emotional exhaustion. This is the penultimate section and should reflect that accordingly: "This is what is happening, and this is how and why its happening" === Types of acting === * Surface acting ** This is where emotive dissonance comes into play ** Leads to emotional exhaustion (Kenworthy et al., 2014) ** "Should" language - in this situation, this is what I ''should'' feel * Deep acting ** People who 'deep act', as in genuinely, intrinsically adjust the way they feel (match the feelings they have with the feelings that are appropriate to display in a given situation) often report positive relationships with emotional labor ** Actual language - in this situation, this is what I ''actually'' feel Discuss the differences in gender norms and expectations in relationships {| class="wikitable" style="margin: auto;" |- ! !! Deep acting !! Surface acting |- | '''Emotional dissonance''' || || X |- | '''Not known to others''' || Hidden area || Unknown |} * === The mediating effect of emotional labor === Surface acting does not mediate the effect (high surface acting leads to emotional exhaustion) whereas the inverse is true for deep acting (Wrobel, 2013) Emotional Expression and Emotional Processing (Hoyt et al., 2024) {{RoundBoxTop|theme=3}} '''What role does emotional labor play in emotional exhaustion''' * In relation to the CORT and JD-R theories, emotional labor acts as a job demand and expenditure of resources to protect existing resources {{RoundBoxBottom}} == Strategies for combating empathy fatigue and emotional exhaustion == Pulling this all together, write a short introduction that describes how to identify which phase you should look at and then use the following table with resources. If you are already emotionally exhausted you need to start at Phase 1. * Talk about emotion regulation (Gross, 1998) and mood regulation strategies as a way to deal with coping (Wrobel, 2013) (Author's note: need more research/references for this section) * Talk about breaks ** Rest and downtime from not performing emotional labor regenerates * How do we reduce the amount of emotional labor we're performing (Jeung et al., 2018) ** Communication/boundaries ** Reducing surface acting (allowing ourselves to feel how we actually feel and reduce emotive dissonance) '''Table 1. Treatment phases of clinical burnout (based on Table 2 from van Dam, A. 2021)''' {| class="wikitable" style="margin: auto; |- ! Phase !! Treatment goal !! Therapeutic interventions |- | '''Phase 1: Crisis''' || Recognition of the patient that the problems are serious, and that serious action is required || Psychoeducation Sick leave Dropping domestic tasks and social obligations Inform social network |- | '''Phase 2: Recovery''' || Recovery of the (physiological) stress system to normal allostatic stress levels || Registration of stress and activities Relaxtion exercises Mindfulness healthy lifestyle advices Graded exercise Gradually resume activities Psychoeducation social network and employer Gradual return to work |- | '''Phase 3: Prevention, learning from the past''' || Acquiring insight and skills to prevent relapse in clinical burnout || Analysing stressful situations Analysing dysfunctional thought patterns Learning new coping skills Learning social skills Job crafting Time management Making choices regarding career and personal life |- | '''Phase 4: Post burnout growth''' || Improving sustainable quality of life. || Setting priorities regarding quality of life and interpersonal relationships. |}{{RoundBoxTop|theme=3}}'''How does understanding emotional labor and empathy fatigue work to reduce the effects of emotional exhaustion''' What about Aelin, what can they do? Work on skills to enhance their knowledge of cognitive empathy and reduce their reliance on affective empathy to provide better outcomes for their patients. *Psychoeducation has been shown to reduce the effects of emotional labor and empathy fatigue on emotional exhaustion {{RoundBoxBottom}} ==Conclusion== * Summarise what empathy fatigue is, and why people become emotionally exhausted (+ the role of emotional labor) * Summarise the symptoms and how to recognise them in yourself and someone else * Emphasise and summarise psychological strategies to support someone going through emotional exhaustion * Summarise ways to prevent emotional exhaustion from happening * Overall message is: *# What is emotional exhaustion *# how do you recognise it *# how do you treat it *# how do you prevent it * Link back to the focus questions ==See also== * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Metacognition_and_emotional_regulation&oldid=2756264 Metacognition and emotional regulation] (Book chapter, 2026) {{ic|Use an internal link style as shown in Tutorial 2}} * [[wikipedia:Conservation_of_resources_theory|Conservation of resources theory]] (Wikipedia) * [[wikipedia:Job_demands-resources_model|Job demands-resources model]] (Wikipedia) ==References== {{Hanging indent|1= Aplin-Houtz, M. (2025). Unraveling the SCARF: How dimensions of the SCARF model influence the relationship between workplace exclusion and turnover intentions. Journal of Management & Organization, 31(6), 2699–2725. https://doi.org/10.1017/jmo.2025.10026 Bošnjaković, J., & Radionov, T. (2018). Empathy: Concepts, Theories and Neuroscientific Basis. Alcoholism (Zagreb), 54(2), 123–150. https://doi.org/10.20471/de..2018.54.02.04 Chan, F. M. (2026). The State of Workplace Burnout 2026: The Workplace Divide. Infinite Potential, Sydney, Australia. https://www.ahri.com.au/wp-content/uploads/The-State-of-Workplace-Burnout-2026-Final.pdf Cui, F., Huang, X., Jing, Y., Luo, Y., Liu, J., & Gu, R. (2022). How resource sharing resists scarcity: the role of cognitive empathy and its neurobiological mechanisms. Cerebral Cortex (New York, N.Y. 1991), 32(23), 5330–5342. https://doi.org/10.1093/cercor/bhac017 De Beer, L. T., van der Vaart, L., Escaffi-Schwarz, M., De Witte, H., & Schaufeli, W. B. (2024). Maslach Burnout Inventory - General Survey: A Systematic Review and Meta-Analysis of Measurement Properties. European journal of psychological assessment : official organ of the European Association of Psychological Assessment, 40(5), 360-375. https://doi.org/10.1027/1015-5759/a000797 Demerouti, E., Bakker, A. B., Nachreiner, F., & Schaufeli, W. B. (2001). The Job Demands-Resources Model of Burnout. Journal of Applied Psychology, 86(3), 499–512. https://doi.org/10.1037/0021-9010.86.3.499 Figley, C. R. (2002). Compassion fatigue: Psychotherapists’ chronic lack of self care. Journal of Clinical Psychology, 58(11), 1433–1441. https://doi.org/10.1002/jclp.10090 Grandey, A. A. (2003). When ‘the show must go on’: surface acting and deep acting as determinants of emotional exhaustion and peer-rated service delivery. Academy of Management Journal, 46(1), 86–96. https://doi.org/10.5465/30040678 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 Halbesleben, J. R. B., Neveu, J.-P., Paustian-Underdahl, S. C., & Westman, M. (2014). Getting to the “COR”: Understanding the Role of Resources in Conservation of Resources Theory. Journal of Management, 40(5), 1334–1364. https://doi.org/10.1177/0149206314527130 Hatfield, E., Cacioppo, J. T., & Rapson, R. L. (1993). Emotional Contagion. Current Directions in Psychological Science : A Journal of the American Psychological Society, 2(3), 96–99. https://doi.org/10.1111/1467-8721.ep10770953 Hollebeek, L. D., Hammedi, W., & Sprott, D. E. (2023). Consumer engagement, stress, and conservation of resources theory: A review, conceptual development, and future research agenda. Psychology & marketing, 40(5), 926-937. https://doi.org/10.1002/mar.21807 Hobfoll, S. E. (1989). Conservation of Resources: A New Attempt at Conceptualizing Stress. The American Psychologist, 44(3), 513–524. https://doi.org/10.1037/0003-066X.44.3.513 Hochschild, A. R. (2012). The managed heart : commercialization of human feeling (Updated, with a new preface.). University of California Press. Jeung, D.-Y., Kim, C., & Chang, S.-J. (2018). Emotional Labor and Burnout: A Review of the Literature. Yonsei Medical Journal, 59(2), 187–193. https://doi.org/10.3349/ymj.2018.59.2.187 Lavelle, J. J., Rupp, D. E., Herda, D. N., Pandey, A., & Lauck, J. R. (2021). Customer Injustice and Employee Performance: Roles of Emotional Exhaustion, Surface Acting, and Emotional Demands–Abilities Fit. Journal of Management, 47(3), Article 0149206319869426. https://doi.org/10.1177/0149206319869426 Lloyd, C., & King, R. (2004). A survey of burnout among Australian mental health occupational therapists and social workers. Social Psychiatry and Psychiatric Epidemiology, 39(9), 752-757. https://doi.org/https://doi.org/10.1007/s00127-004-0808-7 Kenworthy, J., Fay, C., Frame, M., & Petree, R. (2014). A meta-analytic review of the relationship between emotional dissonance and emotional exhaustion. Journal of Applied Social Psychology, 44(2), 94-105. https://doi.org/https://doi.org/10.1111/jasp.12211 Martingano, A. J., Stosic, M. D., & Konrath, S. (2026). Different empathy types show opposing associations with burnout: systematic review and meta-analyses. Psychology, Health & Medicine, 31(3), 798-829. https://doi.org/10.1080/13548506.2025.2591859 Michielsen, H. J., Willemsen, T. M., Croon, M. A., de Vries, J., & Van Heck, G. L. (2004). Determinants of general fatigue and emotional exhaustion: a prospective study. Psychology & health, 19(2), 223-235. https://doi.org/10.1080/08870440310001627135 Panksepp, J. B., & Lahvis, G. P. (2011). Rodent empathy and affective neuroscience. Neuroscience and Biobehavioral Reviews, 35(9), 1864–1875. https://doi.org/10.1016/j.neubiorev.2011.05.013 Pooladvand, S., & Hasanzadeh, S. (2023). Impacts of Stress on Workers’ Risk-Taking Behaviors: Cognitive Tunneling and Impaired Selective Attention. Journal of Construction Engineering and Management, 149(8), 04023060. https://doi.org/doi:10.1061/JCEMD4.COENG-13339 Riess, H. (2017). The Science of Empathy. Journal of Patient Experience, 4(2), 74–77. https://doi.org/10.1177/2374373517699267 Schaufeli, W. B. (2017). Applying the job demands-resources model: a “how to” guide to measuring and tackling work engagement and burnout. Organizational Dynamics, 46(2), 120–132. https://doi.org/10.1016/j.orgdyn.2017.04.008 Schneiderman, N., Ironson, G., & Siegel, S. D. (2005). Stress and Health: Psychological, Behavioral, and Biological Determinants. Annual Review of Clinical Psychology, 1(Volume 1, 2005), 607-628. https://doi.org/https://doi.org/10.1146/annurev.clinpsy.1.102803.144141 Seidler, A., Thinschmidt, M., Deckert, S., Then, F., Hegewald, J., Nieuwenhuijsen, K., & Riedel-Heller, S. G. (2014). The role of psychosocial working conditions on burnout and its core component emotional exhaustion – a systematic review. Journal of Occupational Medicine and Toxicology, 9(1), 10. https://doi.org/10.1186/1745-6673-9-10 Shao, M., Li, L., Li, X., Wei, Z., Wang, J., Hong, M., Liu, X., & Meng, J. (2024). The effect of top-down attention on empathy fatigue. Cerebral Cortex (New York, N.Y. 1991), 34(1), Article bhad441. https://doi.org/10.1093/cercor/bhad441 van Dam, A. (2021). A clinical perspective on burnout: diagnosis, classification, and treatment of clinical burnout. European Journal of Work and Organizational Psychology, 30(5), 732-741. https://doi.org/10.1080/1359432X.2021.1948400 Vial, A. C., & Cowgill, C. M. (2022). Heavier Lies Her Crown: Gendered Patterns of Leader Emotional Labor and Their Downstream Effects. Frontiers in Psychology, 13, 849566. https://doi.org/https://doi.org/10.3389/fpsyg.2022.849566 Wharton, A. S. (2009). The Sociology of Emotional Labor. Annual Review of Sociology, 35(1), 147–165. https://doi.org/10.1146/annurev-soc-070308-115944 Wrobel, M. (2013). Can empathy lead to emotional exhaustion in teachers? The mediating role of emotional labor. International Journal of Occupational Medicine and Environmental Health, 26(4), 581–592. https://doi.org/10.2478/s13382-013-0123-1 }} ==External links== Provide [[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: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Burnout]] [[Category:Motivation and emotion/Book/Empathy]] l96hceno1hh843xy0q7b473ekb4ixoh 2844554 2844549 2026-11-05T02:00:57Z U3143751 3106785 /* Overview */ updated introduction 2844554 wikitext text/x-wiki {{title|Empathy fatigue and emotional exhaustion:<br>How does sustained empathic engagement contribute to emotional exhaustion?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}} [[File:Exhausted smartphone user.JPG|right|thumb|150px|'''Figure 1'''. An emotionally exhausted Aelin at the end of the work day]] ; Scenario Aelin is a 23-year-old social worker employed at a non-profit organisation providing mental health services to teenagers and young adults. Aelin's clients are aged between 18 and 25, many of whom having experienced rough living conditions. They work with approximately 20 clients each week. During Aelin's first six months in the role, they feel highly motivated and are excited about helping people. They are an excellent active listener, and can easily remember details about their clients' lives. During their sessions, Aelin recognises when a client is distressed and adjusts their body language to warm and understanding in an effort to provide a safe and welcoming environment to hold space for their emotions. As Aelin gains more experience working in this role, their caseload increases, and they are exposed to clients with increasing complexity and past traumas. Aelin wants to provide the best possible outcome for all of their clients, and begins to take their work home with them in the evenings, but finds it starts creeping into their weekends as well. After several months of working this way, Aelin begins to notice they feel exhausted every morning - even before the workday begins. Fast-forward a few more months, and Aelin notices a change in how they start responding to people, including clients. Instead of feeling engaged and actively listening, they feel detached and find their thoughts feeling more frustrated than empathetic. After finishing the work day, Aelin is drained and has no energy for friends or family. With increasing irritability, Aelin begins to worry that they won't be able to continue working as a social-worker. {{RoundBoxBottom}} Aelin is experiencing '''emotional exhaustion''' from a sustained empathic response. Due to the nature of Aelin's work, they are constantly engaged in an empathic response. Empathic responses require cognitive and physical resources to sustain. Due to Aelin taking their work home in the evenings and not taking a break on weekends, they do not have a chance to rest. It is important for Aelin to increase their [[wikipedia:Psychoeducation|psychoeducation]] in order to recognise the how, what, and why of their experiences. Without this education, the problem has exacerbated leading to empathy fatigue and emotional exhaustion. Psychological science can support Aelin to recognise and map the symptoms they are experiencing, and provide coping strategies to minimise the current and future effects of empathy fatigue to improve their overall wellbeing. Understanding these symptoms will also support their role as a social worker, enhancing their work engagement and building skills to support their clients on a day-to-day basis. {{RoundBoxTop|theme=3}} '''Focus questions''' * Why do people become emotionally exhausted? * What is the relationship between empathy fatigue and emotional exhaustion? * What role does emotional labor play in emotional exhaustion? * How does understanding emotional labor and empathy fatigue work to reduce the effects of emotional exhaustion? {{RoundBoxBottom}} == Emotional exhaustion == To understand emotional exhaustion, it will be important to first understand burnout. Burnout is a long-term condition of existing in a constant state of high stress and affects millions of people annually. In the early 2000s, a survey from Australia found that more than 50% of social workers and occupational workers reported high levels of burnout symptoms (Llyod and King, 2004) and a recent report from 2026 found that more than one third of all workers in Australia were experiencing burnout (Chan, 2026). Being in a state of high stress for short periods of time is a useful adaptation for humans, so much so that it has been given a name, [[wikipedia:Eustress|Eustress]]. However, as a long-term condition it leads to individual health issues and can be linked to cardiovascular disease, increased vulnerability to upper respiratory diseases, and chronic inflammation (Schneiderman et al., 2005), and cognitive issues such as [[wikipedia:Inattentional_blindness|cognitive tunneling]] and impaired selective attention (Pooladvand and Hasanzadeh, 2023). Burnout can be defined across three dimensions: emotional exhaustion, [[wikipedia:Depersonalization|depersonalisation]], and reduced feelings of personal accomplishment (Maslach, 1993, as cited in Martingano et al., 2026). Emotional exhaustion may be considered the 'core' component of burnout and refers to the depletion of emotional resources, leading to feelings of overwhelm, overextension, and exhaustion (Martingano et al., 2026; De Beer et al., 2024; Seidler et al., 2014). As a major factor, its symptoms are similar to burnout and stress. Although antecedents of burnout are typically agreed upon (Bakker et al., 2014) and divided into individual (e.g. self-efficacy) and situational categories (e.g. work load), there are many theories that attempt to explain why emotional exhaustion occurs. Two of these theories with good evidence are the [[wikipedia:Conservation_of_resources_theory|Conservation of resources stress model]] (Hobfoll, 1989) and the [[wikipedia:Job_demands-resources_model|Job Demands-Resources model]] (Demerouti et al., 2001). === Conservation of resources stress model === The Conservation of resources stress model (CORT; Hobfall, 1989) is a theory that works to understand and explain why stress occurs. Under the model, humans are motivated to maintain, pursue, or attain resources. Stress, defined as "an individual's perceived pressure or strain" (Hollebeek et al., 2023 p.928), occurs when these resources are threatened or lost, or gained at a net loss. As noted above, resources aren't strictly tied to being physical objects. They can also be emotional states, conditions, or even ideas. For example, someone may place value on enjoying a calm, quiet Sunday morning at home. If their neighbour starts using power tools loudly next door, it would disrupt this state and induce stress in the individual. The CORT model focuses primarily on two principles. The first principle is that there is more harm in losing resources than there is in gaining them back (Halbesleben et al., 2014). This is known as loss salience, and some research suggests that the pain of loss is felt twice as strong as the pleasure of gain (Thaler, 2000). Looking at our previous example, imagine that the person who enjoys a quiet Sunday morning at home equally enjoys the sound of a loud car engine. Even though the person enjoys the sound of car engines, if the neighbour started revving their car loudly instead of using power tools, it would still disrupt their desired state of quiet and lead to an increase of stress. The second principle is that people must invest existing resources in order to gain or protect resources (Halbesleben et al., 2014). This resource investment includes recovering from resource loss. Coping is how people manage or overcome a difficult situation, such as protecting or recovering from resource loss. Coping can be broken down into different strategies that people use, but not all strategies are equal (Ito and Brotheridge, 2003). They are divided into two categories: control or escape strategies. Control strategies are considered better at reducing the strain from stress and consist of positive thinking, direct action, and help seeking (Havlovic & Keenan, 1991, as cited in Ito and Brotheridge, 2003). The most useful control strategies incorporate a person's [[locus of control]], emphasise seeking support from co-workers, and highlight the importance of autonomy in decision making. Alarcon (2011) conducted a meta-analysis of 231 studies in a work setting that used the [[wikipedia:Maslach_Burnout_Inventory|Maslach Burnout Inventory]] (MBI) to review the relationship between these principles and burnout to confirm their validity. Although the study had no specific mention of demographics, it was noted that multiple translations of the MBI were used including Dutch (k=51), German (k=7), Italian (k=2), and French (k=2). The findings indicate that workload is positively related to emotional exhaustion (''ρ'' = .49, ''k'' = 86, ''N'' = 51,529) and that control (''ρ'' = − .26, ''k'' = 56, ''N'' = 33,297) and autonomy (''ρ'' = − .24, ''k'' = 34, ''N'' = 17,165) were each negatively associated with emotional exhaustion. The difference of ''ρ'' values is consistent with Thaler (2000) and supports that the benefit of gaining is approximately half that of losing resources. === Job demands-resources model === Another model that attempts to understand why people become emotionally exhausted is the Job Demands-Resources model (JD-R; Demerouti et al., 2001). The JD-R model discusses two concepts: job demands and job resources. Demands and resources are both related to the physical, social, or psychological aspects of a job, but differ in their impact on a person. Job demands require physical or mental effort and exhaust a person's energy supply. Job resources typically support achieving goals, reduce the effort cost of job demands, and stimulate personal growth (Richter and Hacker, 1998, as cited in Demerouti et al., 2001). For example, an unsustainable workload would be considered a job demand, and receiving help and support from others would be considered a resource. The JD-R has two primary processes for how demands and resources interact with one another (Schaufeli, 2017). The stress process occurs when job demands are high and resources are low. As stress increases strain and pressure on an individual, it leads to the exhaustion of their energy eventually, eventually leading to emotional exhaustion. The second process is the motivational process. This process occurs when there is an excess of job resources. Crucially, the motivational process does not require low job demands. With enough job resources, a person will be energised and engaged, reframing job demands as a challenge to overcome. The JD-R model has had extensive research to confirm its validity. A meta-analysis by Lesener et al. (2019) looked at 74 longitudinal studies of the JD-R with a strong average sample size (n=450) against the assumptions as described by Schaufeli (2017). Although most of the studies appear to come from [[wikipedia:Psychology#WEIRD_bias|WEIRD]] countries (k=60) there were a good number from Asia and Africa (k=13) and a mix of genders. The study confirmed that across the literature, job resources predict burnout negatively (''β'' = −.17; ''p'' < .001) and job demands predict burnout over time (''β'' = .10; ''p'' < .001). Although job resources seem like a cure-all for demands, it's important to note that they do not retroactively treat emotional exhaustion. Under the JD-R model, burnout must be reduced by lowering job demands (Gynning et al., 2025). However, once someone has recovered, it is possible to increase job resources alongside job demands to prevent future burnout and support [[wikipedia:Work_engagement|work engagement]]. Work engagement (engagement) has been discussed as the opposite of burnout (Bakker et al., 2014). Engagement is how people express themselves when performing a role. A role being a pattern of behaviour that a person goes through based on the social expectations of that role, similar to how an actor takes on a role for a movie. For example, when attending a typical restaurant, it is socially expected that a person will engage with the waiter and provide them with their meal order to take to the kitchen. This person is performing the script of a restaurant patron. In the context of a workplace, an employee with high engagement may appear to be energised, enthusiastic, and become absorbed in their work losing track of time (Bakkar et al, 2014). {{RoundBoxTop|theme=3}} '''Why do we feel emotionally exhausted''' Under the JD-R and CORT models, emotional exhaustion occurs when the demand cost outweighs the resources available for an extended period of time, eventually depleting someone's energy reserves. Aelin is experiencing burnout because they increased their job demands by taking on more clients, and has experienced a loss in resources by reducing their free time and social support network. {{RoundBoxBottom}} == Empathy fatigue == You may be familiar with the term [[empathy]]. It is often described in terms of supporting others. For example, when someone close to you experiences a loss and is grieving, you may empathise with them and experience a negative emotion akin to grief too. However, empathy is not exclusive to so called negative emotions. For example, compersion is a positive empathic response when a loved one experiences joy. Parents may experience compersion when something joyful happens to their child. At its core, empathy is the individual ability to imagine or understand another person's feelings and/or emotions (Riess, 2017; Shao et al., 2024). Much like typical physical or cognitive processes, such as running or problem solving, empathising will drain energy and resources. Further, and similar to the difference in energy expenditure between running and jogging, there are distinct types of empathy that each have different energy costs. A way to make a distinction between these two types may be considering them as the "what" and "how". Research makes a clear distinction between two types of empathy: affective and cognitive empathy. Affective empathy occurs when a person has an emotional reaction to someone else's mental state (Shao et al., 2024), whereas cognitive empathy occurs when a person understands someone else's feelings or emotions (Cui et al., 2022). === Affective empathy === Affective empathy is the "how" and focuses on the feeling aspect of empathy. This is a more common understanding of empathy, and utilises * Integrating the two - "affective component is the essence of empathy, and the cognitive component is a process through which that essence comes into being" (Bošnjaković & Radionov, 2018) ** (Author's note: change quote into paraphrasing) Emotional contagion is the idea that emotions are contagious. as the theory to moderate surface/deep acting * Emotional contagion (Panksepp and Lahvis, 2011) ** emotions can spread (expand, combines empathy and acting) ** “the tendency to automatically mimic and synchronize expressions, vocalizations, postures, and movements with those of another person’s and, consequently, to converge emotionally.” (Elaine Hatfield, 1993) * Affective process theory looks at unifying the mechanisms of emotional contagion into three types. Convergent linkage, Divergent linkage, and Complementary linkage (Elfenbein, 2014). Storytelling is a way to promote position emotions (Wei et al., 2024). === Cognitive empathy === Cognitive empathy is the "what" and focuses on a person trying to tangibly understand what another person is feeling. This information is processed through a top-down approach (), and primarily utilises the pre-frontal cortex (Shao et al., 2024). This allows the empathiser to feel the emotion from a cognitive, problem solving perspective. As cognitive empathy is processed by the pre-frontal cortex, there is less association with feeling the emotion across the body hence reducing the physiological response and is linked to lower energy expenditure. Paragraph on compassion (use that study in your one-note and look for an additional meta-analysis) === Empathy fatigue and compassion fatigue === * Distinction between empathy and compassion fatigue * Compassion fatigue is closely related to Secondary traumatic stress usually found in healthcare professionals (Figley, 2002) ** Compassion fatigue is to Empathy fatigue as emotional exhaustion is to burnout * Make it clear that Aelin is experiencing Compassion Fatigue vs Empathy fatigue (they are the same but different) (Author's note: include an update to the scenario here!) '''Empathy fatigue questions'''. Choose your answers and click "Submit": <quiz display=simple> {Surface acting is when the observer actually feels the emotion: |type="()"} - True + False {Compassion fatigue and empathy fatigue are synonymous: |type="()"} - True + False </quiz> {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' Under the JD-R and CORT models, a sustained empathic response is a long-term job demand that uses energy to protect existing resources. As Aelin has had their resources is experiencing a * Under the CORT model, a sustained empathic response uses energy to protect existing resources. As resources become low and burnout sets in, cognitive focus narrows leading to empathy fatigue. {{RoundBoxBottom}} == Emotional labor as a mediating factor between empathy fatigue and emotional exhaustion == As a society there are expected [https://en.wikipedia.org/wiki/Social_norm social norms] in behaviour when expressing emotion, regardless if someone is conscious of the process of matching these norms or not. For example, laughing during an emotional speech at a funeral would be considered a rude or offensive expression from the other attendees. As social creatures, humans need a way to regulate socially appropriate emotional expression. This is the essence of Emotional labor: "the regulation of felt and expressed emotions (whether effortful or not) to match these emotional display rules with the objective of fulfilling organizational goals" (Vial & Cowgill, 2022) What is emotional labor (Hochschild, 2012). This section is really where all the above research comes together, and should highlight how the models of stress (JD-R and CORT), relate to the types of empathy (affective/contagion vs cognitive/compassion), and how types of acting (expectations of what you should feel (Emotional labor!!!!) vs accepting what you actually feel) contribute and '''''mediate''''' the relationship between empathy fatigue and emotional exhaustion. This is the penultimate section and should reflect that accordingly: "This is what is happening, and this is how and why its happening" === Types of acting === * Surface acting ** This is where emotive dissonance comes into play ** Leads to emotional exhaustion (Kenworthy et al., 2014) ** "Should" language - in this situation, this is what I ''should'' feel * Deep acting ** People who 'deep act', as in genuinely, intrinsically adjust the way they feel (match the feelings they have with the feelings that are appropriate to display in a given situation) often report positive relationships with emotional labor ** Actual language - in this situation, this is what I ''actually'' feel Discuss the differences in gender norms and expectations in relationships {| class="wikitable" style="margin: auto;" |- ! !! Deep acting !! Surface acting |- | '''Emotional dissonance''' || || X |- | '''Not known to others''' || Hidden area || Unknown |} * === The mediating effect of emotional labor === Surface acting does not mediate the effect (high surface acting leads to emotional exhaustion) whereas the inverse is true for deep acting (Wrobel, 2013) Emotional Expression and Emotional Processing (Hoyt et al., 2024) {{RoundBoxTop|theme=3}} '''What role does emotional labor play in emotional exhaustion''' * In relation to the CORT and JD-R theories, emotional labor acts as a job demand and expenditure of resources to protect existing resources {{RoundBoxBottom}} == Strategies for combating empathy fatigue and emotional exhaustion == Pulling this all together, write a short introduction that describes how to identify which phase you should look at and then use the following table with resources. If you are already emotionally exhausted you need to start at Phase 1. * Talk about emotion regulation (Gross, 1998) and mood regulation strategies as a way to deal with coping (Wrobel, 2013) (Author's note: need more research/references for this section) * Talk about breaks ** Rest and downtime from not performing emotional labor regenerates * How do we reduce the amount of emotional labor we're performing (Jeung et al., 2018) ** Communication/boundaries ** Reducing surface acting (allowing ourselves to feel how we actually feel and reduce emotive dissonance) '''Table 1. Treatment phases of clinical burnout (based on Table 2 from van Dam, A. 2021)''' {| class="wikitable" style="margin: auto; |- ! Phase !! Treatment goal !! Therapeutic interventions |- | '''Phase 1: Crisis''' || Recognition of the patient that the problems are serious, and that serious action is required || Psychoeducation Sick leave Dropping domestic tasks and social obligations Inform social network |- | '''Phase 2: Recovery''' || Recovery of the (physiological) stress system to normal allostatic stress levels || Registration of stress and activities Relaxtion exercises Mindfulness healthy lifestyle advices Graded exercise Gradually resume activities Psychoeducation social network and employer Gradual return to work |- | '''Phase 3: Prevention, learning from the past''' || Acquiring insight and skills to prevent relapse in clinical burnout || Analysing stressful situations Analysing dysfunctional thought patterns Learning new coping skills Learning social skills Job crafting Time management Making choices regarding career and personal life |- | '''Phase 4: Post burnout growth''' || Improving sustainable quality of life. || Setting priorities regarding quality of life and interpersonal relationships. |}{{RoundBoxTop|theme=3}}'''How does understanding emotional labor and empathy fatigue work to reduce the effects of emotional exhaustion''' What about Aelin, what can they do? Work on skills to enhance their knowledge of cognitive empathy and reduce their reliance on affective empathy to provide better outcomes for their patients. *Psychoeducation has been shown to reduce the effects of emotional labor and empathy fatigue on emotional exhaustion {{RoundBoxBottom}} ==Conclusion== * Summarise what empathy fatigue is, and why people become emotionally exhausted (+ the role of emotional labor) * Summarise the symptoms and how to recognise them in yourself and someone else * Emphasise and summarise psychological strategies to support someone going through emotional exhaustion * Summarise ways to prevent emotional exhaustion from happening * Overall message is: *# What is emotional exhaustion *# how do you recognise it *# how do you treat it *# how do you prevent it * Link back to the focus questions ==See also== * [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2025/Metacognition_and_emotional_regulation&oldid=2756264 Metacognition and emotional regulation] (Book chapter, 2026) {{ic|Use an internal link style as shown in Tutorial 2}} * [[wikipedia:Conservation_of_resources_theory|Conservation of resources theory]] (Wikipedia) * [[wikipedia:Job_demands-resources_model|Job demands-resources model]] (Wikipedia) ==References== {{Hanging indent|1= Aplin-Houtz, M. (2025). Unraveling the SCARF: How dimensions of the SCARF model influence the relationship between workplace exclusion and turnover intentions. Journal of Management & Organization, 31(6), 2699–2725. https://doi.org/10.1017/jmo.2025.10026 Bošnjaković, J., & Radionov, T. (2018). Empathy: Concepts, Theories and Neuroscientific Basis. Alcoholism (Zagreb), 54(2), 123–150. https://doi.org/10.20471/de..2018.54.02.04 Chan, F. M. (2026). The State of Workplace Burnout 2026: The Workplace Divide. Infinite Potential, Sydney, Australia. https://www.ahri.com.au/wp-content/uploads/The-State-of-Workplace-Burnout-2026-Final.pdf Cui, F., Huang, X., Jing, Y., Luo, Y., Liu, J., & Gu, R. (2022). How resource sharing resists scarcity: the role of cognitive empathy and its neurobiological mechanisms. Cerebral Cortex (New York, N.Y. 1991), 32(23), 5330–5342. https://doi.org/10.1093/cercor/bhac017 De Beer, L. T., van der Vaart, L., Escaffi-Schwarz, M., De Witte, H., & Schaufeli, W. B. (2024). Maslach Burnout Inventory - General Survey: A Systematic Review and Meta-Analysis of Measurement Properties. European journal of psychological assessment : official organ of the European Association of Psychological Assessment, 40(5), 360-375. https://doi.org/10.1027/1015-5759/a000797 Demerouti, E., Bakker, A. B., Nachreiner, F., & Schaufeli, W. B. (2001). The Job Demands-Resources Model of Burnout. Journal of Applied Psychology, 86(3), 499–512. https://doi.org/10.1037/0021-9010.86.3.499 Figley, C. R. (2002). Compassion fatigue: Psychotherapists’ chronic lack of self care. Journal of Clinical Psychology, 58(11), 1433–1441. https://doi.org/10.1002/jclp.10090 Grandey, A. A. (2003). When ‘the show must go on’: surface acting and deep acting as determinants of emotional exhaustion and peer-rated service delivery. Academy of Management Journal, 46(1), 86–96. https://doi.org/10.5465/30040678 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 Halbesleben, J. R. B., Neveu, J.-P., Paustian-Underdahl, S. C., & Westman, M. (2014). Getting to the “COR”: Understanding the Role of Resources in Conservation of Resources Theory. Journal of Management, 40(5), 1334–1364. https://doi.org/10.1177/0149206314527130 Hatfield, E., Cacioppo, J. T., & Rapson, R. L. (1993). Emotional Contagion. Current Directions in Psychological Science : A Journal of the American Psychological Society, 2(3), 96–99. https://doi.org/10.1111/1467-8721.ep10770953 Hollebeek, L. D., Hammedi, W., & Sprott, D. E. (2023). Consumer engagement, stress, and conservation of resources theory: A review, conceptual development, and future research agenda. Psychology & marketing, 40(5), 926-937. https://doi.org/10.1002/mar.21807 Hobfoll, S. E. (1989). Conservation of Resources: A New Attempt at Conceptualizing Stress. The American Psychologist, 44(3), 513–524. https://doi.org/10.1037/0003-066X.44.3.513 Hochschild, A. R. (2012). The managed heart : commercialization of human feeling (Updated, with a new preface.). University of California Press. Jeung, D.-Y., Kim, C., & Chang, S.-J. (2018). Emotional Labor and Burnout: A Review of the Literature. Yonsei Medical Journal, 59(2), 187–193. https://doi.org/10.3349/ymj.2018.59.2.187 Lavelle, J. J., Rupp, D. E., Herda, D. N., Pandey, A., & Lauck, J. R. (2021). Customer Injustice and Employee Performance: Roles of Emotional Exhaustion, Surface Acting, and Emotional Demands–Abilities Fit. Journal of Management, 47(3), Article 0149206319869426. https://doi.org/10.1177/0149206319869426 Lloyd, C., & King, R. (2004). A survey of burnout among Australian mental health occupational therapists and social workers. Social Psychiatry and Psychiatric Epidemiology, 39(9), 752-757. https://doi.org/https://doi.org/10.1007/s00127-004-0808-7 Kenworthy, J., Fay, C., Frame, M., & Petree, R. (2014). A meta-analytic review of the relationship between emotional dissonance and emotional exhaustion. Journal of Applied Social Psychology, 44(2), 94-105. https://doi.org/https://doi.org/10.1111/jasp.12211 Martingano, A. J., Stosic, M. D., & Konrath, S. (2026). Different empathy types show opposing associations with burnout: systematic review and meta-analyses. Psychology, Health & Medicine, 31(3), 798-829. https://doi.org/10.1080/13548506.2025.2591859 Michielsen, H. J., Willemsen, T. M., Croon, M. A., de Vries, J., & Van Heck, G. L. (2004). Determinants of general fatigue and emotional exhaustion: a prospective study. Psychology & health, 19(2), 223-235. https://doi.org/10.1080/08870440310001627135 Panksepp, J. B., & Lahvis, G. P. (2011). Rodent empathy and affective neuroscience. Neuroscience and Biobehavioral Reviews, 35(9), 1864–1875. https://doi.org/10.1016/j.neubiorev.2011.05.013 Pooladvand, S., & Hasanzadeh, S. (2023). Impacts of Stress on Workers’ Risk-Taking Behaviors: Cognitive Tunneling and Impaired Selective Attention. Journal of Construction Engineering and Management, 149(8), 04023060. https://doi.org/doi:10.1061/JCEMD4.COENG-13339 Riess, H. (2017). The Science of Empathy. Journal of Patient Experience, 4(2), 74–77. https://doi.org/10.1177/2374373517699267 Schaufeli, W. B. (2017). Applying the job demands-resources model: a “how to” guide to measuring and tackling work engagement and burnout. Organizational Dynamics, 46(2), 120–132. https://doi.org/10.1016/j.orgdyn.2017.04.008 Schneiderman, N., Ironson, G., & Siegel, S. D. (2005). Stress and Health: Psychological, Behavioral, and Biological Determinants. Annual Review of Clinical Psychology, 1(Volume 1, 2005), 607-628. https://doi.org/https://doi.org/10.1146/annurev.clinpsy.1.102803.144141 Seidler, A., Thinschmidt, M., Deckert, S., Then, F., Hegewald, J., Nieuwenhuijsen, K., & Riedel-Heller, S. G. (2014). The role of psychosocial working conditions on burnout and its core component emotional exhaustion – a systematic review. Journal of Occupational Medicine and Toxicology, 9(1), 10. https://doi.org/10.1186/1745-6673-9-10 Shao, M., Li, L., Li, X., Wei, Z., Wang, J., Hong, M., Liu, X., & Meng, J. (2024). The effect of top-down attention on empathy fatigue. Cerebral Cortex (New York, N.Y. 1991), 34(1), Article bhad441. https://doi.org/10.1093/cercor/bhad441 van Dam, A. (2021). A clinical perspective on burnout: diagnosis, classification, and treatment of clinical burnout. European Journal of Work and Organizational Psychology, 30(5), 732-741. https://doi.org/10.1080/1359432X.2021.1948400 Vial, A. C., & Cowgill, C. M. (2022). Heavier Lies Her Crown: Gendered Patterns of Leader Emotional Labor and Their Downstream Effects. Frontiers in Psychology, 13, 849566. https://doi.org/https://doi.org/10.3389/fpsyg.2022.849566 Wharton, A. S. (2009). The Sociology of Emotional Labor. Annual Review of Sociology, 35(1), 147–165. https://doi.org/10.1146/annurev-soc-070308-115944 Wrobel, M. (2013). Can empathy lead to emotional exhaustion in teachers? The mediating role of emotional labor. International Journal of Occupational Medicine and Environmental Health, 26(4), 581–592. https://doi.org/10.2478/s13382-013-0123-1 }} ==External links== Provide [[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: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Burnout]] [[Category:Motivation and emotion/Book/Empathy]] oyg65pgd3k0c823pnh0tpfqx74jze2r 2844561 2844554 2026-11-05T02:12:10Z U3143751 3106785 /* External links */ Updating external links and see also 2844561 wikitext text/x-wiki {{title|Empathy fatigue and emotional exhaustion:<br>How does sustained empathic engagement contribute to emotional exhaustion?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}} [[File:Exhausted smartphone user.JPG|right|thumb|150px|'''Figure 1'''. An emotionally exhausted Aelin at the end of the work day]] ; Scenario Aelin is a 23-year-old social worker employed at a non-profit organisation providing mental health services to teenagers and young adults. Aelin's clients are aged between 18 and 25, many of whom having experienced rough living conditions. They work with approximately 20 clients each week. During Aelin's first six months in the role, they feel highly motivated and are excited about helping people. They are an excellent active listener, and can easily remember details about their clients' lives. During their sessions, Aelin recognises when a client is distressed and adjusts their body language to warm and understanding in an effort to provide a safe and welcoming environment to hold space for their emotions. As Aelin gains more experience working in this role, their caseload increases, and they are exposed to clients with increasing complexity and past traumas. Aelin wants to provide the best possible outcome for all of their clients, and begins to take their work home with them in the evenings, but finds it starts creeping into their weekends as well. After several months of working this way, Aelin begins to notice they feel exhausted every morning - even before the workday begins. Fast-forward a few more months, and Aelin notices a change in how they start responding to people, including clients. Instead of feeling engaged and actively listening, they feel detached and find their thoughts feeling more frustrated than empathetic. After finishing the work day, Aelin is drained and has no energy for friends or family. With increasing irritability, Aelin begins to worry that they won't be able to continue working as a social-worker. {{RoundBoxBottom}} Aelin is experiencing '''emotional exhaustion''' from a sustained empathic response. Due to the nature of Aelin's work, they are constantly engaged in an empathic response. Empathic responses require cognitive and physical resources to sustain. Due to Aelin taking their work home in the evenings and not taking a break on weekends, they do not have a chance to rest. It is important for Aelin to increase their [[wikipedia:Psychoeducation|psychoeducation]] in order to recognise the how, what, and why of their experiences. Without this education, the problem has exacerbated leading to empathy fatigue and emotional exhaustion. Psychological science can support Aelin to recognise and map the symptoms they are experiencing, and provide coping strategies to minimise the current and future effects of empathy fatigue to improve their overall wellbeing. Understanding these symptoms will also support their role as a social worker, enhancing their work engagement and building skills to support their clients on a day-to-day basis. {{RoundBoxTop|theme=3}} '''Focus questions''' * Why do people become emotionally exhausted? * What is the relationship between empathy fatigue and emotional exhaustion? * What role does emotional labor play in emotional exhaustion? * How does understanding emotional labor and empathy fatigue work to reduce the effects of emotional exhaustion? {{RoundBoxBottom}} == Emotional exhaustion == To understand emotional exhaustion, it will be important to first understand burnout. Burnout is a long-term condition of existing in a constant state of high stress and affects millions of people annually. In the early 2000s, a survey from Australia found that more than 50% of social workers and occupational workers reported high levels of burnout symptoms (Llyod and King, 2004) and a recent report from 2026 found that more than one third of all workers in Australia were experiencing burnout (Chan, 2026). Being in a state of high stress for short periods of time is a useful adaptation for humans, so much so that it has been given a name, [[wikipedia:Eustress|Eustress]]. However, as a long-term condition it leads to individual health issues and can be linked to cardiovascular disease, increased vulnerability to upper respiratory diseases, and chronic inflammation (Schneiderman et al., 2005), and cognitive issues such as [[wikipedia:Inattentional_blindness|cognitive tunneling]] and impaired selective attention (Pooladvand and Hasanzadeh, 2023). Burnout can be defined across three dimensions: emotional exhaustion, [[wikipedia:Depersonalization|depersonalisation]], and reduced feelings of personal accomplishment (Maslach, 1993, as cited in Martingano et al., 2026). Emotional exhaustion may be considered the 'core' component of burnout and refers to the depletion of emotional resources, leading to feelings of overwhelm, overextension, and exhaustion (Martingano et al., 2026; De Beer et al., 2024; Seidler et al., 2014). As a major factor, its symptoms are similar to burnout and stress. Although antecedents of burnout are typically agreed upon (Bakker et al., 2014) and divided into individual (e.g. self-efficacy) and situational categories (e.g. work load), there are many theories that attempt to explain why emotional exhaustion occurs. Two of these theories with good evidence are the [[wikipedia:Conservation_of_resources_theory|Conservation of resources stress model]] (Hobfoll, 1989) and the [[wikipedia:Job_demands-resources_model|Job Demands-Resources model]] (Demerouti et al., 2001). === Conservation of resources stress model === The Conservation of resources stress model (CORT; Hobfall, 1989) is a theory that works to understand and explain why stress occurs. Under the model, humans are motivated to maintain, pursue, or attain resources. Stress, defined as "an individual's perceived pressure or strain" (Hollebeek et al., 2023 p.928), occurs when these resources are threatened or lost, or gained at a net loss. As noted above, resources aren't strictly tied to being physical objects. They can also be emotional states, conditions, or even ideas. For example, someone may place value on enjoying a calm, quiet Sunday morning at home. If their neighbour starts using power tools loudly next door, it would disrupt this state and induce stress in the individual. The CORT model focuses primarily on two principles. The first principle is that there is more harm in losing resources than there is in gaining them back (Halbesleben et al., 2014). This is known as loss salience, and some research suggests that the pain of loss is felt twice as strong as the pleasure of gain (Thaler, 2000). Looking at our previous example, imagine that the person who enjoys a quiet Sunday morning at home equally enjoys the sound of a loud car engine. Even though the person enjoys the sound of car engines, if the neighbour started revving their car loudly instead of using power tools, it would still disrupt their desired state of quiet and lead to an increase of stress. The second principle is that people must invest existing resources in order to gain or protect resources (Halbesleben et al., 2014). This resource investment includes recovering from resource loss. Coping is how people manage or overcome a difficult situation, such as protecting or recovering from resource loss. Coping can be broken down into different strategies that people use, but not all strategies are equal (Ito and Brotheridge, 2003). They are divided into two categories: control or escape strategies. Control strategies are considered better at reducing the strain from stress and consist of positive thinking, direct action, and help seeking (Havlovic & Keenan, 1991, as cited in Ito and Brotheridge, 2003). The most useful control strategies incorporate a person's [[locus of control]], emphasise seeking support from co-workers, and highlight the importance of autonomy in decision making. Alarcon (2011) conducted a meta-analysis of 231 studies in a work setting that used the [[wikipedia:Maslach_Burnout_Inventory|Maslach Burnout Inventory]] (MBI) to review the relationship between these principles and burnout to confirm their validity. Although the study had no specific mention of demographics, it was noted that multiple translations of the MBI were used including Dutch (k=51), German (k=7), Italian (k=2), and French (k=2). The findings indicate that workload is positively related to emotional exhaustion (''ρ'' = .49, ''k'' = 86, ''N'' = 51,529) and that control (''ρ'' = − .26, ''k'' = 56, ''N'' = 33,297) and autonomy (''ρ'' = − .24, ''k'' = 34, ''N'' = 17,165) were each negatively associated with emotional exhaustion. The difference of ''ρ'' values is consistent with Thaler (2000) and supports that the benefit of gaining is approximately half that of losing resources. === Job demands-resources model === Another model that attempts to understand why people become emotionally exhausted is the Job Demands-Resources model (JD-R; Demerouti et al., 2001). The JD-R model discusses two concepts: job demands and job resources. Demands and resources are both related to the physical, social, or psychological aspects of a job, but differ in their impact on a person. Job demands require physical or mental effort and exhaust a person's energy supply. Job resources typically support achieving goals, reduce the effort cost of job demands, and stimulate personal growth (Richter and Hacker, 1998, as cited in Demerouti et al., 2001). For example, an unsustainable workload would be considered a job demand, and receiving help and support from others would be considered a resource. The JD-R has two primary processes for how demands and resources interact with one another (Schaufeli, 2017). The stress process occurs when job demands are high and resources are low. As stress increases strain and pressure on an individual, it leads to the exhaustion of their energy eventually, eventually leading to emotional exhaustion. The second process is the motivational process. This process occurs when there is an excess of job resources. Crucially, the motivational process does not require low job demands. With enough job resources, a person will be energised and engaged, reframing job demands as a challenge to overcome. The JD-R model has had extensive research to confirm its validity. A meta-analysis by Lesener et al. (2019) looked at 74 longitudinal studies of the JD-R with a strong average sample size (n=450) against the assumptions as described by Schaufeli (2017). Although most of the studies appear to come from [[wikipedia:Psychology#WEIRD_bias|WEIRD]] countries (k=60) there were a good number from Asia and Africa (k=13) and a mix of genders. The study confirmed that across the literature, job resources predict burnout negatively (''β'' = −.17; ''p'' < .001) and job demands predict burnout over time (''β'' = .10; ''p'' < .001). Although job resources seem like a cure-all for demands, it's important to note that they do not retroactively treat emotional exhaustion. Under the JD-R model, burnout must be reduced by lowering job demands (Gynning et al., 2025). However, once someone has recovered, it is possible to increase job resources alongside job demands to prevent future burnout and support [[wikipedia:Work_engagement|work engagement]]. Work engagement (engagement) has been discussed as the opposite of burnout (Bakker et al., 2014). Engagement is how people express themselves when performing a role. A role being a pattern of behaviour that a person goes through based on the social expectations of that role, similar to how an actor takes on a role for a movie. For example, when attending a typical restaurant, it is socially expected that a person will engage with the waiter and provide them with their meal order to take to the kitchen. This person is performing the script of a restaurant patron. In the context of a workplace, an employee with high engagement may appear to be energised, enthusiastic, and become absorbed in their work losing track of time (Bakkar et al, 2014). {{RoundBoxTop|theme=3}} '''Why do we feel emotionally exhausted''' Under the JD-R and CORT models, emotional exhaustion occurs when the demand cost outweighs the resources available for an extended period of time, eventually depleting someone's energy reserves. Aelin is experiencing burnout because they increased their job demands by taking on more clients, and has experienced a loss in resources by reducing their free time and social support network. {{RoundBoxBottom}} == Empathy fatigue == You may be familiar with the term [[empathy]]. It is often described in terms of supporting others. For example, when someone close to you experiences a loss and is grieving, you may empathise with them and experience a negative emotion akin to grief too. However, empathy is not exclusive to so called negative emotions. For example, compersion is a positive empathic response when a loved one experiences joy. Parents may experience compersion when something joyful happens to their child. At its core, empathy is the individual ability to imagine or understand another person's feelings and/or emotions (Riess, 2017; Shao et al., 2024). Much like typical physical or cognitive processes, such as running or problem solving, empathising will drain energy and resources. Further, and similar to the difference in energy expenditure between running and jogging, there are distinct types of empathy that each have different energy costs. A way to make a distinction between these two types may be considering them as the "what" and "how". Research makes a clear distinction between two types of empathy: affective and cognitive empathy. Affective empathy occurs when a person has an emotional reaction to someone else's mental state (Shao et al., 2024), whereas cognitive empathy occurs when a person understands someone else's feelings or emotions (Cui et al., 2022). === Affective empathy === Affective empathy is the "how" and focuses on the feeling aspect of empathy. This is a more common understanding of empathy, and utilises * Integrating the two - "affective component is the essence of empathy, and the cognitive component is a process through which that essence comes into being" (Bošnjaković & Radionov, 2018) ** (Author's note: change quote into paraphrasing) Emotional contagion is the idea that emotions are contagious. as the theory to moderate surface/deep acting * Emotional contagion (Panksepp and Lahvis, 2011) ** emotions can spread (expand, combines empathy and acting) ** “the tendency to automatically mimic and synchronize expressions, vocalizations, postures, and movements with those of another person’s and, consequently, to converge emotionally.” (Elaine Hatfield, 1993) * Affective process theory looks at unifying the mechanisms of emotional contagion into three types. Convergent linkage, Divergent linkage, and Complementary linkage (Elfenbein, 2014). Storytelling is a way to promote position emotions (Wei et al., 2024). === Cognitive empathy === Cognitive empathy is the "what" and focuses on a person trying to tangibly understand what another person is feeling. This information is processed through a top-down approach (), and primarily utilises the pre-frontal cortex (Shao et al., 2024). This allows the empathiser to feel the emotion from a cognitive, problem solving perspective. As cognitive empathy is processed by the pre-frontal cortex, there is less association with feeling the emotion across the body hence reducing the physiological response and is linked to lower energy expenditure. Paragraph on compassion (use that study in your one-note and look for an additional meta-analysis) === Empathy fatigue and compassion fatigue === * Distinction between empathy and compassion fatigue * Compassion fatigue is closely related to Secondary traumatic stress usually found in healthcare professionals (Figley, 2002) ** Compassion fatigue is to Empathy fatigue as emotional exhaustion is to burnout * Make it clear that Aelin is experiencing Compassion Fatigue vs Empathy fatigue (they are the same but different) (Author's note: include an update to the scenario here!) '''Empathy fatigue questions'''. Choose your answers and click "Submit": <quiz display=simple> {Surface acting is when the observer actually feels the emotion: |type="()"} - True + False {Compassion fatigue and empathy fatigue are synonymous: |type="()"} - True + False </quiz> {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' Under the JD-R and CORT models, a sustained empathic response is a long-term job demand that uses energy to protect existing resources. As Aelin has had their resources is experiencing a * Under the CORT model, a sustained empathic response uses energy to protect existing resources. As resources become low and burnout sets in, cognitive focus narrows leading to empathy fatigue. {{RoundBoxBottom}} == Emotional labor as a mediating factor between empathy fatigue and emotional exhaustion == As a society there are expected [https://en.wikipedia.org/wiki/Social_norm social norms] in behaviour when expressing emotion, regardless if someone is conscious of the process of matching these norms or not. For example, laughing during an emotional speech at a funeral would be considered a rude or offensive expression from the other attendees. As social creatures, humans need a way to regulate socially appropriate emotional expression. This is the essence of Emotional labor: "the regulation of felt and expressed emotions (whether effortful or not) to match these emotional display rules with the objective of fulfilling organizational goals" (Vial & Cowgill, 2022) What is emotional labor (Hochschild, 2012). This section is really where all the above research comes together, and should highlight how the models of stress (JD-R and CORT), relate to the types of empathy (affective/contagion vs cognitive/compassion), and how types of acting (expectations of what you should feel (Emotional labor!!!!) vs accepting what you actually feel) contribute and '''''mediate''''' the relationship between empathy fatigue and emotional exhaustion. This is the penultimate section and should reflect that accordingly: "This is what is happening, and this is how and why its happening" === Types of acting === * Surface acting ** This is where emotive dissonance comes into play ** Leads to emotional exhaustion (Kenworthy et al., 2014) ** "Should" language - in this situation, this is what I ''should'' feel * Deep acting ** People who 'deep act', as in genuinely, intrinsically adjust the way they feel (match the feelings they have with the feelings that are appropriate to display in a given situation) often report positive relationships with emotional labor ** Actual language - in this situation, this is what I ''actually'' feel Discuss the differences in gender norms and expectations in relationships {| class="wikitable" style="margin: auto;" |- ! !! Deep acting !! Surface acting |- | '''Emotional dissonance''' || || X |- | '''Not known to others''' || Hidden area || Unknown |} * === The mediating effect of emotional labor === Surface acting does not mediate the effect (high surface acting leads to emotional exhaustion) whereas the inverse is true for deep acting (Wrobel, 2013) Emotional Expression and Emotional Processing (Hoyt et al., 2024) {{RoundBoxTop|theme=3}} '''What role does emotional labor play in emotional exhaustion''' * In relation to the CORT and JD-R theories, emotional labor acts as a job demand and expenditure of resources to protect existing resources {{RoundBoxBottom}} == Strategies for combating empathy fatigue and emotional exhaustion == Pulling this all together, write a short introduction that describes how to identify which phase you should look at and then use the following table with resources. If you are already emotionally exhausted you need to start at Phase 1. * Talk about emotion regulation (Gross, 1998) and mood regulation strategies as a way to deal with coping (Wrobel, 2013) (Author's note: need more research/references for this section) * Talk about breaks ** Rest and downtime from not performing emotional labor regenerates * How do we reduce the amount of emotional labor we're performing (Jeung et al., 2018) ** Communication/boundaries ** Reducing surface acting (allowing ourselves to feel how we actually feel and reduce emotive dissonance) '''Table 1. Treatment phases of clinical burnout (based on Table 2 from van Dam, A. 2021)''' {| class="wikitable" style="margin: auto; |- ! Phase !! Treatment goal !! Therapeutic interventions |- | '''Phase 1: Crisis''' || Recognition of the patient that the problems are serious, and that serious action is required || Psychoeducation Sick leave Dropping domestic tasks and social obligations Inform social network |- | '''Phase 2: Recovery''' || Recovery of the (physiological) stress system to normal allostatic stress levels || Registration of stress and activities Relaxtion exercises Mindfulness healthy lifestyle advices Graded exercise Gradually resume activities Psychoeducation social network and employer Gradual return to work |- | '''Phase 3: Prevention, learning from the past''' || Acquiring insight and skills to prevent relapse in clinical burnout || Analysing stressful situations Analysing dysfunctional thought patterns Learning new coping skills Learning social skills Job crafting Time management Making choices regarding career and personal life |- | '''Phase 4: Post burnout growth''' || Improving sustainable quality of life. || Setting priorities regarding quality of life and interpersonal relationships. |}{{RoundBoxTop|theme=3}}'''How does understanding emotional labor and empathy fatigue work to reduce the effects of emotional exhaustion''' What about Aelin, what can they do? Work on skills to enhance their knowledge of cognitive empathy and reduce their reliance on affective empathy to provide better outcomes for their patients. *Psychoeducation has been shown to reduce the effects of emotional labor and empathy fatigue on emotional exhaustion {{RoundBoxBottom}} ==Conclusion== * Summarise what empathy fatigue is, and why people become emotionally exhausted (+ the role of emotional labor) * Summarise the symptoms and how to recognise them in yourself and someone else * Emphasise and summarise psychological strategies to support someone going through emotional exhaustion * Summarise ways to prevent emotional exhaustion from happening * Overall message is: *# What is emotional exhaustion *# how do you recognise it *# how do you treat it *# how do you prevent it * Link back to the focus questions ==See also== * [[Motivation and emotion/Book/2025/Metacognition and emotional regulation|Metacognition and emotional regulation]] (Book chapter, 2025) * [[wikipedia:Conservation_of_resources_theory|Conservation of resources theory]] (Wikipedia) * [[wikipedia:Job_demands-resources_model|Job demands-resources model]] (Wikipedia) * Emotional Contagion * Perspective taking * Emotion regulation * ==References== {{Hanging indent|1= Bošnjaković, J., & Radionov, T. (2018). Empathy: Concepts, Theories and Neuroscientific Basis. Alcoholism (Zagreb), 54(2), 123–150. https://doi.org/10.20471/de..2018.54.02.04 Chan, F. M. (2026). The State of Workplace Burnout 2026: The Workplace Divide. Infinite Potential, Sydney, Australia. https://www.ahri.com.au/wp-content/uploads/The-State-of-Workplace-Burnout-2026-Final.pdf Cui, F., Huang, X., Jing, Y., Luo, Y., Liu, J., & Gu, R. (2022). How resource sharing resists scarcity: the role of cognitive empathy and its neurobiological mechanisms. Cerebral Cortex (New York, N.Y. 1991), 32(23), 5330–5342. https://doi.org/10.1093/cercor/bhac017 De Beer, L. T., van der Vaart, L., Escaffi-Schwarz, M., De Witte, H., & Schaufeli, W. B. (2024). Maslach Burnout Inventory - General Survey: A Systematic Review and Meta-Analysis of Measurement Properties. European journal of psychological assessment : official organ of the European Association of Psychological Assessment, 40(5), 360-375. https://doi.org/10.1027/1015-5759/a000797 Demerouti, E., Bakker, A. B., Nachreiner, F., & Schaufeli, W. B. (2001). The Job Demands-Resources Model of Burnout. Journal of Applied Psychology, 86(3), 499–512. https://doi.org/10.1037/0021-9010.86.3.499 Figley, C. R. (2002). Compassion fatigue: Psychotherapists’ chronic lack of self care. Journal of Clinical Psychology, 58(11), 1433–1441. https://doi.org/10.1002/jclp.10090 Grandey, A. A. (2003). When ‘the show must go on’: surface acting and deep acting as determinants of emotional exhaustion and peer-rated service delivery. Academy of Management Journal, 46(1), 86–96. https://doi.org/10.5465/30040678 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 Halbesleben, J. R. B., Neveu, J.-P., Paustian-Underdahl, S. C., & Westman, M. (2014). Getting to the “COR”: Understanding the Role of Resources in Conservation of Resources Theory. Journal of Management, 40(5), 1334–1364. https://doi.org/10.1177/0149206314527130 Hatfield, E., Cacioppo, J. T., & Rapson, R. L. (1993). Emotional Contagion. Current Directions in Psychological Science : A Journal of the American Psychological Society, 2(3), 96–99. https://doi.org/10.1111/1467-8721.ep10770953 Hollebeek, L. D., Hammedi, W., & Sprott, D. E. (2023). Consumer engagement, stress, and conservation of resources theory: A review, conceptual development, and future research agenda. Psychology & marketing, 40(5), 926-937. https://doi.org/10.1002/mar.21807 Hobfoll, S. E. (1989). Conservation of Resources: A New Attempt at Conceptualizing Stress. The American Psychologist, 44(3), 513–524. https://doi.org/10.1037/0003-066X.44.3.513 Hochschild, A. R. (2012). The managed heart : commercialization of human feeling (Updated, with a new preface.). University of California Press. Jeung, D.-Y., Kim, C., & Chang, S.-J. (2018). Emotional Labor and Burnout: A Review of the Literature. Yonsei Medical Journal, 59(2), 187–193. https://doi.org/10.3349/ymj.2018.59.2.187 Lavelle, J. J., Rupp, D. E., Herda, D. N., Pandey, A., & Lauck, J. R. (2021). Customer Injustice and Employee Performance: Roles of Emotional Exhaustion, Surface Acting, and Emotional Demands–Abilities Fit. Journal of Management, 47(3), Article 0149206319869426. https://doi.org/10.1177/0149206319869426 Lloyd, C., & King, R. (2004). A survey of burnout among Australian mental health occupational therapists and social workers. Social Psychiatry and Psychiatric Epidemiology, 39(9), 752-757. https://doi.org/https://doi.org/10.1007/s00127-004-0808-7 Kenworthy, J., Fay, C., Frame, M., & Petree, R. (2014). A meta-analytic review of the relationship between emotional dissonance and emotional exhaustion. Journal of Applied Social Psychology, 44(2), 94-105. https://doi.org/https://doi.org/10.1111/jasp.12211 Martingano, A. J., Stosic, M. D., & Konrath, S. (2026). Different empathy types show opposing associations with burnout: systematic review and meta-analyses. Psychology, Health & Medicine, 31(3), 798-829. https://doi.org/10.1080/13548506.2025.2591859 Michielsen, H. J., Willemsen, T. M., Croon, M. A., de Vries, J., & Van Heck, G. L. (2004). Determinants of general fatigue and emotional exhaustion: a prospective study. Psychology & health, 19(2), 223-235. https://doi.org/10.1080/08870440310001627135 Panksepp, J. B., & Lahvis, G. P. (2011). Rodent empathy and affective neuroscience. Neuroscience and Biobehavioral Reviews, 35(9), 1864–1875. https://doi.org/10.1016/j.neubiorev.2011.05.013 Pooladvand, S., & Hasanzadeh, S. (2023). Impacts of Stress on Workers’ Risk-Taking Behaviors: Cognitive Tunneling and Impaired Selective Attention. Journal of Construction Engineering and Management, 149(8), 04023060. https://doi.org/doi:10.1061/JCEMD4.COENG-13339 Riess, H. (2017). The Science of Empathy. Journal of Patient Experience, 4(2), 74–77. https://doi.org/10.1177/2374373517699267 Schaufeli, W. B. (2017). Applying the job demands-resources model: a “how to” guide to measuring and tackling work engagement and burnout. Organizational Dynamics, 46(2), 120–132. https://doi.org/10.1016/j.orgdyn.2017.04.008 Schneiderman, N., Ironson, G., & Siegel, S. D. (2005). Stress and Health: Psychological, Behavioral, and Biological Determinants. Annual Review of Clinical Psychology, 1(Volume 1, 2005), 607-628. https://doi.org/https://doi.org/10.1146/annurev.clinpsy.1.102803.144141 Seidler, A., Thinschmidt, M., Deckert, S., Then, F., Hegewald, J., Nieuwenhuijsen, K., & Riedel-Heller, S. G. (2014). The role of psychosocial working conditions on burnout and its core component emotional exhaustion – a systematic review. Journal of Occupational Medicine and Toxicology, 9(1), 10. https://doi.org/10.1186/1745-6673-9-10 Shao, M., Li, L., Li, X., Wei, Z., Wang, J., Hong, M., Liu, X., & Meng, J. (2024). The effect of top-down attention on empathy fatigue. Cerebral Cortex (New York, N.Y. 1991), 34(1), Article bhad441. https://doi.org/10.1093/cercor/bhad441 van Dam, A. (2021). A clinical perspective on burnout: diagnosis, classification, and treatment of clinical burnout. European Journal of Work and Organizational Psychology, 30(5), 732-741. https://doi.org/10.1080/1359432X.2021.1948400 Vial, A. C., & Cowgill, C. M. (2022). Heavier Lies Her Crown: Gendered Patterns of Leader Emotional Labor and Their Downstream Effects. Frontiers in Psychology, 13, 849566. https://doi.org/https://doi.org/10.3389/fpsyg.2022.849566 Wharton, A. S. (2009). The Sociology of Emotional Labor. Annual Review of Sociology, 35(1), 147–165. https://doi.org/10.1146/annurev-soc-070308-115944 Wrobel, M. (2013). Can empathy lead to emotional exhaustion in teachers? The mediating role of emotional labor. International Journal of Occupational Medicine and Environmental Health, 26(4), 581–592. https://doi.org/10.2478/s13382-013-0123-1 }} ==External links== '''Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order.''' * [https://www.youtube.com/watch?v=MvSvdGArhxc How to train your empathy muscles] | (YouTube.com | Tedx talks) * [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Burnout]] [[Category:Motivation and emotion/Book/Empathy]] plmarvt6s14ux79kd2ha06n9ym0fqv9 2844566 2844561 2026-11-05T02:17:28Z U3143751 3106785 /* See also */ 2844566 wikitext text/x-wiki {{title|Empathy fatigue and emotional exhaustion:<br>How does sustained empathic engagement contribute to emotional exhaustion?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}} [[File:Exhausted smartphone user.JPG|right|thumb|150px|'''Figure 1'''. An emotionally exhausted Aelin at the end of the work day]] ; Scenario Aelin is a 23-year-old social worker employed at a non-profit organisation providing mental health services to teenagers and young adults. Aelin's clients are aged between 18 and 25, many of whom having experienced rough living conditions. They work with approximately 20 clients each week. During Aelin's first six months in the role, they feel highly motivated and are excited about helping people. They are an excellent active listener, and can easily remember details about their clients' lives. During their sessions, Aelin recognises when a client is distressed and adjusts their body language to warm and understanding in an effort to provide a safe and welcoming environment to hold space for their emotions. As Aelin gains more experience working in this role, their caseload increases, and they are exposed to clients with increasing complexity and past traumas. Aelin wants to provide the best possible outcome for all of their clients, and begins to take their work home with them in the evenings, but finds it starts creeping into their weekends as well. After several months of working this way, Aelin begins to notice they feel exhausted every morning - even before the workday begins. Fast-forward a few more months, and Aelin notices a change in how they start responding to people, including clients. Instead of feeling engaged and actively listening, they feel detached and find their thoughts feeling more frustrated than empathetic. After finishing the work day, Aelin is drained and has no energy for friends or family. With increasing irritability, Aelin begins to worry that they won't be able to continue working as a social-worker. {{RoundBoxBottom}} Aelin is experiencing '''emotional exhaustion''' from a sustained empathic response. Due to the nature of Aelin's work, they are constantly engaged in an empathic response. Empathic responses require cognitive and physical resources to sustain. Due to Aelin taking their work home in the evenings and not taking a break on weekends, they do not have a chance to rest. It is important for Aelin to increase their [[wikipedia:Psychoeducation|psychoeducation]] in order to recognise the how, what, and why of their experiences. Without this education, the problem has exacerbated leading to empathy fatigue and emotional exhaustion. Psychological science can support Aelin to recognise and map the symptoms they are experiencing, and provide coping strategies to minimise the current and future effects of empathy fatigue to improve their overall wellbeing. Understanding these symptoms will also support their role as a social worker, enhancing their work engagement and building skills to support their clients on a day-to-day basis. {{RoundBoxTop|theme=3}} '''Focus questions''' * Why do people become emotionally exhausted? * What is the relationship between empathy fatigue and emotional exhaustion? * What role does emotional labor play in emotional exhaustion? * How does understanding emotional labor and empathy fatigue work to reduce the effects of emotional exhaustion? {{RoundBoxBottom}} == Emotional exhaustion == To understand emotional exhaustion, it will be important to first understand burnout. Burnout is a long-term condition of existing in a constant state of high stress and affects millions of people annually. In the early 2000s, a survey from Australia found that more than 50% of social workers and occupational workers reported high levels of burnout symptoms (Llyod and King, 2004) and a recent report from 2026 found that more than one third of all workers in Australia were experiencing burnout (Chan, 2026). Being in a state of high stress for short periods of time is a useful adaptation for humans, so much so that it has been given a name, [[wikipedia:Eustress|Eustress]]. However, as a long-term condition it leads to individual health issues and can be linked to cardiovascular disease, increased vulnerability to upper respiratory diseases, and chronic inflammation (Schneiderman et al., 2005), and cognitive issues such as [[wikipedia:Inattentional_blindness|cognitive tunneling]] and impaired selective attention (Pooladvand and Hasanzadeh, 2023). Burnout can be defined across three dimensions: emotional exhaustion, [[wikipedia:Depersonalization|depersonalisation]], and reduced feelings of personal accomplishment (Maslach, 1993, as cited in Martingano et al., 2026). Emotional exhaustion may be considered the 'core' component of burnout and refers to the depletion of emotional resources, leading to feelings of overwhelm, overextension, and exhaustion (Martingano et al., 2026; De Beer et al., 2024; Seidler et al., 2014). As a major factor, its symptoms are similar to burnout and stress. Although antecedents of burnout are typically agreed upon (Bakker et al., 2014) and divided into individual (e.g. self-efficacy) and situational categories (e.g. work load), there are many theories that attempt to explain why emotional exhaustion occurs. Two of these theories with good evidence are the [[wikipedia:Conservation_of_resources_theory|Conservation of resources stress model]] (Hobfoll, 1989) and the [[wikipedia:Job_demands-resources_model|Job Demands-Resources model]] (Demerouti et al., 2001). === Conservation of resources stress model === The Conservation of resources stress model (CORT; Hobfall, 1989) is a theory that works to understand and explain why stress occurs. Under the model, humans are motivated to maintain, pursue, or attain resources. Stress, defined as "an individual's perceived pressure or strain" (Hollebeek et al., 2023 p.928), occurs when these resources are threatened or lost, or gained at a net loss. As noted above, resources aren't strictly tied to being physical objects. They can also be emotional states, conditions, or even ideas. For example, someone may place value on enjoying a calm, quiet Sunday morning at home. If their neighbour starts using power tools loudly next door, it would disrupt this state and induce stress in the individual. The CORT model focuses primarily on two principles. The first principle is that there is more harm in losing resources than there is in gaining them back (Halbesleben et al., 2014). This is known as loss salience, and some research suggests that the pain of loss is felt twice as strong as the pleasure of gain (Thaler, 2000). Looking at our previous example, imagine that the person who enjoys a quiet Sunday morning at home equally enjoys the sound of a loud car engine. Even though the person enjoys the sound of car engines, if the neighbour started revving their car loudly instead of using power tools, it would still disrupt their desired state of quiet and lead to an increase of stress. The second principle is that people must invest existing resources in order to gain or protect resources (Halbesleben et al., 2014). This resource investment includes recovering from resource loss. Coping is how people manage or overcome a difficult situation, such as protecting or recovering from resource loss. Coping can be broken down into different strategies that people use, but not all strategies are equal (Ito and Brotheridge, 2003). They are divided into two categories: control or escape strategies. Control strategies are considered better at reducing the strain from stress and consist of positive thinking, direct action, and help seeking (Havlovic & Keenan, 1991, as cited in Ito and Brotheridge, 2003). The most useful control strategies incorporate a person's [[locus of control]], emphasise seeking support from co-workers, and highlight the importance of autonomy in decision making. Alarcon (2011) conducted a meta-analysis of 231 studies in a work setting that used the [[wikipedia:Maslach_Burnout_Inventory|Maslach Burnout Inventory]] (MBI) to review the relationship between these principles and burnout to confirm their validity. Although the study had no specific mention of demographics, it was noted that multiple translations of the MBI were used including Dutch (k=51), German (k=7), Italian (k=2), and French (k=2). The findings indicate that workload is positively related to emotional exhaustion (''ρ'' = .49, ''k'' = 86, ''N'' = 51,529) and that control (''ρ'' = − .26, ''k'' = 56, ''N'' = 33,297) and autonomy (''ρ'' = − .24, ''k'' = 34, ''N'' = 17,165) were each negatively associated with emotional exhaustion. The difference of ''ρ'' values is consistent with Thaler (2000) and supports that the benefit of gaining is approximately half that of losing resources. === Job demands-resources model === Another model that attempts to understand why people become emotionally exhausted is the Job Demands-Resources model (JD-R; Demerouti et al., 2001). The JD-R model discusses two concepts: job demands and job resources. Demands and resources are both related to the physical, social, or psychological aspects of a job, but differ in their impact on a person. Job demands require physical or mental effort and exhaust a person's energy supply. Job resources typically support achieving goals, reduce the effort cost of job demands, and stimulate personal growth (Richter and Hacker, 1998, as cited in Demerouti et al., 2001). For example, an unsustainable workload would be considered a job demand, and receiving help and support from others would be considered a resource. The JD-R has two primary processes for how demands and resources interact with one another (Schaufeli, 2017). The stress process occurs when job demands are high and resources are low. As stress increases strain and pressure on an individual, it leads to the exhaustion of their energy eventually, eventually leading to emotional exhaustion. The second process is the motivational process. This process occurs when there is an excess of job resources. Crucially, the motivational process does not require low job demands. With enough job resources, a person will be energised and engaged, reframing job demands as a challenge to overcome. The JD-R model has had extensive research to confirm its validity. A meta-analysis by Lesener et al. (2019) looked at 74 longitudinal studies of the JD-R with a strong average sample size (n=450) against the assumptions as described by Schaufeli (2017). Although most of the studies appear to come from [[wikipedia:Psychology#WEIRD_bias|WEIRD]] countries (k=60) there were a good number from Asia and Africa (k=13) and a mix of genders. The study confirmed that across the literature, job resources predict burnout negatively (''β'' = −.17; ''p'' < .001) and job demands predict burnout over time (''β'' = .10; ''p'' < .001). Although job resources seem like a cure-all for demands, it's important to note that they do not retroactively treat emotional exhaustion. Under the JD-R model, burnout must be reduced by lowering job demands (Gynning et al., 2025). However, once someone has recovered, it is possible to increase job resources alongside job demands to prevent future burnout and support [[wikipedia:Work_engagement|work engagement]]. Work engagement (engagement) has been discussed as the opposite of burnout (Bakker et al., 2014). Engagement is how people express themselves when performing a role. A role being a pattern of behaviour that a person goes through based on the social expectations of that role, similar to how an actor takes on a role for a movie. For example, when attending a typical restaurant, it is socially expected that a person will engage with the waiter and provide them with their meal order to take to the kitchen. This person is performing the script of a restaurant patron. In the context of a workplace, an employee with high engagement may appear to be energised, enthusiastic, and become absorbed in their work losing track of time (Bakkar et al, 2014). {{RoundBoxTop|theme=3}} '''Why do we feel emotionally exhausted''' Under the JD-R and CORT models, emotional exhaustion occurs when the demand cost outweighs the resources available for an extended period of time, eventually depleting someone's energy reserves. Aelin is experiencing burnout because they increased their job demands by taking on more clients, and has experienced a loss in resources by reducing their free time and social support network. {{RoundBoxBottom}} == Empathy fatigue == You may be familiar with the term [[empathy]]. It is often described in terms of supporting others. For example, when someone close to you experiences a loss and is grieving, you may empathise with them and experience a negative emotion akin to grief too. However, empathy is not exclusive to so called negative emotions. For example, compersion is a positive empathic response when a loved one experiences joy. Parents may experience compersion when something joyful happens to their child. At its core, empathy is the individual ability to imagine or understand another person's feelings and/or emotions (Riess, 2017; Shao et al., 2024). Much like typical physical or cognitive processes, such as running or problem solving, empathising will drain energy and resources. Further, and similar to the difference in energy expenditure between running and jogging, there are distinct types of empathy that each have different energy costs. A way to make a distinction between these two types may be considering them as the "what" and "how". Research makes a clear distinction between two types of empathy: affective and cognitive empathy. Affective empathy occurs when a person has an emotional reaction to someone else's mental state (Shao et al., 2024), whereas cognitive empathy occurs when a person understands someone else's feelings or emotions (Cui et al., 2022). === Affective empathy === Affective empathy is the "how" and focuses on the feeling aspect of empathy. This is a more common understanding of empathy, and utilises * Integrating the two - "affective component is the essence of empathy, and the cognitive component is a process through which that essence comes into being" (Bošnjaković & Radionov, 2018) ** (Author's note: change quote into paraphrasing) Emotional contagion is the idea that emotions are contagious. as the theory to moderate surface/deep acting * Emotional contagion (Panksepp and Lahvis, 2011) ** emotions can spread (expand, combines empathy and acting) ** “the tendency to automatically mimic and synchronize expressions, vocalizations, postures, and movements with those of another person’s and, consequently, to converge emotionally.” (Elaine Hatfield, 1993) * Affective process theory looks at unifying the mechanisms of emotional contagion into three types. Convergent linkage, Divergent linkage, and Complementary linkage (Elfenbein, 2014). Storytelling is a way to promote position emotions (Wei et al., 2024). === Cognitive empathy === Cognitive empathy is the "what" and focuses on a person trying to tangibly understand what another person is feeling. This information is processed through a top-down approach (), and primarily utilises the pre-frontal cortex (Shao et al., 2024). This allows the empathiser to feel the emotion from a cognitive, problem solving perspective. As cognitive empathy is processed by the pre-frontal cortex, there is less association with feeling the emotion across the body hence reducing the physiological response and is linked to lower energy expenditure. Paragraph on compassion (use that study in your one-note and look for an additional meta-analysis) === Empathy fatigue and compassion fatigue === * Distinction between empathy and compassion fatigue * Compassion fatigue is closely related to Secondary traumatic stress usually found in healthcare professionals (Figley, 2002) ** Compassion fatigue is to Empathy fatigue as emotional exhaustion is to burnout * Make it clear that Aelin is experiencing Compassion Fatigue vs Empathy fatigue (they are the same but different) (Author's note: include an update to the scenario here!) '''Empathy fatigue questions'''. Choose your answers and click "Submit": <quiz display=simple> {Surface acting is when the observer actually feels the emotion: |type="()"} - True + False {Compassion fatigue and empathy fatigue are synonymous: |type="()"} - True + False </quiz> {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' Under the JD-R and CORT models, a sustained empathic response is a long-term job demand that uses energy to protect existing resources. As Aelin has had their resources is experiencing a * Under the CORT model, a sustained empathic response uses energy to protect existing resources. As resources become low and burnout sets in, cognitive focus narrows leading to empathy fatigue. {{RoundBoxBottom}} == Emotional labor as a mediating factor between empathy fatigue and emotional exhaustion == As a society there are expected [https://en.wikipedia.org/wiki/Social_norm social norms] in behaviour when expressing emotion, regardless if someone is conscious of the process of matching these norms or not. For example, laughing during an emotional speech at a funeral would be considered a rude or offensive expression from the other attendees. As social creatures, humans need a way to regulate socially appropriate emotional expression. This is the essence of Emotional labor: "the regulation of felt and expressed emotions (whether effortful or not) to match these emotional display rules with the objective of fulfilling organizational goals" (Vial & Cowgill, 2022) What is emotional labor (Hochschild, 2012). This section is really where all the above research comes together, and should highlight how the models of stress (JD-R and CORT), relate to the types of empathy (affective/contagion vs cognitive/compassion), and how types of acting (expectations of what you should feel (Emotional labor!!!!) vs accepting what you actually feel) contribute and '''''mediate''''' the relationship between empathy fatigue and emotional exhaustion. This is the penultimate section and should reflect that accordingly: "This is what is happening, and this is how and why its happening" === Types of acting === Surface acting * This is where emotive dissonance comes into play * Leads to emotional exhaustion (Kenworthy et al., 2014) * "Should" language - in this situation, this is what I ''should'' feel * Deep acting ** People who 'deep act', as in genuinely, intrinsically adjust the way they feel (match the feelings they have with the feelings that are appropriate to display in a given situation) often report positive relationships with emotional labor ** Actual language - in this situation, this is what I ''actually'' feel === The mediating effect of emotional labor === Surface acting does not mediate the effect (high surface acting leads to emotional exhaustion) whereas the inverse is true for deep acting (Wrobel, 2013) Discuss the differences in gender norms and expectations in relationships Emotional Expression and Emotional Processing (Hoyt et al., 2024) {{RoundBoxTop|theme=3}} '''What role does emotional labor play in emotional exhaustion''' * In relation to the CORT and JD-R theories, emotional labor acts as a job demand and expenditure of resources to protect existing resources {{RoundBoxBottom}} == Strategies for combating empathy fatigue and emotional exhaustion == Pulling this all together, write a short introduction that describes how to identify which phase you should look at and then use the following table with resources. If you are already emotionally exhausted you need to start at Phase 1. * Talk about emotion regulation (Gross, 1998) and mood regulation strategies as a way to deal with coping (Wrobel, 2013) (Author's note: need more research/references for this section) * Talk about breaks ** Rest and downtime from not performing emotional labor regenerates * How do we reduce the amount of emotional labor we're performing (Jeung et al., 2018) ** Communication/boundaries ** Reducing surface acting (allowing ourselves to feel how we actually feel and reduce emotive dissonance) '''Table 1. Treatment phases of clinical burnout (based on Table 2 from van Dam, A. 2021)''' {| class="wikitable" style="margin: auto; |- ! Phase !! Treatment goal !! Therapeutic interventions |- | '''Phase 1: Crisis''' || Recognition of the patient that the problems are serious, and that serious action is required || Psychoeducation Sick leave Dropping domestic tasks and social obligations Inform social network |- | '''Phase 2: Recovery''' || Recovery of the (physiological) stress system to normal allostatic stress levels || Registration of stress and activities Relaxtion exercises Mindfulness healthy lifestyle advices Graded exercise Gradually resume activities Psychoeducation social network and employer Gradual return to work |- | '''Phase 3: Prevention, learning from the past''' || Acquiring insight and skills to prevent relapse in clinical burnout || Analysing stressful situations Analysing dysfunctional thought patterns Learning new coping skills Learning social skills Job crafting Time management Making choices regarding career and personal life |- | '''Phase 4: Post burnout growth''' || Improving sustainable quality of life. || Setting priorities regarding quality of life and interpersonal relationships. |}{{RoundBoxTop|theme=3}}'''How does understanding emotional labor and empathy fatigue work to reduce the effects of emotional exhaustion''' What about Aelin, what can they do? Work on skills to enhance their knowledge of cognitive empathy and reduce their reliance on affective empathy to provide better outcomes for their patients. *Psychoeducation has been shown to reduce the effects of emotional labor and empathy fatigue on emotional exhaustion {{RoundBoxBottom}} ==Conclusion== * Summarise what empathy fatigue is, and why people become emotionally exhausted (+ the role of emotional labor) * Summarise the symptoms and how to recognise them in yourself and someone else * Emphasise and summarise psychological strategies to support someone going through emotional exhaustion * Summarise ways to prevent emotional exhaustion from happening * Overall message is: *# What is emotional exhaustion *# how do you recognise it *# how do you treat it *# how do you prevent it * Link back to the focus questions ==See also== * [[Motivation and emotion/Book/2025/Metacognition and emotional regulation|Metacognition and emotional regulation]] (Book chapter, 2025) * [[wikipedia:Conservation_of_resources_theory|Conservation of resources theory]] (Wikipedia) * [[wikipedia:Job_demands-resources_model|Job demands-resources model]] (Wikipedia) * Emotional Contagion * Perspective taking * Emotion regulation *[[wikipedia:List_of_psychological_research_methods|List of psychological research methods]] (Wikipedia) ==References== {{Hanging indent|1= Bošnjaković, J., & Radionov, T. (2018). Empathy: Concepts, Theories and Neuroscientific Basis. Alcoholism (Zagreb), 54(2), 123–150. https://doi.org/10.20471/de..2018.54.02.04 Chan, F. M. (2026). The State of Workplace Burnout 2026: The Workplace Divide. Infinite Potential, Sydney, Australia. https://www.ahri.com.au/wp-content/uploads/The-State-of-Workplace-Burnout-2026-Final.pdf Cui, F., Huang, X., Jing, Y., Luo, Y., Liu, J., & Gu, R. (2022). How resource sharing resists scarcity: the role of cognitive empathy and its neurobiological mechanisms. Cerebral Cortex (New York, N.Y. 1991), 32(23), 5330–5342. https://doi.org/10.1093/cercor/bhac017 De Beer, L. T., van der Vaart, L., Escaffi-Schwarz, M., De Witte, H., & Schaufeli, W. B. (2024). Maslach Burnout Inventory - General Survey: A Systematic Review and Meta-Analysis of Measurement Properties. European journal of psychological assessment : official organ of the European Association of Psychological Assessment, 40(5), 360-375. https://doi.org/10.1027/1015-5759/a000797 Demerouti, E., Bakker, A. B., Nachreiner, F., & Schaufeli, W. B. (2001). The Job Demands-Resources Model of Burnout. Journal of Applied Psychology, 86(3), 499–512. https://doi.org/10.1037/0021-9010.86.3.499 Figley, C. R. (2002). Compassion fatigue: Psychotherapists’ chronic lack of self care. Journal of Clinical Psychology, 58(11), 1433–1441. https://doi.org/10.1002/jclp.10090 Grandey, A. A. (2003). When ‘the show must go on’: surface acting and deep acting as determinants of emotional exhaustion and peer-rated service delivery. Academy of Management Journal, 46(1), 86–96. https://doi.org/10.5465/30040678 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 Halbesleben, J. R. B., Neveu, J.-P., Paustian-Underdahl, S. C., & Westman, M. (2014). Getting to the “COR”: Understanding the Role of Resources in Conservation of Resources Theory. Journal of Management, 40(5), 1334–1364. https://doi.org/10.1177/0149206314527130 Hatfield, E., Cacioppo, J. T., & Rapson, R. L. (1993). Emotional Contagion. Current Directions in Psychological Science : A Journal of the American Psychological Society, 2(3), 96–99. https://doi.org/10.1111/1467-8721.ep10770953 Hollebeek, L. D., Hammedi, W., & Sprott, D. E. (2023). Consumer engagement, stress, and conservation of resources theory: A review, conceptual development, and future research agenda. Psychology & marketing, 40(5), 926-937. https://doi.org/10.1002/mar.21807 Hobfoll, S. E. (1989). Conservation of Resources: A New Attempt at Conceptualizing Stress. The American Psychologist, 44(3), 513–524. https://doi.org/10.1037/0003-066X.44.3.513 Hochschild, A. R. (2012). The managed heart : commercialization of human feeling (Updated, with a new preface.). University of California Press. Jeung, D.-Y., Kim, C., & Chang, S.-J. (2018). Emotional Labor and Burnout: A Review of the Literature. Yonsei Medical Journal, 59(2), 187–193. https://doi.org/10.3349/ymj.2018.59.2.187 Lavelle, J. J., Rupp, D. E., Herda, D. N., Pandey, A., & Lauck, J. R. (2021). Customer Injustice and Employee Performance: Roles of Emotional Exhaustion, Surface Acting, and Emotional Demands–Abilities Fit. Journal of Management, 47(3), Article 0149206319869426. https://doi.org/10.1177/0149206319869426 Lloyd, C., & King, R. (2004). A survey of burnout among Australian mental health occupational therapists and social workers. Social Psychiatry and Psychiatric Epidemiology, 39(9), 752-757. https://doi.org/https://doi.org/10.1007/s00127-004-0808-7 Kenworthy, J., Fay, C., Frame, M., & Petree, R. (2014). A meta-analytic review of the relationship between emotional dissonance and emotional exhaustion. Journal of Applied Social Psychology, 44(2), 94-105. https://doi.org/https://doi.org/10.1111/jasp.12211 Martingano, A. J., Stosic, M. D., & Konrath, S. (2026). Different empathy types show opposing associations with burnout: systematic review and meta-analyses. Psychology, Health & Medicine, 31(3), 798-829. https://doi.org/10.1080/13548506.2025.2591859 Michielsen, H. J., Willemsen, T. M., Croon, M. A., de Vries, J., & Van Heck, G. L. (2004). Determinants of general fatigue and emotional exhaustion: a prospective study. Psychology & health, 19(2), 223-235. https://doi.org/10.1080/08870440310001627135 Panksepp, J. B., & Lahvis, G. P. (2011). Rodent empathy and affective neuroscience. Neuroscience and Biobehavioral Reviews, 35(9), 1864–1875. https://doi.org/10.1016/j.neubiorev.2011.05.013 Pooladvand, S., & Hasanzadeh, S. (2023). Impacts of Stress on Workers’ Risk-Taking Behaviors: Cognitive Tunneling and Impaired Selective Attention. Journal of Construction Engineering and Management, 149(8), 04023060. https://doi.org/doi:10.1061/JCEMD4.COENG-13339 Riess, H. (2017). The Science of Empathy. Journal of Patient Experience, 4(2), 74–77. https://doi.org/10.1177/2374373517699267 Schaufeli, W. B. (2017). Applying the job demands-resources model: a “how to” guide to measuring and tackling work engagement and burnout. Organizational Dynamics, 46(2), 120–132. https://doi.org/10.1016/j.orgdyn.2017.04.008 Schneiderman, N., Ironson, G., & Siegel, S. D. (2005). Stress and Health: Psychological, Behavioral, and Biological Determinants. Annual Review of Clinical Psychology, 1(Volume 1, 2005), 607-628. https://doi.org/https://doi.org/10.1146/annurev.clinpsy.1.102803.144141 Seidler, A., Thinschmidt, M., Deckert, S., Then, F., Hegewald, J., Nieuwenhuijsen, K., & Riedel-Heller, S. G. (2014). The role of psychosocial working conditions on burnout and its core component emotional exhaustion – a systematic review. Journal of Occupational Medicine and Toxicology, 9(1), 10. https://doi.org/10.1186/1745-6673-9-10 Shao, M., Li, L., Li, X., Wei, Z., Wang, J., Hong, M., Liu, X., & Meng, J. (2024). The effect of top-down attention on empathy fatigue. Cerebral Cortex (New York, N.Y. 1991), 34(1), Article bhad441. https://doi.org/10.1093/cercor/bhad441 van Dam, A. (2021). A clinical perspective on burnout: diagnosis, classification, and treatment of clinical burnout. European Journal of Work and Organizational Psychology, 30(5), 732-741. https://doi.org/10.1080/1359432X.2021.1948400 Vial, A. C., & Cowgill, C. M. (2022). Heavier Lies Her Crown: Gendered Patterns of Leader Emotional Labor and Their Downstream Effects. Frontiers in Psychology, 13, 849566. https://doi.org/https://doi.org/10.3389/fpsyg.2022.849566 Wharton, A. S. (2009). The Sociology of Emotional Labor. Annual Review of Sociology, 35(1), 147–165. https://doi.org/10.1146/annurev-soc-070308-115944 Wrobel, M. (2013). Can empathy lead to emotional exhaustion in teachers? The mediating role of emotional labor. International Journal of Occupational Medicine and Environmental Health, 26(4), 581–592. https://doi.org/10.2478/s13382-013-0123-1 }} ==External links== '''Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order.''' * [https://www.youtube.com/watch?v=MvSvdGArhxc How to train your empathy muscles] | (YouTube.com | Tedx talks) * [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Burnout]] [[Category:Motivation and emotion/Book/Empathy]] g26ozi1f71bv0i95p8lzhy258db0m27 2844571 2844566 2026-11-05T02:35:44Z U3143751 3106785 Updated empathy fatigue section 2844571 wikitext text/x-wiki {{title|Empathy fatigue and emotional exhaustion:<br>How does sustained empathic engagement contribute to emotional exhaustion?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}} [[File:Exhausted smartphone user.JPG|right|thumb|150px|'''Figure 1'''. An emotionally exhausted Aelin at the end of the work day]] ; Scenario Aelin is a 23-year-old social worker employed at a non-profit organisation providing mental health services to teenagers and young adults. Aelin's clients are aged between 18 and 25, many of whom having experienced rough living conditions. They work with approximately 20 clients each week. During Aelin's first six months in the role, they feel highly motivated and are excited about helping people. They are an excellent active listener, and can easily remember details about their clients' lives. During their sessions, Aelin recognises when a client is distressed and adjusts their body language to warm and understanding in an effort to provide a safe and welcoming environment to hold space for their emotions. As Aelin gains more experience working in this role, their caseload increases, and they are exposed to clients with increasing complexity and past traumas. Aelin wants to provide the best possible outcome for all of their clients, and begins to take their work home with them in the evenings, but finds it starts creeping into their weekends as well. After several months of working this way, Aelin begins to notice they feel exhausted every morning - even before the workday begins. Fast-forward a few more months, and Aelin notices a change in how they start responding to people, including clients. Instead of feeling engaged and actively listening, they feel detached and find their thoughts feeling more frustrated than empathetic. After finishing the work day, Aelin is drained and has no energy for friends or family. With increasing irritability, Aelin begins to worry that they won't be able to continue working as a social-worker. {{RoundBoxBottom}} Aelin is experiencing '''emotional exhaustion''' from a sustained empathic response. Due to the nature of Aelin's work, they are constantly engaged in an empathic response. Empathic responses require cognitive and physical resources to sustain. Due to Aelin taking their work home in the evenings and not taking a break on weekends, they do not have a chance to rest. It is important for Aelin to increase their [[wikipedia:Psychoeducation|psychoeducation]] in order to recognise the how, what, and why of their experiences. Without this education, the problem has exacerbated leading to empathy fatigue and emotional exhaustion. Psychological science can support Aelin to recognise and map the symptoms they are experiencing, and provide coping strategies to minimise the current and future effects of empathy fatigue to improve their overall wellbeing. Understanding these symptoms will also support their role as a social worker, enhancing their work engagement and building skills to support their clients on a day-to-day basis. {{RoundBoxTop|theme=3}} '''Focus questions''' * Why do people become emotionally exhausted? * What is the relationship between empathy fatigue and emotional exhaustion? * What role does emotional labor play in emotional exhaustion? * How does understanding emotional labor and empathy fatigue work to reduce the effects of emotional exhaustion? {{RoundBoxBottom}} == Emotional exhaustion == To understand emotional exhaustion, it will be important to first understand burnout. Burnout is a long-term condition of existing in a constant state of high stress and affects millions of people annually. In the early 2000s, a survey from Australia found that more than 50% of social workers and occupational workers reported high levels of burnout symptoms (Llyod and King, 2004) and a recent report from 2026 found that more than one third of all workers in Australia were experiencing burnout (Chan, 2026). Being in a state of high stress for short periods of time is a useful adaptation for humans, so much so that it has been given a name, [[wikipedia:Eustress|Eustress]]. However, as a long-term condition it leads to individual health issues and can be linked to cardiovascular disease, increased vulnerability to upper respiratory diseases, and chronic inflammation (Schneiderman et al., 2005), and cognitive issues such as [[wikipedia:Inattentional_blindness|cognitive tunneling]] and impaired selective attention (Pooladvand and Hasanzadeh, 2023). Burnout can be defined across three dimensions: emotional exhaustion, [[wikipedia:Depersonalization|depersonalisation]], and reduced feelings of personal accomplishment (Maslach, 1993, as cited in Martingano et al., 2026). Emotional exhaustion may be considered the 'core' component of burnout and refers to the depletion of emotional resources, leading to feelings of overwhelm, overextension, and exhaustion (Martingano et al., 2026; De Beer et al., 2024; Seidler et al., 2014). As a major factor, its symptoms are similar to burnout and stress. Although antecedents of burnout are typically agreed upon (Bakker et al., 2014) and divided into individual (e.g. self-efficacy) and situational categories (e.g. work load), there are many theories that attempt to explain why emotional exhaustion occurs. Two of these theories with good evidence are the [[wikipedia:Conservation_of_resources_theory|Conservation of resources stress model]] (Hobfoll, 1989) and the [[wikipedia:Job_demands-resources_model|Job Demands-Resources model]] (Demerouti et al., 2001). === Conservation of resources stress model === The Conservation of resources stress model (CORT; Hobfall, 1989) is a theory that works to understand and explain why stress occurs. Under the model, humans are motivated to maintain, pursue, or attain resources. Stress, defined as "an individual's perceived pressure or strain" (Hollebeek et al., 2023 p.928), occurs when these resources are threatened or lost, or gained at a net loss. As noted above, resources aren't strictly tied to being physical objects. They can also be emotional states, conditions, or even ideas. For example, someone may place value on enjoying a calm, quiet Sunday morning at home. If their neighbour starts using power tools loudly next door, it would disrupt this state and induce stress in the individual. The CORT model focuses primarily on two principles. The first principle is that there is more harm in losing resources than there is in gaining them back (Halbesleben et al., 2014). This is known as loss salience, and some research suggests that the pain of loss is felt twice as strong as the pleasure of gain (Thaler, 2000). Looking at our previous example, imagine that the person who enjoys a quiet Sunday morning at home equally enjoys the sound of a loud car engine. Even though the person enjoys the sound of car engines, if the neighbour started revving their car loudly instead of using power tools, it would still disrupt their desired state of quiet and lead to an increase of stress. The second principle is that people must invest existing resources in order to gain or protect resources (Halbesleben et al., 2014). This resource investment includes recovering from resource loss. Coping is how people manage or overcome a difficult situation, such as protecting or recovering from resource loss. Coping can be broken down into different strategies that people use, but not all strategies are equal (Ito and Brotheridge, 2003). They are divided into two categories: control or escape strategies. Control strategies are considered better at reducing the strain from stress and consist of positive thinking, direct action, and help seeking (Havlovic & Keenan, 1991, as cited in Ito and Brotheridge, 2003). The most useful control strategies incorporate a person's [[locus of control]], emphasise seeking support from co-workers, and highlight the importance of autonomy in decision making. Alarcon (2011) conducted a meta-analysis of 231 studies in a work setting that used the [[wikipedia:Maslach_Burnout_Inventory|Maslach Burnout Inventory]] (MBI) to review the relationship between these principles and burnout to confirm their validity. Although the study had no specific mention of demographics, it was noted that multiple translations of the MBI were used including Dutch (k=51), German (k=7), Italian (k=2), and French (k=2). The findings indicate that workload is positively related to emotional exhaustion (''ρ'' = .49, ''k'' = 86, ''N'' = 51,529) and that control (''ρ'' = − .26, ''k'' = 56, ''N'' = 33,297) and autonomy (''ρ'' = − .24, ''k'' = 34, ''N'' = 17,165) were each negatively associated with emotional exhaustion. The difference of ''ρ'' values is consistent with Thaler (2000) and supports that the benefit of gaining is approximately half that of losing resources. === Job demands-resources model === Another model that attempts to understand why people become emotionally exhausted is the Job Demands-Resources model (JD-R; Demerouti et al., 2001). The JD-R model discusses two concepts: job demands and job resources. Demands and resources are both related to the physical, social, or psychological aspects of a job, but differ in their impact on a person. Job demands require physical or mental effort and exhaust a person's energy supply. Job resources typically support achieving goals, reduce the effort cost of job demands, and stimulate personal growth (Richter and Hacker, 1998, as cited in Demerouti et al., 2001). For example, an unsustainable workload would be considered a job demand, and receiving help and support from others would be considered a resource. The JD-R has two primary processes for how demands and resources interact with one another (Schaufeli, 2017). The stress process occurs when job demands are high and resources are low. As stress increases strain and pressure on an individual, it leads to the exhaustion of their energy eventually, eventually leading to emotional exhaustion. The second process is the motivational process. This process occurs when there is an excess of job resources. Crucially, the motivational process does not require low job demands. With enough job resources, a person will be energised and engaged, reframing job demands as a challenge to overcome. The JD-R model has had extensive research to confirm its validity. A meta-analysis by Lesener et al. (2019) looked at 74 longitudinal studies of the JD-R with a strong average sample size (n=450) against the assumptions as described by Schaufeli (2017). Although most of the studies appear to come from [[wikipedia:Psychology#WEIRD_bias|WEIRD]] countries (k=60) there were a good number from Asia and Africa (k=13) and a mix of genders. The study confirmed that across the literature, job resources predict burnout negatively (''β'' = −.17; ''p'' < .001) and job demands predict burnout over time (''β'' = .10; ''p'' < .001). Although job resources seem like a cure-all for demands, it's important to note that they do not retroactively treat emotional exhaustion. Under the JD-R model, burnout must be reduced by lowering job demands (Gynning et al., 2025). However, once someone has recovered, it is possible to increase job resources alongside job demands to prevent future burnout and support [[wikipedia:Work_engagement|work engagement]]. Work engagement (engagement) has been discussed as the opposite of burnout (Bakker et al., 2014). Engagement is how people express themselves when performing a role. A role being a pattern of behaviour that a person goes through based on the social expectations of the situation, similar to how an actor takes on a role for a movie. For example, when attending a typical restaurant, it is socially expected that a person will engage with the waiter and provide them with their meal order to take to the kitchen. This person is performing the script of a restaurant patron. In the context of a workplace, an employee with high engagement may appear to be energised, enthusiastic, and become absorbed in their work losing track of time (Bakkar et al, 2014). {{RoundBoxTop|theme=3}} '''Why do we feel emotionally exhausted''' Under the JD-R and CORT models, emotional exhaustion occurs when the demand cost outweighs the resources available for an extended period of time, eventually depleting someone's energy reserves. Aelin is experiencing burnout because they increased their job demands by taking on more clients and spending more time working, and has experienced a loss in resources by reducing their free time and social support network. {{RoundBoxBottom}} == Empathy fatigue == You may be familiar with the term [[empathy]]. It is often described in terms of supporting others by understanding or mirroring their emotional experience. For example, when someone close to you experiences a loss and is grieving, you may empathise with them and personally experience a negative emotion akin to grief too. However, empathy is not exclusive to so called negative emotions. For example, compersion is a positive empathic response when a loved one experiences joy. Parents may experience compersion when something joyful happens to their child. At its core, empathy is the individual ability to imagine or understand another person's feelings and/or emotions (Riess, 2017; Shao et al., 2024). Much like typical physical or cognitive processes, such as running or problem solving, empathising will drain energy and resources. Further, and similar to the difference in energy expenditure between running and jogging, there are distinct types of empathy that each have different energy costs. A way to make a distinction between these two types may be considering them as the "what" and "how". Research makes a clear distinction between two types of empathy: affective and cognitive empathy. Affective empathy occurs when a person has an emotional reaction to someone else's mental state (Shao et al., 2024), whereas cognitive empathy occurs when a person understands someone else's feelings or emotions (Cui et al., 2022). Integrating the two - "affective component is the essence of empathy, and the cognitive component is a process through which that essence comes into being" (Bošnjaković & Radionov, 2018) === Affective empathy === Affective empathy is the "how" and focuses on the feeling aspect of empathy. Personally experiencing another person's grief would be considered as affective empathy. Research links it to emotional contagion. Emotional contagion (Panksepp and Lahvis, 2011) is a theory that suggests that emotions are contagious and can spread from one person to another. Hatfield (1993) coined the term and suggested that emotional contagion is an automatic mimicry process that synchronises expressions, vocalisations, postures, and movements with those of another person. Affective process theory looks at organising emotional contagion into three processes. Convergent linkage, Divergent linkage, and Complementary linkage (Elfenbein, 2014). Storytelling is a way to promote position emotions (Wei et al., 2024). === Cognitive empathy === Cognitive empathy is the "what" and focuses on a person trying to tangibly understand what another person is feeling. This information is processed through a top-down approach (), and primarily utilises the pre-frontal cortex (Shao et al., 2024). This allows the empathiser to feel the emotion from a cognitive, problem solving perspective. As cognitive empathy is processed by the pre-frontal cortex, there is less association with feeling the emotion across the body hence reducing the physiological response and is linked to lower energy expenditure. Our example from before, compersion could be considered an empathic response that results from cognitive empathy. Cognitive empathy also lends itself to compassion (use that study in your one-note and look for an additional meta-analysis). === Compassion fatigue === Often compassion fatigue is referred to as empathy fatigue, but is a distinct theory in and of itself. Compassion fatigue is closely related to, and could be more aligned to becoming synonymous with, secondary traumatic stress. Usually found in healthcare professionals (Figley, 2002), secondary traumatic stress occurs when ... '''Test your knowledge''' Choose your answers and click "Submit": <quiz display=simple> {Surface acting is when the observer actually feels the emotion: |type="()"} - True + False {Compassion fatigue and empathy fatigue are synonymous: |type="()"} - True + False </quiz> {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' Under the JD-R and CORT models, a sustained empathic response is a long-term job demand that uses energy to protect existing resources. As Aelin has had their resources is experiencing a * Under the CORT model, a sustained empathic response uses energy to protect existing resources. As resources become low and burnout sets in, cognitive focus narrows leading to empathy fatigue. {{RoundBoxBottom}} == Emotional labor as a mediating factor between empathy fatigue and emotional exhaustion == As a society there are expected [https://en.wikipedia.org/wiki/Social_norm social norms] in behaviour when expressing emotion, regardless if someone is conscious of the process of matching these norms or not. For example, laughing during an emotional speech at a funeral would be considered a rude or offensive expression from the other attendees. As social creatures, humans need a way to regulate socially appropriate emotional expression. This is the essence of Emotional labor: "the regulation of felt and expressed emotions (whether effortful or not) to match these emotional display rules with the objective of fulfilling organizational goals" (Vial & Cowgill, 2022) What is emotional labor (Hochschild, 2012). This section is really where all the above research comes together, and should highlight how the models of stress (JD-R and CORT), relate to the types of empathy (affective/contagion vs cognitive/compassion), and how types of acting (expectations of what you should feel (Emotional labor!!!!) vs accepting what you actually feel) contribute and '''''mediate''''' the relationship between empathy fatigue and emotional exhaustion. This is the penultimate section and should reflect that accordingly: "This is what is happening, and this is how and why its happening" === Types of acting === Surface acting * This is where emotive dissonance comes into play * Leads to emotional exhaustion (Kenworthy et al., 2014) * "Should" language - in this situation, this is what I ''should'' feel * Deep acting ** People who 'deep act', as in genuinely, intrinsically adjust the way they feel (match the feelings they have with the feelings that are appropriate to display in a given situation) often report positive relationships with emotional labor ** Actual language - in this situation, this is what I ''actually'' feel === The mediating effect of emotional labor === Surface acting does not mediate the effect (high surface acting leads to emotional exhaustion) whereas the inverse is true for deep acting (Wrobel, 2013) Discuss the differences in gender norms and expectations in relationships Emotional Expression and Emotional Processing (Hoyt et al., 2024) {{RoundBoxTop|theme=3}} '''What role does emotional labor play in emotional exhaustion''' * In relation to the CORT and JD-R theories, emotional labor acts as a job demand and expenditure of resources to protect existing resources {{RoundBoxBottom}} == Strategies for combating empathy fatigue and emotional exhaustion == Pulling this all together, write a short introduction that describes how to identify which phase you should look at and then use the following table with resources. If you are already emotionally exhausted you need to start at Phase 1. * Talk about emotion regulation (Gross, 1998) and mood regulation strategies as a way to deal with coping (Wrobel, 2013) (Author's note: need more research/references for this section) * Talk about breaks ** Rest and downtime from not performing emotional labor regenerates * How do we reduce the amount of emotional labor we're performing (Jeung et al., 2018) ** Communication/boundaries ** Reducing surface acting (allowing ourselves to feel how we actually feel and reduce emotive dissonance) '''Table 1. Treatment phases of clinical burnout (based on Table 2 from van Dam, A. 2021)''' {| class="wikitable" style="margin: auto; |- ! Phase !! Treatment goal !! Therapeutic interventions |- | '''Phase 1: Crisis''' || Recognition of the patient that the problems are serious, and that serious action is required || Psychoeducation Sick leave Dropping domestic tasks and social obligations Inform social network |- | '''Phase 2: Recovery''' || Recovery of the (physiological) stress system to normal allostatic stress levels || Registration of stress and activities Relaxtion exercises Mindfulness healthy lifestyle advices Graded exercise Gradually resume activities Psychoeducation social network and employer Gradual return to work |- | '''Phase 3: Prevention, learning from the past''' || Acquiring insight and skills to prevent relapse in clinical burnout || Analysing stressful situations Analysing dysfunctional thought patterns Learning new coping skills Learning social skills Job crafting Time management Making choices regarding career and personal life |- | '''Phase 4: Post burnout growth''' || Improving sustainable quality of life. || Setting priorities regarding quality of life and interpersonal relationships. |}{{RoundBoxTop|theme=3}}'''How does understanding emotional labor and empathy fatigue work to reduce the effects of emotional exhaustion''' What about Aelin, what can they do? Work on skills to enhance their knowledge of cognitive empathy and reduce their reliance on affective empathy to provide better outcomes for their patients. *Psychoeducation has been shown to reduce the effects of emotional labor and empathy fatigue on emotional exhaustion {{RoundBoxBottom}} ==Conclusion== * Summarise what empathy fatigue is, and why people become emotionally exhausted (+ the role of emotional labor) * Summarise the symptoms and how to recognise them in yourself and someone else * Emphasise and summarise psychological strategies to support someone going through emotional exhaustion * Summarise ways to prevent emotional exhaustion from happening * Overall message is: *# What is emotional exhaustion *# how do you recognise it *# how do you treat it *# how do you prevent it * Link back to the focus questions ==See also== * [[Motivation and emotion/Book/2025/Metacognition and emotional regulation|Metacognition and emotional regulation]] (Book chapter, 2025) * [[wikipedia:Conservation_of_resources_theory|Conservation of resources theory]] (Wikipedia) * [[wikipedia:Job_demands-resources_model|Job demands-resources model]] (Wikipedia) * Emotional Contagion * Perspective taking * Emotion regulation *[[wikipedia:List_of_psychological_research_methods|List of psychological research methods]] (Wikipedia) ==References== {{Hanging indent|1= Bošnjaković, J., & Radionov, T. (2018). Empathy: Concepts, Theories and Neuroscientific Basis. Alcoholism (Zagreb), 54(2), 123–150. https://doi.org/10.20471/de..2018.54.02.04 Chan, F. M. (2026). The State of Workplace Burnout 2026: The Workplace Divide. Infinite Potential, Sydney, Australia. https://www.ahri.com.au/wp-content/uploads/The-State-of-Workplace-Burnout-2026-Final.pdf Cui, F., Huang, X., Jing, Y., Luo, Y., Liu, J., & Gu, R. (2022). How resource sharing resists scarcity: the role of cognitive empathy and its neurobiological mechanisms. Cerebral Cortex (New York, N.Y. 1991), 32(23), 5330–5342. https://doi.org/10.1093/cercor/bhac017 De Beer, L. T., van der Vaart, L., Escaffi-Schwarz, M., De Witte, H., & Schaufeli, W. B. (2024). Maslach Burnout Inventory - General Survey: A Systematic Review and Meta-Analysis of Measurement Properties. European journal of psychological assessment : official organ of the European Association of Psychological Assessment, 40(5), 360-375. https://doi.org/10.1027/1015-5759/a000797 Demerouti, E., Bakker, A. B., Nachreiner, F., & Schaufeli, W. B. (2001). The Job Demands-Resources Model of Burnout. Journal of Applied Psychology, 86(3), 499–512. https://doi.org/10.1037/0021-9010.86.3.499 Figley, C. R. (2002). Compassion fatigue: Psychotherapists’ chronic lack of self care. Journal of Clinical Psychology, 58(11), 1433–1441. https://doi.org/10.1002/jclp.10090 Grandey, A. A. (2003). When ‘the show must go on’: surface acting and deep acting as determinants of emotional exhaustion and peer-rated service delivery. Academy of Management Journal, 46(1), 86–96. https://doi.org/10.5465/30040678 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 Halbesleben, J. R. B., Neveu, J.-P., Paustian-Underdahl, S. C., & Westman, M. (2014). Getting to the “COR”: Understanding the Role of Resources in Conservation of Resources Theory. Journal of Management, 40(5), 1334–1364. https://doi.org/10.1177/0149206314527130 Hatfield, E., Cacioppo, J. T., & Rapson, R. L. (1993). Emotional Contagion. Current Directions in Psychological Science : A Journal of the American Psychological Society, 2(3), 96–99. https://doi.org/10.1111/1467-8721.ep10770953 Hollebeek, L. D., Hammedi, W., & Sprott, D. E. (2023). Consumer engagement, stress, and conservation of resources theory: A review, conceptual development, and future research agenda. Psychology & marketing, 40(5), 926-937. https://doi.org/10.1002/mar.21807 Hobfoll, S. E. (1989). Conservation of Resources: A New Attempt at Conceptualizing Stress. The American Psychologist, 44(3), 513–524. https://doi.org/10.1037/0003-066X.44.3.513 Hochschild, A. R. (2012). The managed heart : commercialization of human feeling (Updated, with a new preface.). University of California Press. Jeung, D.-Y., Kim, C., & Chang, S.-J. (2018). Emotional Labor and Burnout: A Review of the Literature. Yonsei Medical Journal, 59(2), 187–193. https://doi.org/10.3349/ymj.2018.59.2.187 Lavelle, J. J., Rupp, D. E., Herda, D. N., Pandey, A., & Lauck, J. R. (2021). Customer Injustice and Employee Performance: Roles of Emotional Exhaustion, Surface Acting, and Emotional Demands–Abilities Fit. Journal of Management, 47(3), Article 0149206319869426. https://doi.org/10.1177/0149206319869426 Lloyd, C., & King, R. (2004). A survey of burnout among Australian mental health occupational therapists and social workers. Social Psychiatry and Psychiatric Epidemiology, 39(9), 752-757. https://doi.org/https://doi.org/10.1007/s00127-004-0808-7 Kenworthy, J., Fay, C., Frame, M., & Petree, R. (2014). A meta-analytic review of the relationship between emotional dissonance and emotional exhaustion. Journal of Applied Social Psychology, 44(2), 94-105. https://doi.org/https://doi.org/10.1111/jasp.12211 Martingano, A. J., Stosic, M. D., & Konrath, S. (2026). Different empathy types show opposing associations with burnout: systematic review and meta-analyses. Psychology, Health & Medicine, 31(3), 798-829. https://doi.org/10.1080/13548506.2025.2591859 Michielsen, H. J., Willemsen, T. M., Croon, M. A., de Vries, J., & Van Heck, G. L. (2004). Determinants of general fatigue and emotional exhaustion: a prospective study. Psychology & health, 19(2), 223-235. https://doi.org/10.1080/08870440310001627135 Panksepp, J. B., & Lahvis, G. P. (2011). Rodent empathy and affective neuroscience. Neuroscience and Biobehavioral Reviews, 35(9), 1864–1875. https://doi.org/10.1016/j.neubiorev.2011.05.013 Pooladvand, S., & Hasanzadeh, S. (2023). Impacts of Stress on Workers’ Risk-Taking Behaviors: Cognitive Tunneling and Impaired Selective Attention. Journal of Construction Engineering and Management, 149(8), 04023060. https://doi.org/doi:10.1061/JCEMD4.COENG-13339 Riess, H. (2017). The Science of Empathy. Journal of Patient Experience, 4(2), 74–77. https://doi.org/10.1177/2374373517699267 Schaufeli, W. B. (2017). Applying the job demands-resources model: a “how to” guide to measuring and tackling work engagement and burnout. Organizational Dynamics, 46(2), 120–132. https://doi.org/10.1016/j.orgdyn.2017.04.008 Schneiderman, N., Ironson, G., & Siegel, S. D. (2005). Stress and Health: Psychological, Behavioral, and Biological Determinants. Annual Review of Clinical Psychology, 1(Volume 1, 2005), 607-628. https://doi.org/https://doi.org/10.1146/annurev.clinpsy.1.102803.144141 Seidler, A., Thinschmidt, M., Deckert, S., Then, F., Hegewald, J., Nieuwenhuijsen, K., & Riedel-Heller, S. G. (2014). The role of psychosocial working conditions on burnout and its core component emotional exhaustion – a systematic review. Journal of Occupational Medicine and Toxicology, 9(1), 10. https://doi.org/10.1186/1745-6673-9-10 Shao, M., Li, L., Li, X., Wei, Z., Wang, J., Hong, M., Liu, X., & Meng, J. (2024). The effect of top-down attention on empathy fatigue. Cerebral Cortex (New York, N.Y. 1991), 34(1), Article bhad441. https://doi.org/10.1093/cercor/bhad441 van Dam, A. (2021). A clinical perspective on burnout: diagnosis, classification, and treatment of clinical burnout. European Journal of Work and Organizational Psychology, 30(5), 732-741. https://doi.org/10.1080/1359432X.2021.1948400 Vial, A. C., & Cowgill, C. M. (2022). Heavier Lies Her Crown: Gendered Patterns of Leader Emotional Labor and Their Downstream Effects. Frontiers in Psychology, 13, 849566. https://doi.org/https://doi.org/10.3389/fpsyg.2022.849566 Wharton, A. S. (2009). The Sociology of Emotional Labor. Annual Review of Sociology, 35(1), 147–165. https://doi.org/10.1146/annurev-soc-070308-115944 Wrobel, M. (2013). Can empathy lead to emotional exhaustion in teachers? The mediating role of emotional labor. International Journal of Occupational Medicine and Environmental Health, 26(4), 581–592. https://doi.org/10.2478/s13382-013-0123-1 }} ==External links== '''Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order.''' * [https://www.youtube.com/watch?v=MvSvdGArhxc How to train your empathy muscles] | (YouTube.com | Tedx talks) * [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Burnout]] [[Category:Motivation and emotion/Book/Empathy]] 8oro4lep2kk6ouh9rw1w9bgp4j3ecje 2844573 2844571 2026-11-05T02:38:29Z U3143751 3106785 2844573 wikitext text/x-wiki {{title|Empathy fatigue and emotional exhaustion:<br>How does sustained empathic engagement contribute to emotional exhaustion?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}} [[File:Exhausted smartphone user.JPG|right|thumb|150px|'''Figure 1'''. An emotionally exhausted Aelin at the end of the work day]] ; Scenario Aelin is a 23-year-old social worker employed at a non-profit organisation providing mental health services to teenagers and young adults. Aelin's clients are aged between 18 and 25, many of whom having experienced rough living conditions. They work with approximately 20 clients each week. During Aelin's first six months in the role, they feel highly motivated and are excited about helping people. They are an excellent active listener, and can easily remember details about their clients' lives. During their sessions, Aelin recognises when a client is distressed and adjusts their body language to warm and understanding in an effort to provide a safe and welcoming environment to hold space for their emotions. As Aelin gains more experience working in this role, their caseload increases, and they are exposed to clients with increasing complexity and past traumas. Aelin wants to provide the best possible outcome for all of their clients, and begins to take their work home with them in the evenings, but finds it starts creeping into their weekends as well. After several months of working this way, Aelin begins to notice they feel exhausted every morning - even before the workday begins. Fast-forward a few more months, and Aelin notices a change in how they start responding to people, including clients. Instead of feeling engaged and actively listening, they feel detached and find their thoughts feeling more frustrated than empathetic. After finishing the work day, Aelin is drained and has no energy for friends or family. With increasing irritability, Aelin begins to worry that they won't be able to continue working as a social-worker. {{RoundBoxBottom}} Aelin is experiencing '''emotional exhaustion''' from a sustained empathic response. Due to the nature of Aelin's work, they are constantly engaged in an empathic response. Empathic responses require cognitive and physical resources to sustain. Due to Aelin taking their work home in the evenings and not taking a break on weekends, they do not have a chance to rest. It is important for Aelin to increase their [[wikipedia:Psychoeducation|psychoeducation]] in order to recognise the how, what, and why of their experiences. Without this education, the problem has exacerbated leading to empathy fatigue and emotional exhaustion. Psychological science can support Aelin to recognise and map the symptoms they are experiencing, and provide coping strategies to minimise the current and future effects of empathy fatigue to improve their overall wellbeing. Understanding these symptoms will also support their role as a social worker, enhancing their work engagement and building skills to support their clients on a day-to-day basis. {{RoundBoxTop|theme=3}} '''Focus questions''' * Why do people become emotionally exhausted? * What is the relationship between empathy fatigue and emotional exhaustion? * What role does emotional labor play in emotional exhaustion? * How does understanding emotional labor and empathy fatigue work to reduce the effects of emotional exhaustion? {{RoundBoxBottom}} == Emotional exhaustion == To understand emotional exhaustion, it will be important to first understand burnout. Burnout is a long-term condition of existing in a constant state of high stress and affects millions of people annually. In the early 2000s, a survey from Australia found that more than 50% of social workers and occupational workers reported high levels of burnout symptoms (Llyod and King, 2004) and a recent report from 2026 found that more than one third of all workers in Australia were experiencing burnout (Chan, 2026). Being in a state of high stress for short periods of time is a useful adaptation for humans, so much so that it has been given a name, [[wikipedia:Eustress|Eustress]]. However, as a long-term condition it leads to individual health issues and can be linked to cardiovascular disease, increased vulnerability to upper respiratory diseases, and chronic inflammation (Schneiderman et al., 2005), and cognitive issues such as [[wikipedia:Inattentional_blindness|cognitive tunneling]] and impaired selective attention (Pooladvand and Hasanzadeh, 2023). Burnout can be defined across three dimensions: emotional exhaustion, [[wikipedia:Depersonalization|depersonalisation]], and reduced feelings of personal accomplishment (Maslach, 1993, as cited in Martingano et al., 2026). Emotional exhaustion may be considered the 'core' component of burnout and refers to the depletion of emotional resources, leading to feelings of overwhelm, overextension, and exhaustion (Martingano et al., 2026; De Beer et al., 2024; Seidler et al., 2014). As a major factor, its symptoms are similar to burnout and stress. Although antecedents of burnout are typically agreed upon (Bakker et al., 2014) and divided into individual (e.g. self-efficacy) and situational categories (e.g. work load), there are many theories that attempt to explain why emotional exhaustion occurs. Two of these theories with good evidence are the [[wikipedia:Conservation_of_resources_theory|Conservation of resources stress model]] (Hobfoll, 1989) and the [[wikipedia:Job_demands-resources_model|Job Demands-Resources model]] (Demerouti et al., 2001). === Conservation of resources stress model === The Conservation of resources stress model (CORT; Hobfall, 1989) is a theory that works to understand and explain why stress occurs. Under the model, humans are motivated to maintain, pursue, or attain resources. Stress, defined as "an individual's perceived pressure or strain" (Hollebeek et al., 2023 p.928), occurs when these resources are threatened or lost, or gained at a net loss. As noted above, resources aren't strictly tied to being physical objects. They can also be emotional states, conditions, or even ideas. For example, someone may place value on enjoying a calm, quiet Sunday morning at home. If their neighbour starts using power tools loudly next door, it would disrupt this state and induce stress in the individual. The CORT model focuses primarily on two principles. The first principle is that there is more harm in losing resources than there is in gaining them back (Halbesleben et al., 2014). This is known as loss salience, and some research suggests that the pain of loss is felt twice as strong as the pleasure of gain (Thaler, 2000). Looking at our previous example, imagine that the person who enjoys a quiet Sunday morning at home equally enjoys the sound of a loud car engine. Even though the person enjoys the sound of car engines, if the neighbour started revving their car loudly instead of using power tools, it would still disrupt their desired state of quiet and lead to an increase of stress. The second principle is that people must invest existing resources in order to gain or protect resources (Halbesleben et al., 2014). This resource investment includes recovering from resource loss. Coping is how people manage or overcome a difficult situation, such as protecting or recovering from resource loss. Coping can be broken down into different strategies that people use, but not all strategies are equal (Ito and Brotheridge, 2003). They are divided into two categories: control or escape strategies. Control strategies are considered better at reducing the strain from stress and consist of positive thinking, direct action, and help seeking (Havlovic & Keenan, 1991, as cited in Ito and Brotheridge, 2003). The most useful control strategies incorporate a person's [[locus of control]], emphasise seeking support from co-workers, and highlight the importance of autonomy in decision making. Alarcon (2011) conducted a meta-analysis of 231 studies in a work setting that used the [[wikipedia:Maslach_Burnout_Inventory|Maslach Burnout Inventory]] (MBI) to review the relationship between these principles and burnout to confirm their validity. Although the study had no specific mention of demographics, it was noted that multiple translations of the MBI were used including Dutch (k=51), German (k=7), Italian (k=2), and French (k=2). The findings indicate that workload is positively related to emotional exhaustion (''ρ'' = .49, ''k'' = 86, ''N'' = 51,529) and that control (''ρ'' = − .26, ''k'' = 56, ''N'' = 33,297) and autonomy (''ρ'' = − .24, ''k'' = 34, ''N'' = 17,165) were each negatively associated with emotional exhaustion. The difference of ''ρ'' values is consistent with Thaler (2000) and supports that the benefit of gaining is approximately half that of losing resources. === Job demands-resources model === Another model that attempts to understand why people become emotionally exhausted is the Job Demands-Resources model (JD-R; Demerouti et al., 2001). The JD-R model discusses two concepts: job demands and job resources. Demands and resources are both related to the physical, social, or psychological aspects of a job, but differ in their impact on a person. Job demands require physical or mental effort and exhaust a person's energy supply. Job resources typically support achieving goals, reduce the effort cost of job demands, and stimulate personal growth (Richter and Hacker, 1998, as cited in Demerouti et al., 2001). For example, an unsustainable workload would be considered a job demand, and receiving help and support from others would be considered a resource. The JD-R has two primary processes for how demands and resources interact with one another (Schaufeli, 2017). The stress process occurs when job demands are high and resources are low. As stress increases strain and pressure on an individual, it leads to the exhaustion of their energy eventually, eventually leading to emotional exhaustion. The second process is the motivational process. This process occurs when there is an excess of job resources. Crucially, the motivational process does not require low job demands. With enough job resources, a person will be energised and engaged, reframing job demands as a challenge to overcome. The JD-R model has had extensive research to confirm its validity. A meta-analysis by Lesener et al. (2019) looked at 74 longitudinal studies of the JD-R with a strong average sample size (n=450) against the assumptions as described by Schaufeli (2017). Although most of the studies appear to come from [[wikipedia:Psychology#WEIRD_bias|WEIRD]] countries (k=60) there were a good number from Asia and Africa (k=13) and a mix of genders. The study confirmed that across the literature, job resources predict burnout negatively (''β'' = −.17; ''p'' < .001) and job demands predict burnout over time (''β'' = .10; ''p'' < .001). Although job resources seem like a cure-all for demands, it's important to note that they do not retroactively treat emotional exhaustion. Under the JD-R model, burnout must be reduced by lowering job demands (Gynning et al., 2025). However, once someone has recovered, it is possible to increase job resources alongside job demands to prevent future burnout and support [[wikipedia:Work_engagement|work engagement]]. Work engagement (engagement) has been discussed as the opposite of burnout (Bakker et al., 2014). Engagement is how people express themselves when performing a role. A role being a pattern of behaviour that a person goes through based on the social expectations of the situation, similar to how an actor takes on a role for a movie. For example, when attending a typical restaurant, it is socially expected that a person will engage with the waiter and provide them with their meal order to take to the kitchen. This person is performing the script of a restaurant patron. In the context of a workplace, an employee with high engagement may appear to be energised, enthusiastic, and become absorbed in their work losing track of time (Bakkar et al, 2014). {{RoundBoxTop|theme=3}} '''Why do we feel emotionally exhausted''' Under the JD-R and CORT models, emotional exhaustion occurs when the demand cost outweighs the resources available for an extended period of time, eventually depleting someone's energy reserves. Aelin is experiencing burnout because they increased their job demands by taking on more clients and spending more time working, and has experienced a loss in resources by reducing their free time and social support network. {{RoundBoxBottom}} == Empathy fatigue == You may be familiar with the term [[empathy]]. It is often described in terms of supporting others by understanding or mirroring their emotional experience. For example, when someone close to you experiences a loss and is grieving, you may empathise with them and personally experience a negative emotion akin to grief too. However, empathy is not exclusive to so called negative emotions. For example, compersion is a positive empathic response when a loved one experiences joy. Parents may experience compersion when something joyful happens to their child. At its core, empathy is the individual ability to imagine or understand another person's feelings and/or emotions (Riess, 2017; Shao et al., 2024). Much like typical physical or cognitive processes, such as running or problem solving, empathising will drain energy and resources. Further, and similar to the difference in energy expenditure between running and jogging, there are distinct types of empathy that each have different energy costs. A way to make a distinction between these two types may be considering them as the "what" and "how". Research makes a clear distinction between two types of empathy: affective and cognitive empathy. Affective empathy occurs when a person has an emotional reaction to someone else's mental state (Shao et al., 2024), whereas cognitive empathy occurs when a person understands someone else's feelings or emotions (Cui et al., 2022). Integrating the two - "affective component is the essence of empathy, and the cognitive component is a process through which that essence comes into being" (Bošnjaković & Radionov, 2018) === Affective empathy === Affective empathy is the "how" and focuses on the feeling aspect of empathy. Personally experiencing another person's grief would be considered as affective empathy. Research links it to emotional contagion. Emotional contagion (Panksepp and Lahvis, 2011) is a theory that suggests that emotions are contagious and can spread from one person to another. Hatfield (1993) coined the term and suggested that emotional contagion is an automatic mimicry process that synchronises expressions, vocalisations, postures, and movements with those of another person. Affective process theory looks at organising emotional contagion into three processes. Convergent linkage, Divergent linkage, and Complementary linkage (Elfenbein, 2014). Storytelling is a way to promote position emotions (Wei et al., 2024). === Cognitive empathy === Cognitive empathy is the "what" and focuses on a person trying to tangibly understand what another person is feeling. This information is processed through a top-down approach (), and primarily utilises the pre-frontal cortex (Shao et al., 2024). This allows the empathiser to feel the emotion from a cognitive, problem solving perspective. As cognitive empathy is processed by the pre-frontal cortex, there is less association with feeling the emotion across the body hence reducing the physiological response and is linked to lower energy expenditure. Our example from before, compersion could be considered an empathic response that results from cognitive empathy. Cognitive empathy also lends itself to compassion (use that study in your one-note and look for an additional meta-analysis). === Compassion fatigue === Often compassion fatigue is referred to as empathy fatigue, but is a distinct theory in and of itself. Compassion fatigue is closely related to, and could be more aligned to becoming synonymous with, secondary traumatic stress. Usually found in healthcare professionals (Figley, 2002), secondary traumatic stress occurs when ... {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' Under the JD-R and CORT models, a sustained empathic response is a long-term job demand that uses energy to protect existing resources. As Aelin has had their resources is experiencing a * Under the CORT model, a sustained empathic response uses energy to protect existing resources. As resources become low and burnout sets in, cognitive focus narrows leading to empathy fatigue. {{RoundBoxBottom}} == Emotional labor as a mediating factor between empathy fatigue and emotional exhaustion == As a society there are expected [https://en.wikipedia.org/wiki/Social_norm social norms] in behaviour when expressing emotion, regardless if someone is conscious of the process of matching these norms or not. For example, laughing during an emotional speech at a funeral would be considered a rude or offensive expression from the other attendees. As social creatures, humans need a way to regulate socially appropriate emotional expression. This is the essence of Emotional labor: "the regulation of felt and expressed emotions (whether effortful or not) to match these emotional display rules with the objective of fulfilling organizational goals" (Vial & Cowgill, 2022) What is emotional labor (Hochschild, 2012). This section is really where all the above research comes together, and should highlight how the models of stress (JD-R and CORT), relate to the types of empathy (affective/contagion vs cognitive/compassion), and how types of acting (expectations of what you should feel (Emotional labor!!!!) vs accepting what you actually feel) contribute and '''''mediate''''' the relationship between empathy fatigue and emotional exhaustion. This is the penultimate section and should reflect that accordingly: "This is what is happening, and this is how and why its happening" === Types of acting === Surface acting * This is where emotive dissonance comes into play * Leads to emotional exhaustion (Kenworthy et al., 2014) * "Should" language - in this situation, this is what I ''should'' feel * Deep acting ** People who 'deep act', as in genuinely, intrinsically adjust the way they feel (match the feelings they have with the feelings that are appropriate to display in a given situation) often report positive relationships with emotional labor ** Actual language - in this situation, this is what I ''actually'' feel === The mediating effect of emotional labor === Surface acting does not mediate the effect (high surface acting leads to emotional exhaustion) whereas the inverse is true for deep acting (Wrobel, 2013) Discuss the differences in gender norms and expectations in relationships Emotional Expression and Emotional Processing (Hoyt et al., 2024) {{RoundBoxTop|theme=3}} '''What role does emotional labor play in emotional exhaustion''' * In relation to the CORT and JD-R theories, emotional labor acts as a job demand and expenditure of resources to protect existing resources '''Test your knowledge''' Choose your answers and click "Submit":<quiz display=simple> {Surface acting is when the observer actually feels the emotion: |type="()"} - True + False {Compassion fatigue and empathy fatigue are synonymous: |type="()"} - True + False </quiz> {{RoundBoxBottom}} == Strategies for combating empathy fatigue and emotional exhaustion == Pulling this all together, write a short introduction that describes how to identify which phase you should look at and then use the following table with resources. If you are already emotionally exhausted you need to start at Phase 1. * Talk about emotion regulation (Gross, 1998) and mood regulation strategies as a way to deal with coping (Wrobel, 2013) (Author's note: need more research/references for this section) * Talk about breaks ** Rest and downtime from not performing emotional labor regenerates * How do we reduce the amount of emotional labor we're performing (Jeung et al., 2018) ** Communication/boundaries ** Reducing surface acting (allowing ourselves to feel how we actually feel and reduce emotive dissonance) '''Table 1. Treatment phases of clinical burnout (based on Table 2 from van Dam, A. 2021)''' {| class="wikitable" style="margin: auto; |- ! Phase !! Treatment goal !! Therapeutic interventions |- | '''Phase 1: Crisis''' || Recognition of the patient that the problems are serious, and that serious action is required || Psychoeducation Sick leave Dropping domestic tasks and social obligations Inform social network |- | '''Phase 2: Recovery''' || Recovery of the (physiological) stress system to normal allostatic stress levels || Registration of stress and activities Relaxtion exercises Mindfulness healthy lifestyle advices Graded exercise Gradually resume activities Psychoeducation social network and employer Gradual return to work |- | '''Phase 3: Prevention, learning from the past''' || Acquiring insight and skills to prevent relapse in clinical burnout || Analysing stressful situations Analysing dysfunctional thought patterns Learning new coping skills Learning social skills Job crafting Time management Making choices regarding career and personal life |- | '''Phase 4: Post burnout growth''' || Improving sustainable quality of life. || Setting priorities regarding quality of life and interpersonal relationships. |}{{RoundBoxTop|theme=3}}'''How does understanding emotional labor and empathy fatigue work to reduce the effects of emotional exhaustion''' What about Aelin, what can they do? Work on skills to enhance their knowledge of cognitive empathy and reduce their reliance on affective empathy to provide better outcomes for their patients. *Psychoeducation has been shown to reduce the effects of emotional labor and empathy fatigue on emotional exhaustion {{RoundBoxBottom}} ==Conclusion== * Summarise what empathy fatigue is, and why people become emotionally exhausted (+ the role of emotional labor) * Summarise the symptoms and how to recognise them in yourself and someone else * Emphasise and summarise psychological strategies to support someone going through emotional exhaustion * Summarise ways to prevent emotional exhaustion from happening * Overall message is: *# What is emotional exhaustion *# how do you recognise it *# how do you treat it *# how do you prevent it * Link back to the focus questions ==See also== * [[Motivation and emotion/Book/2025/Metacognition and emotional regulation|Metacognition and emotional regulation]] (Book chapter, 2025) * [[wikipedia:Conservation_of_resources_theory|Conservation of resources theory]] (Wikipedia) * [[wikipedia:Job_demands-resources_model|Job demands-resources model]] (Wikipedia) * Emotional Contagion * Perspective taking * Emotion regulation *[[wikipedia:List_of_psychological_research_methods|List of psychological research methods]] (Wikipedia) ==References== {{Hanging indent|1= Bošnjaković, J., & Radionov, T. (2018). Empathy: Concepts, Theories and Neuroscientific Basis. Alcoholism (Zagreb), 54(2), 123–150. https://doi.org/10.20471/de..2018.54.02.04 Chan, F. M. (2026). The State of Workplace Burnout 2026: The Workplace Divide. Infinite Potential, Sydney, Australia. https://www.ahri.com.au/wp-content/uploads/The-State-of-Workplace-Burnout-2026-Final.pdf Cui, F., Huang, X., Jing, Y., Luo, Y., Liu, J., & Gu, R. (2022). How resource sharing resists scarcity: the role of cognitive empathy and its neurobiological mechanisms. Cerebral Cortex (New York, N.Y. 1991), 32(23), 5330–5342. https://doi.org/10.1093/cercor/bhac017 De Beer, L. T., van der Vaart, L., Escaffi-Schwarz, M., De Witte, H., & Schaufeli, W. B. (2024). Maslach Burnout Inventory - General Survey: A Systematic Review and Meta-Analysis of Measurement Properties. European journal of psychological assessment : official organ of the European Association of Psychological Assessment, 40(5), 360-375. https://doi.org/10.1027/1015-5759/a000797 Demerouti, E., Bakker, A. B., Nachreiner, F., & Schaufeli, W. B. (2001). The Job Demands-Resources Model of Burnout. Journal of Applied Psychology, 86(3), 499–512. https://doi.org/10.1037/0021-9010.86.3.499 Figley, C. R. (2002). Compassion fatigue: Psychotherapists’ chronic lack of self care. Journal of Clinical Psychology, 58(11), 1433–1441. https://doi.org/10.1002/jclp.10090 Grandey, A. A. (2003). When ‘the show must go on’: surface acting and deep acting as determinants of emotional exhaustion and peer-rated service delivery. Academy of Management Journal, 46(1), 86–96. https://doi.org/10.5465/30040678 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 Halbesleben, J. R. B., Neveu, J.-P., Paustian-Underdahl, S. C., & Westman, M. (2014). Getting to the “COR”: Understanding the Role of Resources in Conservation of Resources Theory. Journal of Management, 40(5), 1334–1364. https://doi.org/10.1177/0149206314527130 Hatfield, E., Cacioppo, J. T., & Rapson, R. L. (1993). Emotional Contagion. Current Directions in Psychological Science : A Journal of the American Psychological Society, 2(3), 96–99. https://doi.org/10.1111/1467-8721.ep10770953 Hollebeek, L. D., Hammedi, W., & Sprott, D. E. (2023). Consumer engagement, stress, and conservation of resources theory: A review, conceptual development, and future research agenda. Psychology & marketing, 40(5), 926-937. https://doi.org/10.1002/mar.21807 Hobfoll, S. E. (1989). Conservation of Resources: A New Attempt at Conceptualizing Stress. The American Psychologist, 44(3), 513–524. https://doi.org/10.1037/0003-066X.44.3.513 Hochschild, A. R. (2012). The managed heart : commercialization of human feeling (Updated, with a new preface.). University of California Press. Jeung, D.-Y., Kim, C., & Chang, S.-J. (2018). Emotional Labor and Burnout: A Review of the Literature. Yonsei Medical Journal, 59(2), 187–193. https://doi.org/10.3349/ymj.2018.59.2.187 Lavelle, J. J., Rupp, D. E., Herda, D. N., Pandey, A., & Lauck, J. R. (2021). Customer Injustice and Employee Performance: Roles of Emotional Exhaustion, Surface Acting, and Emotional Demands–Abilities Fit. Journal of Management, 47(3), Article 0149206319869426. https://doi.org/10.1177/0149206319869426 Lloyd, C., & King, R. (2004). A survey of burnout among Australian mental health occupational therapists and social workers. Social Psychiatry and Psychiatric Epidemiology, 39(9), 752-757. https://doi.org/https://doi.org/10.1007/s00127-004-0808-7 Kenworthy, J., Fay, C., Frame, M., & Petree, R. (2014). A meta-analytic review of the relationship between emotional dissonance and emotional exhaustion. Journal of Applied Social Psychology, 44(2), 94-105. https://doi.org/https://doi.org/10.1111/jasp.12211 Martingano, A. J., Stosic, M. D., & Konrath, S. (2026). Different empathy types show opposing associations with burnout: systematic review and meta-analyses. Psychology, Health & Medicine, 31(3), 798-829. https://doi.org/10.1080/13548506.2025.2591859 Michielsen, H. J., Willemsen, T. M., Croon, M. A., de Vries, J., & Van Heck, G. L. (2004). Determinants of general fatigue and emotional exhaustion: a prospective study. Psychology & health, 19(2), 223-235. https://doi.org/10.1080/08870440310001627135 Panksepp, J. B., & Lahvis, G. P. (2011). Rodent empathy and affective neuroscience. Neuroscience and Biobehavioral Reviews, 35(9), 1864–1875. https://doi.org/10.1016/j.neubiorev.2011.05.013 Pooladvand, S., & Hasanzadeh, S. (2023). Impacts of Stress on Workers’ Risk-Taking Behaviors: Cognitive Tunneling and Impaired Selective Attention. Journal of Construction Engineering and Management, 149(8), 04023060. https://doi.org/doi:10.1061/JCEMD4.COENG-13339 Riess, H. (2017). The Science of Empathy. Journal of Patient Experience, 4(2), 74–77. https://doi.org/10.1177/2374373517699267 Schaufeli, W. B. (2017). Applying the job demands-resources model: a “how to” guide to measuring and tackling work engagement and burnout. Organizational Dynamics, 46(2), 120–132. https://doi.org/10.1016/j.orgdyn.2017.04.008 Schneiderman, N., Ironson, G., & Siegel, S. D. (2005). Stress and Health: Psychological, Behavioral, and Biological Determinants. Annual Review of Clinical Psychology, 1(Volume 1, 2005), 607-628. https://doi.org/https://doi.org/10.1146/annurev.clinpsy.1.102803.144141 Seidler, A., Thinschmidt, M., Deckert, S., Then, F., Hegewald, J., Nieuwenhuijsen, K., & Riedel-Heller, S. G. (2014). The role of psychosocial working conditions on burnout and its core component emotional exhaustion – a systematic review. Journal of Occupational Medicine and Toxicology, 9(1), 10. https://doi.org/10.1186/1745-6673-9-10 Shao, M., Li, L., Li, X., Wei, Z., Wang, J., Hong, M., Liu, X., & Meng, J. (2024). The effect of top-down attention on empathy fatigue. Cerebral Cortex (New York, N.Y. 1991), 34(1), Article bhad441. https://doi.org/10.1093/cercor/bhad441 van Dam, A. (2021). A clinical perspective on burnout: diagnosis, classification, and treatment of clinical burnout. European Journal of Work and Organizational Psychology, 30(5), 732-741. https://doi.org/10.1080/1359432X.2021.1948400 Vial, A. C., & Cowgill, C. M. (2022). Heavier Lies Her Crown: Gendered Patterns of Leader Emotional Labor and Their Downstream Effects. Frontiers in Psychology, 13, 849566. https://doi.org/https://doi.org/10.3389/fpsyg.2022.849566 Wharton, A. S. (2009). The Sociology of Emotional Labor. Annual Review of Sociology, 35(1), 147–165. https://doi.org/10.1146/annurev-soc-070308-115944 Wrobel, M. (2013). Can empathy lead to emotional exhaustion in teachers? The mediating role of emotional labor. International Journal of Occupational Medicine and Environmental Health, 26(4), 581–592. https://doi.org/10.2478/s13382-013-0123-1 }} ==External links== '''Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order.''' * [https://www.youtube.com/watch?v=MvSvdGArhxc How to train your empathy muscles] | (YouTube.com | Tedx talks) * [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Burnout]] [[Category:Motivation and emotion/Book/Empathy]] 7d0uca3zh07ofrqgixffuhkfcw4mfqg 1 331432 2844555 2844064 2026-11-05T02:03:54Z Jtneill 10242 Remove genai query 2844555 wikitext text/x-wiki == Scenario Suggestion - From U3275775 == For your overview, I think you could add a scenario about a coffee shop worker and their regular customer who comes in everyday. At first they would only make small talk but as the customer comes in each day, their talks get longer. This would demonstrate how self-disclosure develops gradually over-time, and how this leads to emotional intimacy. [[User:U3275775|U3275775]] ([[User talk:U3275775|discuss]] • [[Special:Contributions/U3275775|contribs]]) 04:56, 24 August 2026 (UTC) == Resource == Hello! I am loving the start to this topic it looks good so far and I'm interested to see how it develops. I found a resource that may be helpful, it's an article about brain hyperscanning to better understand the brain mechanisms that underlie emotional intimacy through self-disclosure. I'll paste the link and I hope it is helpful or at least points you to some extra resorces. Cheers! https://link.springer.com/article/10.1186/s40359-026-04257-3 [[User:J.M.A Watson|J.M.A Watson]] ([[User talk:J.M.A Watson|discuss]] • [[Special:Contributions/J.M.A Watson|contribs]]) 21:54, 27 August 2026 (UTC) == Extra Resource == Hey, great job so far! I noticed out topics had some similarities and thought you might benefit from this source I've had a look at :) Reis, H. T., & Shaver, P. (1988). <nowiki>''</nowiki>Intimacy as an Interpersonal Process<nowiki>''</nowiki> (pp. 367–389). Routledge. <nowiki>https://www.researchgate.net/publication/347687013_Intimacy_as_an_interpersonal_process</nowiki> [[User:U3282586|U3282586]] ([[User talk:U3282586|discuss]] • [[Special:Contributions/U3282586|contribs]]) 10:55, 28 August 2026 (UTC) == Heading casing == {| style="float: center; background:transparent;color:inherit;" |- | [[File:Crystal Clear app ktip.svg|48px|left]] | {{#if:U3283302|Hi [[User:U3283302|U3283302]].|}} 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> 05:28, 31 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 --> # Promising [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development (e.g., consider using subheadings) # Avoid having sections with only 1 sub-heading – use 0 or 2+ sub-headings <!-- Alignment with focus questions --> # Good alignment between sub-title, focus questions, and heading structure, but there is room for improvement |3= <!-- Overview--> # Scenario, image, evocative description of the problem/topic, and focus questions <!-- Description --> # A promising description of the problem/topic is planned or presented <!-- Focus questions --> # Reasonably good alignment between focus questions and heading structure, but consider closer alignment |4= <!-- Key points--> <!-- Overall --> # Promising development - basic development of bullet-points with some citations for the proposed sections # Highlight the most relevant theories and synthesise the best research on the topic # Provide more detailed edit summaries <!-- Scope --> # The scope is excellent (i.e., not too little/narrow or too big/broad) <!-- Conclusion --> # Conclusion is well underway |5= <!-- Figure --> # Relevant figure(s) are presented and captioned <!-- Cite --> # Cite each figure at least once in the main text using APA style (e.g., see Figure 1) <!-- Size --> # Consider increasing image size(s) (especially if they have text) to make them easier to view |6= <!-- Learning feature --> <!-- Interwiki links ---> # Two in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]]. Also embed links to [[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) <!-- Tables --> # Also consider using tables to summarise key information |7= <!-- References --> <!-- Overall --> # Very good # Only cite sources you have consulted <!-- 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]: ## use dois where available instead of other links ## include hyperlinked dois |8= <!-- Resources --> <!-- See also --> # See also ## Excellent <!-- External links --> # External links ## Excellent |9= <!-- User page --> # Used effectively <!-- 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 --> # 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> 05:28, 31 August 2026 (UTC) <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is a basic chapter; it is strong in theory, weak in research, and moderate in style/expression # The main area(s) for potential improvement: #* 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 #* more substantial engagement in social contribution <!-- Overall - Edit summaries --> #* Insufficient use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit <!-- 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 – Copy edits --> # For additional feedback, see the following comments and [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FSelf-disclosure_and_emotional_intimacy&diff=2844059&oldid=2839579 these copy edits] |2= <!-- Overview – Comments... --> # Excellent <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario # The Figure 1 caption could be more 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; minor areas for improvements <!-- 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 <!-- Theory – Depth --> # Very good 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 --> # In many places, the chapter could be strengthened by including more academic, peer-reviewed citations (e.g., see the {{f}} tags). 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This would help to demonstrate a deeper understanding of relevant psychological science about the topic. <!-- 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, country, 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... --> # 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 # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> # Good summary and conclusion # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are summarised in a basic way <!-- Conclusion – Focus questions --> # The focus questions are addressed <!-- Conclusion – Take-home messages --> # Clear take-home message(s) |7= <!-- Style comments... --> <!-- Style – Written expression --> # Written expression ## The quality of written expression is basic <!-- Style – Sentences --> ## The clarity and grammar of many sentences could be improved; consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression <!-- Style – Paragraphs --> ## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences. <!-- Style – 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. <!-- Style – Layout --> # Layout ## The chapter structure is underdeveloped ## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings ## Remove abbreviations/citations from headings <!-- Style – 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 ## Use past tense when describing research studies, although implications of findings could be in the present tense <!-- Style – 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) <!-- Style – 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] <!-- Style – 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)] <!-- Style – 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) <!-- Style – Figures --> ## Figures ### Reasonably well captioned <!-- Style – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style <!-- Style – Tables --> ## Table(s) ### Well captioned <!-- Style – Tables - Citations --> ### Each Table is referred to at least once within the main text using APA style <!-- Style – Citations --> # In many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags) # Use ampersand (&) inside [[w:Bracket#Parentheses|parentheses]] and "and" outside parentheses <!-- Style – References --> ## References use very good to excellent APA style: ### 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 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 --> # Excellent use of figure(s) # Very good use of table(s) # Very good use of feature box(es) # Very good use of scenarios, case studies, or examples <!-- Learning features – Quizzes --> # No 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 --> # Insufficient use of the [[{{PAGENAME}}#External links|External links]] section ## Link to the top 3-6 external resources about this topic |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. # For additional social contribution credit, list uploads to Wikimedia Commons. ## This file is poorly named: [[:File:Ipm r and s.png]] (I've requested that it be renamed) }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 23:12, 2 November 2026 (UTC) i99weyzdtwcym9hj9jcyweswaayqbrm 2844557 2844555 2026-11-05T02:05:29Z Jtneill 10242 Remove Australian spelling comment 2844557 wikitext text/x-wiki == Scenario Suggestion - From U3275775 == For your overview, I think you could add a scenario about a coffee shop worker and their regular customer who comes in everyday. At first they would only make small talk but as the customer comes in each day, their talks get longer. This would demonstrate how self-disclosure develops gradually over-time, and how this leads to emotional intimacy. [[User:U3275775|U3275775]] ([[User talk:U3275775|discuss]] • [[Special:Contributions/U3275775|contribs]]) 04:56, 24 August 2026 (UTC) == Resource == Hello! I am loving the start to this topic it looks good so far and I'm interested to see how it develops. I found a resource that may be helpful, it's an article about brain hyperscanning to better understand the brain mechanisms that underlie emotional intimacy through self-disclosure. I'll paste the link and I hope it is helpful or at least points you to some extra resorces. Cheers! https://link.springer.com/article/10.1186/s40359-026-04257-3 [[User:J.M.A Watson|J.M.A Watson]] ([[User talk:J.M.A Watson|discuss]] • [[Special:Contributions/J.M.A Watson|contribs]]) 21:54, 27 August 2026 (UTC) == Extra Resource == Hey, great job so far! I noticed out topics had some similarities and thought you might benefit from this source I've had a look at :) Reis, H. T., & Shaver, P. (1988). <nowiki>''</nowiki>Intimacy as an Interpersonal Process<nowiki>''</nowiki> (pp. 367–389). Routledge. <nowiki>https://www.researchgate.net/publication/347687013_Intimacy_as_an_interpersonal_process</nowiki> [[User:U3282586|U3282586]] ([[User talk:U3282586|discuss]] • [[Special:Contributions/U3282586|contribs]]) 10:55, 28 August 2026 (UTC) == Heading casing == {| style="float: center; background:transparent;color:inherit;" |- | [[File:Crystal Clear app ktip.svg|48px|left]] | {{#if:U3283302|Hi [[User:U3283302|U3283302]].|}} 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> 05:28, 31 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 --> # Promising [[Motivation and emotion/Assessment/Major project/Structure|1-level heading structure]] – could benefit from further development (e.g., consider using subheadings) # Avoid having sections with only 1 sub-heading – use 0 or 2+ sub-headings <!-- Alignment with focus questions --> # Good alignment between sub-title, focus questions, and heading structure, but there is room for improvement |3= <!-- Overview--> # Scenario, image, evocative description of the problem/topic, and focus questions <!-- Description --> # A promising description of the problem/topic is planned or presented <!-- Focus questions --> # Reasonably good alignment between focus questions and heading structure, but consider closer alignment |4= <!-- Key points--> <!-- Overall --> # Promising development - basic development of bullet-points with some citations for the proposed sections # Highlight the most relevant theories and synthesise the best research on the topic # Provide more detailed edit summaries <!-- Scope --> # The scope is excellent (i.e., not too little/narrow or too big/broad) <!-- Conclusion --> # Conclusion is well underway |5= <!-- Figure --> # Relevant figure(s) are presented and captioned <!-- Cite --> # Cite each figure at least once in the main text using APA style (e.g., see Figure 1) <!-- Size --> # Consider increasing image size(s) (especially if they have text) to make them easier to view |6= <!-- Learning feature --> <!-- Interwiki links ---> # Two in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]]. Also embed links to [[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) <!-- Tables --> # Also consider using tables to summarise key information |7= <!-- References --> <!-- Overall --> # Very good # Only cite sources you have consulted <!-- 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]: ## use dois where available instead of other links ## include hyperlinked dois |8= <!-- Resources --> <!-- See also --> # See also ## Excellent <!-- External links --> # External links ## Excellent |9= <!-- User page --> # Used effectively <!-- 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 --> # 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> 05:28, 31 August 2026 (UTC) <!-- Official book chapter feedback --> {{MEBF/2026 |1= <!-- Overall comments... --> # This is a basic chapter; it is strong in theory, weak in research, and moderate in style/expression # The main area(s) for potential improvement: #* 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 #* more substantial engagement in social contribution <!-- Overall - Edit summaries --> #* Insufficient use of [[Motivation and emotion/Wikiversity/Edit summary|edit summaries]] to describe the changes made in each edit <!-- 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 – Copy edits --> # For additional feedback, see the following comments and [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FSelf-disclosure_and_emotional_intimacy&diff=2844059&oldid=2839579 these copy edits] |2= <!-- Overview – Comments... --> # Excellent <!-- Overview – Scenario --> <!-- Overview – Scenario - Description --> # Engaging scenario or case study <!-- Overview – Scenario - Figure --> # Figure 1 is relevant to the scenario # The Figure 1 caption could be more 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; minor areas for improvements <!-- 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 <!-- Theory – Depth --> # Very good 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 --> # 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 --> # Very good/ use of 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 # 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 --> # Insufficient [[w:Critical thinking|critical thinking]] about relevant research is evident # [[w:Critical thinking|Critical thinking]] about research could be further evidenced by: ## describing the methodology (e.g., sample, country, measures) in important studies ## considering the strength of relationships ## acknowledging limitations ## pointing out specific critiques/counterarguments ## suggesting ''specific'' directions for future research |5= <!-- Integration comments... --> # 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 # Insufficient integration with related [[Motivation and emotion/Book|chapters]] |6= <!-- Conclusion comments... --> # Good summary and conclusion # Highlight the importance of the problem or phenomenon of interest <!-- Conclusion – Key points --> # Key points are summarised in a basic way <!-- Conclusion – Focus questions --> # The focus questions are addressed <!-- Conclusion – Take-home messages --> # Clear take-home message(s) |7= <!-- Style comments... --> <!-- Style – Written expression --> # Written expression ## The quality of written expression is basic <!-- Style – Sentences --> ## The clarity and grammar of many sentences could be improved; consider using [https://www.studiosity.com/students/ Studiosity] for targetted feedback to help improve the quality of written expression <!-- Style – Paragraphs --> ## Some paragraphs are overly long. Communicate one key idea per paragraph in three to five sentences. <!-- Style – 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. <!-- Style – Layout --> # Layout ## The chapter structure is underdeveloped ## Avoid having sections with 1 sub-heading – use 0 or 2+ sub-headings ## Remove abbreviations/citations from headings <!-- Style – 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 ## Use past tense when describing research studies, although implications of findings could be in the present tense <!-- Style – 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] <!-- Style – 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)] <!-- Style – 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) <!-- Style – Figures --> ## Figures ### Reasonably well captioned <!-- Style – Figures - Citations --> ### Each Figure is referred to at least once within the main text using APA style <!-- Style – Tables --> ## Table(s) ### Well captioned <!-- Style – Tables - Citations --> ### Each Table is referred to at least once within the main text using APA style <!-- Style – Citations --> # In many places, better use could be made of academic, peer-reviewed citations (e.g., see the {{f}} tags) # Use ampersand (&) inside [[w:Bracket#Parentheses|parentheses]] and "and" outside parentheses <!-- Style – References --> ## References use very good to excellent APA style: ### 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 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 --> # Excellent use of figure(s) # Very good use of table(s) # Very good use of feature box(es) # Very good use of scenarios, case studies, or examples <!-- Learning features – Quizzes --> # No 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 --> # Insufficient use of the [[{{PAGENAME}}#External links|External links]] section ## Link to the top 3-6 external resources about this topic |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. # For additional social contribution credit, list uploads to Wikimedia Commons. ## This file is poorly named: [[:File:Ipm r and s.png]] (I've requested that it be renamed) }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 23:12, 2 November 2026 (UTC) jn3it7r85r16t9wd8xmbtya91p7t4fs 2 331770 2844599 2830272 2026-11-05T10:58:30Z Dick Bos 24466 subpages 2844599 wikitext text/x-wiki [[David Ricardo]] (1772–1823) is one of the most important economists of the classical period, and his thoughts and influences are still frequently debated in recent times. For instance by Nat Dyer (2024): ''Ricardo's Dream : How Economists Forgot the Real World and Led us Astray.'' I would like to introduce the works of Ricardo here. * First of all: ''[[David Ricardo/An Essay on Profits]]'' (1815), with a couple of subpages (firstly based on my doctoral thesis 1984): ** [[David Ricardo/An Essay on Profits/Graphical|Graphical interpretation]] of the Ricardian theory in the ''Essay'' ** [[David Ricardo/An Essay on Profits/Mathematical|Mathematical interpretation]] ** [[David Ricardo/An Essay on Profits/Nature|Some remarks]] concerning the role of "nature" in the ''Essay'' asieia0pf1guyyic1bzeo06rmq3im7n 0 331771 2844602 2841560 2026-11-05T11:23:09Z Dick Bos 24466 Principles new subpage 2844602 wikitext text/x-wiki [[File:Portrait of David Ricardo by Thomas Phillips.jpg|left|250px|thumb|Portrait of David Ricardo by Thomas Phillips (around 1820)]] {{20% done}} Welcome to the pages about '''David Ricardo'''. David Ricardo (1772–1823) was an English stockbroker, economist and politician. He made particularly significant contributions to the development of classical political economy, and later of economy as a science and he is often mentioned in the same breath as the "founder of economics", [[Adam Smith]] (1723–1790). He maintained close contact with contemporaries such as [[Thomas Robert Malthus]] and [[James Mill]] and his son [[John Stuart Mill]]. == Biographical notes == David Ricardo was born in London. His father, a Sephardic Jew of Portuguese origin, was a successful stockbroker. David Ricardo had a traditional Jewish upbringing, and received little formal education. In the years from 1783 to 1785 he stayed in Amsterdam. After he returned to London, he started working with his father at the age of 14. When he was 21 years old, he learned to know Priscilla Anne Wilkinson. She was a Quaker. And when David Ricardo married her, that led to a major rift with his family. He went into business of his own. "As a trustee of the early modern Stock Exchange, he was a member of the Committee of Proprietors that decided in 1801 to reconstitute it on a private subscription basis and subsequently helped oversee the execution of the plan. He later played a leading role in syndicates contracting loans that financed the Napoleonic wars. He made the bulk of his fortune speculating on the outcome of the Battle of Waterloo in 1815, when he reportedly ‘netted upwards of a million sterling’–an impressive sum at the time."<ref name=Comerford>{{cite web | last = Comerford | first = Patrick | url = https://www.patrickcomerford.com/2025/06/a-discreet-plaque-in-london-remembers.html | title = A discreet plaque in London remembers David Ricardo | website = Comerford Way |access-date = 2026-10-14 | archive-url = https://web.archive.org/web/20260622011359/https://www.patrickcomerford.com/2025/06/a-discreet-plaque-in-london-remembers.html | archive-date = 2026-06-22 }}</ref> Ricardo retired immediately and bought Gatcombe Park, an estate in Gloucestershire now owned by Princess Anne. == Works == His first work in print: * {{cite book | last = Ricardo | first = David | authorlink = David Ricardo | date = 1810 | title = The High Price of Bullion, a Proof of the Depreciation of Bank Notes | location = London | publisher = John Murray }}<ref name=bul>''The High Price of Bullion'' (1810): [[:s:en:The High Price of Bullion | Third edition ("with additions", 1810)]] on [[Image:wikisource-logo.svg|16x16px]] WikiSource.</ref> After his (very early) retirement Ricardo fully developed his interest in economic theory. He read Adam Smith's ''Wealth of Nations'' and subsequently published some articles in newspapers and pamphlets: * —— (1815). ''[[David Ricardo/An Essay on Profits|An Essay on the Influence of a Low Price of Corn on the Profits of Stock]]'' (often shortly named ''(An) Essay on Profits''). A few years later his Magnum Opus was first published: * —— (1817). ''[[David Ricardo/Principles|On the Principles of Political Economy; And Taxation]]'' (often shortly named ''(The) Principles''). His collected works were first published in 1852 by John Ramsay McCulloch: * {{cite book |last=Ricardo|first=David|editor-last = McCulloch | editor-first = J.R.|editor-link=:w:en:John Ramsay McCulloch |title=The Works of David Ricardo : with a Notice of the Life and Writings of the Author |url=https://preserver.beic.it:443/delivery/DeliveryManagerServlet?dps_pid=IE6470169 |year=1852|publisher=John Murray|location=London}}<ref name=works18868>[https://catalog.hathitrust.org/Record/100340294 1886] and [https://archive.org/details/worksofdavidrica01rica 1888] editions of McCulloch's ''The Works of David Ricardo''.</ref> His works and writings were collected from 1951 onwards by Piero Sraffa, with the collaboration of Maurice Dobb: * {{Cite book |last=Ricardo|first=David| editor1-last = Sraffa | editor1-first = Piero | editor1-link = :w:en:Piero Sraffa |editor2-last = Dobb |editor2-first = Maurice Herbert | editor2-link = :w:en:Maurice Dobb |title=The Works and Correspondence of David Ricardo |date=1981|origyear=1951 |location=Cambridge |publisher=Cambridge University Press |isbn=0521285054 |oclc=10251383}} (first paperback edition, in 11 volumes) This is the collection, that is mostly used nowadays: === ''The Works and Correspondence of David Ricardo'' === ''The Works and Correspondence of David Ricardo,'' in 11 volumes (1951-1973) was edited by [[w:en:Piero Sraffa|Piero Sraffa]], with the cooperation of [[w:en:Maurice Dobb|Maurice Dobb]]. It was published for the Royal Economic Society, by Cambridge University Press.<ref name=OLL>See also: [https://oll.libertyfund.org/titles/ricardo-the-works-and-correspondence-of-david-ricardo-11-vols-sraffa-ed The Works etc. in 11 vols.] at the Online Library of Liberty (OLL).</ref> * The first volume (1951)<ref name=Works01>[https://archive.org/details/workscorresponde0001rica_n2f1/page/n3/mode/1up Works etc. vol. 1] online available after free registration at [[:w:en:Internet Archive|Internet Archive]]. See also [https://archive.org/details/workscorresponde0001unse/page/n6/mode/1up this copy of Works vol. 1] online (at IA).</ref> holds: ** An "'Introduction" by Sraffa (p. xiii – lxii + 2 tables) ** ''The Principles of Political Economy, and Taxation'' (the edition follows the 3rd edition of 1821)<ref name=principlesed>For other editions of the ''Principles'' see: …..e.g. first edition in ws…</ref> * vol. 2 (1957): Notes on Malthus<ref name=Works02>[https://archive.org/details/workscorresponde0000rica/page/n8/mode/1up Works vol. 2] online.</ref> * vol. 3 : Pamphlets and Papers, 1809–1811<ref name=Works03>vol. 3 online.</ref> * vol. 4 (1962): Pamphlets and Papers, 1815-1823<ref name=Works04>[https://archive.org/details/workscorresponde0004rica Works vol. 4] online.</ref> * vol. 5 (1962): Speeches and Evidence<ref name=Works05>[https://archive.org/details/workscorresponde0000rica_c0y0/page/n8/mode/1up Works vol. 5] online. See also [https://archive.org/details/workscorresponde0005unse/page/n8/mode/1up this copy of Works vol. 5] online (at IA)</ref> * vol. 6 (1962):Letters 1810–1815<ref name=Works06>[https://archive.org/details/workscorresponde0000rica_v5n8/page/n6/mode/1up Works vol. 6] online. See also [https://archive.org/details/workscorresponde0006unse_i2q2/mode/1up this copy of Works vol. 6] online (at IA).</ref> * vol. 7: Letters 1816–1818 * vol. 8: Letters 1819–June 1821 * vol. 9 (1962): Letters, July 1821–1823<ref name=Works09>[https://archive.org/details/workscorresponde0009unse Works vol. 9] online. See also [https://archive.org/details/workscorresponde0009rica this copy of Works vol. 9] online (at IA).</ref> * vol. 10: Biographical Miscellany * vol. 11 (1973): General Index<ref name=Works11>[https://archive.org/details/workscorresponde0011rica Works vol. 11 (Index)] online.</ref> == Philosophy and Method == van der Pol makes some remarks on the "philosophical premises" of Ricardo's theory. He says, that among scholars, there is "little disagreement regarding its content, although opinions differ as to the influence of those premises on the various parts of Ricardo's theory."{{sfnp|van der Pol|1981|p=20}} Ricardo does not explicitly address the philosophical foundations of his theory anywhere. It can, however, be inferred from various passages that Ricardo regards the economic process as one governed by the laws of nature. In Ricardo’s theory, the economic process–which encompasses production, distribution and exchange–does not unfold in a random and chaotic manner, but according to an order determined by the laws of nature. These laws form the framework within which economic activities take place.{{sfnp|van der Pol|1981|p=20}} For instance concerning the (fall of the) rate of profit: "This will necessarily be rendered permanent by the laws of nature, which have limited the productive powers of the land."{{sfnp|Ricardo|1817|p=126}} Ricardo's approach implies that there is little room for the analysis of historically specific phenomena and characteristics.{{sfnp|van der Pol|1981|p=20–21}} The first thing that must be mentioned, concerning Ricardo's '''method''' is his strong deductive approach. It has led many scholars to what is sometimes called the "Ricardian vice," or "Ricardo's dream." Writing about the ''Principles'' Huf<ref name=Huf2017>{{cite web | url = https://history.cass.anu.edu.au/news/rents-profits-wages-bicentenary-ricardos-principles-and-making-economics | date = 2017-04-03 | title = "Rents, Profits, Wages": The Bicentenary of Ricardo's Principles and the making of economics | website = Australian National University / School of History }}</ref> says: "The book was notable for its brevity and theoretical, ‘deductive method’, contrasting greatly with Smith’s broad sociological-historical analysis (only aspects of which Ricardo saw himself refining), and to the prophetic appeal of Marx’s Capital or philosophical breadth of J.S. Mill’s Principles, both of which were deeply indebted to Ricardo’s framework." == Additional resources == * [[:w:en:David Ricardo|David Ricardo]] at English [[File:Wikipedia-logo-v2.svg|16px|alt=16 px Wikipedia logo]] Wikipedia * [[:s:en:Author:David Ricardo|David Ricardo]] at English [[Image:wikisource-logo.svg|16x16px]] Wikisource * [https://onlinebooks.library.upenn.edu/webbin/book/lookupname?key=Ricardo%2c%20David%2c%201772%2d1823&c=x David Ricardo] at The Online Books Page == Sources and further reading == * {{cite book | last = van der Pol | first = Kick | date = 1981 | title = Marx contra Ricardo : De wet van de dalende winstvoet | language = nl | location = Groningen | publisher = Wolters-Noordhoff | ISBN = 90 01 71290 8 | ref = harv }} == References == {{references}} {{DEFAULTSORT:Ricardo, David}} [[Category:Economics]] [[Category:History of Economic Thought]] 5733a5zahmrcs2ulzkj9mdpr0x84t74 0 331787 2844480 2842959 2026-11-04T15:42:34Z Dick Bos 24466 /* Sources */ Blaug corr. italics 2844480 wikitext text/x-wiki [[File:Ricardo Essay on Profits 1815 Titlepage.png|thumb|250px|right]] {{Right|"It is doubtful whether it is possible<br />to understand the Principles without<br />knowledge of the contents of the Essay."<br />({{harvp|O'Brien|1975|p=139}}). }} <div style="line-height:8em;">&nbsp;</div> This resource gives information on '''[[David Ricardo]]'s''' '''''An Essay on the Influence of a Low Price of Corn on the Profits of Stock''''' (often abridged to '''''(An) Essay on Profits'''''). Two years before the publication of the first edition of his Magnum Opus, ''On the Principles of Political Economy and Taxation'' (often abridged to ''(The) Principles''), in 1815, Ricardo published a pamphlet entitled ''An Essay on the Influence of a Low Price of Corn on the Profits of Stock''. Full title: :::''An Essay on the Influence of a Low Price of Corn on the Profits of Stock; Shewing the Inexpediency of Restrictions on Importation; With Remarks on Mr. Malthus' Two Last Publications: "An Inquiry into the Nature and Progress of Rent;" and "The Grounds of an Opinion on the Policy of Restricting the Importation of Foreign Corn."''<ref name=Essay1>We will use the following edition of ''The Essay on Profits'': * 1815: 2nd edition, online available at (English) [[Image:wikisource-logo.svg|16x16px]] Wikisource: [[:s:en:An Essay on the Influence of a Low Price of Corn on the Profits of Stock|An Essay on Profits (2nd ed.)]] (We will quote from this edition).<br /> Another edition, is the one published in the ''Works and Correspondence of David Ricardo'' in 1962, edited by Piero Sraffa, with the collaboration of Maurice Dobb, volume 4 (Pamphlets and Papers, 1815-1823), pp. 9-41. With note by P. Sraffa (p. 1-8). Available to borrow at [https://archive.org/details/workscorresponde0004rica The Internet Archive] after (free) subscription.</ref> Opinions among Ricardo scholars differ as to the precise significance of this "Essay on Profits". Among other things, there are subtle differences regarding the interpretation of the main variable under investigation,<ref>According to Hollander 1973, p. 273, "the main variable under investigation in the Essay (…) is the general rate of profit".<br />According to Stigler 1952, pp. 201–202, the Essay contains "two main elements of the Ricardian system: the theory of rent and the dominant influence of diminishing returns in agriculture upon the rate of profit."<br />According to Eatwell 1975, p. 186, the Essay is "the first complete statement of a theory of distribution based on the concept of surplus; that is to say, of a deductive relationship between wages and profits".</ref> and there are disagreements about the relationship between the "Essay" and the "Principles".<ref>Compare, for example, the discussion between Hollander and Eatwell on the significance of the "corn model" in Hollander 1973 and 1975, and Eatwell 1975.</ref> In the ''Essay on Profits'' Ricardo presents a comprehensive theory of ground rent within the context of a "corn model".<ref>This model is set out on pages 10 to 18 of the ''Essay on Profits.'' From page 18 onwards, the grain is (suddenly) assigned a price.</ref> Ricardo examines economic relationships, which are aggregated in terms of quantities of corn. He employs a single-output model, in which corn is produced through the application of labour to land. Inputs, outputs and income shares are measured in quantities of corn. <br /> This so-called "corn model" must be viewed against the background of Ricardo’s deductive method.<ref>For an analysis of the importance of the deductive method of Ricardo see e.g. {{harvp|O'Brien|1975|p=42f., 68f.}}</ref> <br /> The model has very limited validity.<ref name=valid>{{harvp|Hollander|1973|p=275}}: Even in the event that an adequate relationship is defined between the agricultural and general rates of profit the model would be unsuitable for the treatment of the income shares, and in fact Ricardo nowhere states the belief that there is a parallel movement between the ''aggregate'' income shares and the shares in ''agricultural output'' as yielded by the model."</ref> This resource will be followed by some pages, in which aspects of some issues in the ''Essay on Profits'' will be examined a bit deeper: * [[David Ricardo/An Essay on Profits/Graphical|Graphical interpretation]] of the Ricardian theory of rent * [[David Ricardo/An Essay on Profits/Mathematical|Mathematical interpretation]] of the Ricardian theory of rent * [[David Ricardo/An Essay on Profits/Nature|Some remarks]] concerning the role of "nature" in the Ricardian theory == Rent == To start with, Ricardo uses a definition of rent that is taken directly from [[:w:en:Thomas Robert Malthus|Thomas Robert Malthus]]: {{Bq|Mr. Malthus very correctly defines, "the rent of land to be that portion of the value of the whole produce which remains to the owner, after all the outgoings belonging to its cultivation, of whatever kind, have been paid, including the profits of the capital employed, estimated according to the usual and ordinary rate of the profits of agricultural stock at the time being."<ref name=rent1>{{harvp|Ricardo|1815}}, [[:s:en:An Essay on the Influence of a Low Price of Corn on the Profits of Stock#3|p. 3]]. Ricardo quotes: {{harvp|Malthus|1815|p=1–2}}.</ref>}} When land is first brought into cultivation, and land is freely available, the total output will consist of costs and profits, without ground rent. As long as there is sufficient land of equal fertility available, and that land is always equally well situated, this will remain the case. The net product (‘neat produce’) will consist entirely of profit. If we now assume that no improvements take place in agriculture, and that capital and population grow at the same rate – and thus that real wages remain constant – and if, moreover, it is assumed that the rate of profit in agriculture, commerce and industry is the same, it is possible to ascertain what particular effects occur as capital accumulation takes place, the population grows and more land is brought into use, particularly land situated further away and/or less fertile land. Where land is equally fertile but situated further away, more capital and labour will have to be employed to produce the same quantity of goods available for consumption from the same area. Consequently, less profit will be made on this newly cultivated land. And because profits are determined by the profit made on the ‘marginal’ land, a lower profit will also be realised on the land that was cultivated earlier. The portion of the net product that thereby becomes available on this land – and which was previously appropriated as profit by the entrepreneurs or tenants – will now constitute ground rent. This process will occur each time new land of poorer quality, or less favourably situated, is brought into use: "Thus by bringing successively land of a worse quality, or less favourably situated into cultivation, rent would rise on the land previously cultivated, and precisely in the same degree would profits fall; and if the smallness of profits do not check accumulation, there are hardly any limits to the rise of rent, and the fall of profit." (p. [[:s:en:An Essay on the Influence of a Low Price of Corn on the Profits of Stock#9|9]]–10) == The table in the ''Essay on Profits'' == To illustrate his line of reasoning and clarify how the process he describes works, Ricardo uses a table: the "Table, shewing the Progress of Rent and Profit under an assumed Augmentation of Capital".<ref name=tablerem>The table can be found on p. [[:s:en:An Essay on the Influence of a Low Price of Corn on the Profits of Stock#14|14]]. Two remarks on this table: * 1 quarter = 8 bushel = about 291 litre. * Although the "Essay" refers to wheat, in line with the common usage amongst economists who specialise in Ricardo, the term “corn” will be used throughout this resource.</ref> This 'model' demonstrates how the economy might develop as new land is brought into use in succession. The table is constructed as follows: it is assumed that, when only the first plot of land is brought into use, a profit of one hundred quarters is made on a capital of two hundred quarters; when a second plot of land is brought into use, a capital of 210 quarters must be invested in it to achieve a profit of 90 q., and so on. The total output from each plot is therefore always 300 q. of wheat. The ‘net produce’ (total output minus production costs) amounts to 100 q. for the first plot; 90 for the second; 80 for the third; and so on. {| style="border-collapse: collapse; border: 1px solid; margin: auto; text-align: center;" |+ '''''TABLE, showing the Progress of Rent and Profit under an assumed Augmentation of Capital.''''' |- style="border: 1px solid; font-size: 70%; hyphens: auto; text-align: center;" ! style="border: 1px solid;" | Capital estimated in quarters of wheat. ! style="border: 1px solid;" | Profit per cent. ! style="border: 1px solid;" | Neat produce in quarters of wheat, after paying the cost of production on each capital. ! style="border: 1px solid;" | Profit of 1st portion of land in quarters of wheat. ! style="border: 1px solid;" | Rent of 1st portion of land in quarters of wheat. ! style="border: 1px solid;" | Profit of 2d portion of land in quarters of wheat. ! style="border: 1px solid;" | Rent of 2d portion of land in quarters of wheat. ! style="border: 1px solid;" | Profit of 3d portion of land in quarters of wheat. ! style="border: 1px solid;" | Rent of 3d portion of land in quarters of wheat. ! style="border: 1px solid;" | Profit of 4th portion of land in quarters of wheat. ! style="border: 1px solid;" | Rent of 4th portion of land in quarters of wheat. ! style="border: 1px solid;" | Profit of 5th portion of land in quarters of wheat. ! style="border: 1px solid;" | Rent of 5th portion of land in quarters of wheat. ! style="border: 1px solid;" | Profit of 6th portion of land in quarters of wheat. ! style="border: 1px solid;" | Rent of 6th portion of land in quarters of wheat. ! style="border: 1px solid;" | Profit of 7th portion of land in quarters of wheat. ! style="border: 1px solid;" | Rent of 7th portion of land in quarters of wheat. ! style="border: 1px solid;" | Profit of 8th portion of land in quarters of wheat. |- | style="border-left: 1px solid; border-right: 1px solid;" | 200 | style="border-left: 1px solid; border-right: 1px solid;" | 50 | style="border-left: 1px solid; border-right: 1px solid;" | 100 | style="border-left: 1px solid; border-right: 1px solid;" | 100 | style="border-left: 1px solid; border-right: 1px solid;" | none. | style="border-left: 1px solid; border-right: 1px solid;" | | style="border-left: 1px solid; border-right: 1px solid;" | | style="border-left: 1px solid; border-right: 1px solid;" | | style="border-left: 1px solid; border-right: 1px solid;" | | style="border-left: 1px solid; border-right: 1px solid;" | | style="border-left: 1px solid; border-right: 1px solid;" | | style="border-left: 1px solid; border-right: 1px solid;" | | style="border-left: 1px solid; border-right: 1px solid;" | | style="border-left: 1px solid; border-right: 1px solid;" | | style="border-left: 1px solid; border-right: 1px solid;" | | style="border-left: 1px solid; border-right: 1px solid;" | | style="border-left: 1px solid; border-right: 1px solid;" | | style="border-left: 1px solid; border-right: 1px solid;" | |- | style="border-left: 1px solid; border-right: 1px solid;" | 210 | style="border-left: 1px solid; border-right: 1px solid;" | 43 | style="border-left: 1px solid; border-right: 1px solid;" | 90 | style="border-left: 1px solid; border-right: 1px solid;" | 86 | style="border-left: 1px solid; border-right: 1px solid;" | 14 | style="border-left: 1px solid; border-right: 1px solid;" | 90 | style="border-left: 1px solid; border-right: 1px solid;" | none. | style="border-left: 1px solid; border-right: 1px solid;" | | style="border-left: 1px solid; border-right: 1px solid;" | | style="border-left: 1px solid; border-right: 1px solid;" | | style="border-left: 1px solid; border-right: 1px solid;" | | style="border-left: 1px solid; border-right: 1px solid;" | | style="border-left: 1px solid; border-right: 1px solid;" | | style="border-left: 1px solid; border-right: 1px solid;" | | style="border-left: 1px solid; border-right: 1px solid;" | | style="border-left: 1px solid; border-right: 1px solid;" | | style="border-left: 1px solid; border-right: 1px solid;" | | style="border-left: 1px solid; border-right: 1px solid;" | |- | style="border-left: 1px solid; border-right: 1px solid;" | 220 | style="border-left: 1px solid; border-right: 1px solid;" | 36 | style="border-left: 1px solid; border-right: 1px solid;" | 80 | style="border-left: 1px solid; border-right: 1px solid;" | 72 | style="border-left: 1px solid; border-right: 1px solid;" | 28 | style="border-left: 1px solid; border-right: 1px solid;" | 76 | style="border-left: 1px solid; border-right: 1px solid;" | 14 | style="border-left: 1px solid; border-right: 1px solid;" | 80 | style="border-left: 1px solid; border-right: 1px solid;" | none. | style="border-left: 1px solid; border-right: 1px solid;" | | style="border-left: 1px solid; border-right: 1px solid;" | | style="border-left: 1px solid; border-right: 1px solid;" | | style="border-left: 1px solid; border-right: 1px solid;" | | style="border-left: 1px solid; border-right: 1px solid;" | | style="border-left: 1px solid; border-right: 1px solid;" | | style="border-left: 1px solid; border-right: 1px solid;" | | style="border-left: 1px solid; border-right: 1px solid;" | | style="border-left: 1px solid; border-right: 1px solid;" | |- | style="border-left: 1px solid; border-right: 1px solid;" | 230 | style="border-left: 1px solid; border-right: 1px solid;" | 30 | style="border-left: 1px solid; border-right: 1px solid;" | 70 | style="border-left: 1px solid; border-right: 1px solid;" | 60 | style="border-left: 1px solid; border-right: 1px solid;" | 40 | style="border-left: 1px solid; border-right: 1px solid;" | 63 | style="border-left: 1px solid; border-right: 1px solid;" | 27 | style="border-left: 1px solid; border-right: 1px solid;" | 66 | style="border-left: 1px solid; border-right: 1px solid;" | 14 | style="border-left: 1px solid; border-right: 1px solid;" | 70 | style="border-left: 1px solid; border-right: 1px solid;" | none. | style="border-left: 1px solid; border-right: 1px solid;" | | style="border-left: 1px solid; border-right: 1px solid;" | | style="border-left: 1px solid; border-right: 1px solid;" | | style="border-left: 1px solid; border-right: 1px solid;" | | style="border-left: 1px solid; border-right: 1px solid;" | | style="border-left: 1px solid; border-right: 1px solid;" | | style="border-left: 1px solid; border-right: 1px solid;" | |- | style="border-left: 1px solid; border-right: 1px solid;" | 240 | style="border-left: 1px solid; border-right: 1px solid;" | 25 | style="border-left: 1px solid; border-right: 1px solid;" | 60 | style="border-left: 1px solid; border-right: 1px solid;" | 50 | style="border-left: 1px solid; border-right: 1px solid;" | 50 | style="border-left: 1px solid; border-right: 1px solid;" | 52½ | style="border-left: 1px solid; border-right: 1px solid;" | 37½ | style="border-left: 1px solid; border-right: 1px solid;" | 55 | style="border-left: 1px solid; border-right: 1px solid;" | 25 | style="border-left: 1px solid; border-right: 1px solid;" | 57½ | style="border-left: 1px solid; border-right: 1px solid;" | 12½ | style="border-left: 1px solid; border-right: 1px solid;" | 60 | style="border-left: 1px solid; border-right: 1px solid;" | none. | style="border-left: 1px solid; border-right: 1px solid;" | | style="border-left: 1px solid; border-right: 1px solid;" | | style="border-left: 1px solid; border-right: 1px solid;" | | style="border-left: 1px solid; border-right: 1px solid;" | | style="border-left: 1px solid; border-right: 1px solid;" | |- | style="border-left: 1px solid; border-right: 1px solid;" | 250 | style="border-left: 1px solid; border-right: 1px solid;" | 20 | style="border-left: 1px solid; border-right: 1px solid;" | 50 | style="border-left: 1px solid; border-right: 1px solid;" | 40 | style="border-left: 1px solid; border-right: 1px solid;" | 60 | style="border-left: 1px solid; border-right: 1px solid;" | 42 | style="border-left: 1px solid; border-right: 1px solid;" | 48 | style="border-left: 1px solid; border-right: 1px solid;" | 44 | style="border-left: 1px solid; border-right: 1px solid;" | 36 | style="border-left: 1px solid; border-right: 1px solid;" | 46 | style="border-left: 1px solid; border-right: 1px solid;" | 24 | style="border-left: 1px solid; border-right: 1px solid;" | 48 | style="border-left: 1px solid; border-right: 1px solid;" | 12 | style="border-left: 1px solid; border-right: 1px solid;" | 50 | style="border-left: 1px solid; border-right: 1px solid;" | none. | style="border-left: 1px solid; border-right: 1px solid;" | | style="border-left: 1px solid; border-right: 1px solid;" | | style="border-left: 1px solid; border-right: 1px solid;" | |- | style="border-left: 1px solid; border-right: 1px solid;" | 260 | style="border-left: 1px solid; border-right: 1px solid;" | 15 | style="border-left: 1px solid; border-right: 1px solid;" | 40 | style="border-left: 1px solid; border-right: 1px solid;" | 30 | style="border-left: 1px solid; border-right: 1px solid;" | 70 | style="border-left: 1px solid; border-right: 1px solid;" | 31½ | style="border-left: 1px solid; border-right: 1px solid;" | 58½ | style="border-left: 1px solid; border-right: 1px solid;" | 33 | style="border-left: 1px solid; border-right: 1px solid;" | 47 | style="border-left: 1px solid; border-right: 1px solid;" | 34½ | style="border-left: 1px solid; border-right: 1px solid;" | 35½ | style="border-left: 1px solid; border-right: 1px solid;" | 36 | style="border-left: 1px solid; border-right: 1px solid;" | 24 | style="border-left: 1px solid; border-right: 1px solid;" | 37½ | style="border-left: 1px solid; border-right: 1px solid;" | 12½ | style="border-left: 1px solid; border-right: 1px solid;" | 40 | style="border-left: 1px solid; border-right: 1px solid;" | none. | style="border-left: 1px solid; border-right: 1px solid;" | |- | style="border-left: 1px solid; border-right: 1px solid;" | 270 | style="border-left: 1px solid; border-right: 1px solid;" | 11 | style="border-left: 1px solid; border-right: 1px solid;" | 30 | style="border-left: 1px solid; border-right: 1px solid;" | 22 | style="border-left: 1px solid; border-right: 1px solid;" | 78 | style="border-left: 1px solid; border-right: 1px solid;" | 23 | style="border-left: 1px solid; border-right: 1px solid;" | 67 | style="border-left: 1px solid; border-right: 1px solid;" | 24 | style="border-left: 1px solid; border-right: 1px solid;" | 56 | style="border-left: 1px solid; border-right: 1px solid;" | 25.3 | style="border-left: 1px solid; border-right: 1px solid;" | 44.7 | style="border-left: 1px solid; border-right: 1px solid;" | 26.4 | style="border-left: 1px solid; border-right: 1px solid;" | 33.6 | style="border-left: 1px solid; border-right: 1px solid;" | 27½ | style="border-left: 1px solid; border-right: 1px solid;" | 22½ | style="border-left: 1px solid; border-right: 1px solid;" | 27.6 | style="border-left: 1px solid; border-right: 1px solid;" | 12.4 | style="border-left: 1px solid; border-right: 1px solid;" | 29.7 |} {| style="border-collapse: collapse; border: 1px solid; margin: auto; text-align: center; width:90%;" |- style="border: 1px solid; font-size: 70%; hyphens: auto; text-align: center;" ! style="width: 50px;" | ! style="border: 1px solid; width: 40px;" | When the whole capital employed is ! style="border: 1px solid; width: 40px;" | Whole amount of rent received by landlords in quarter of wheat. ! style="border: 1px solid; width: 40px;" | Whole amount of profits in quarters received by owners of stock. ! style="border: 1px solid; width: 40px;" | Profit per cent. on the whole capital. ! style="border: 1px solid; width: 40px;" | Rent per cent, on the whole capital. ! style="border: 1px solid; width: 40px;" | Total produce in quarters of wheat, after paying the cost of production. |- | style="border-left: 1px solid; border-right: 1px solid;" | 1st&nbsp;Period | style="border-left: 1px solid; border-right: 1px solid;" | 200 | style="border-left: 1px solid; border-right: 1px solid;" | none. | style="border-left: 1px solid; border-right: 1px solid;" | 100 | style="border-left: 1px solid; border-right: 1px solid;" | 50 | style="border-left: 1px solid; border-right: 1px solid;" | | style="border-left: 1px solid; border-right: 1px solid;" | 100 |- | style="border-left: 1px solid; border-right: 1px solid;" | 2d Ditto | style="border-left: 1px solid; border-right: 1px solid;" | 410 | style="border-left: 1px solid; border-right: 1px solid;" | 14 | style="border-left: 1px solid; border-right: 1px solid;" | 176 | style="border-left: 1px solid; border-right: 1px solid;" | 43 | style="border-left: 1px solid; border-right: 1px solid;" | 3½ | style="border-left: 1px solid; border-right: 1px solid;" | 190 |- | style="border-left: 1px solid; border-right: 1px solid;" | 3d Ditto | style="border-left: 1px solid; border-right: 1px solid;" | 630 | style="border-left: 1px solid; border-right: 1px solid;" | 42 | style="border-left: 1px solid; border-right: 1px solid;" | 228 | style="border-left: 1px solid; border-right: 1px solid;" | 36 | style="border-left: 1px solid; border-right: 1px solid;" | 6½ | style="border-left: 1px solid; border-right: 1px solid;" | 270 |- | style="border-left: 1px solid; border-right: 1px solid;" | 4th Ditto | style="border-left: 1px solid; border-right: 1px solid;" | 860 | style="border-left: 1px solid; border-right: 1px solid;" | 81 | style="border-left: 1px solid; border-right: 1px solid;" | 259 | style="border-left: 1px solid; border-right: 1px solid;" | 30 | style="border-left: 1px solid; border-right: 1px solid;" | 9½ | style="border-left: 1px solid; border-right: 1px solid;" | 340 |- | style="border-left: 1px solid; border-right: 1px solid;" | 5th Ditto | style="border-left: 1px solid; border-right: 1px solid;" | 1100 | style="border-left: 1px solid; border-right: 1px solid;" | 125 | style="border-left: 1px solid; border-right: 1px solid;" | 275 | style="border-left: 1px solid; border-right: 1px solid;" | 25 | style="border-left: 1px solid; border-right: 1px solid;" | 11½ | style="border-left: 1px solid; border-right: 1px solid;" | 400 |- | style="border-left: 1px solid; border-right: 1px solid;" | 6th Ditto | style="border-left: 1px solid; border-right: 1px solid;" | 1350 | style="border-left: 1px solid; border-right: 1px solid;" | 180 | style="border-left: 1px solid; border-right: 1px solid;" | 270 | style="border-left: 1px solid; border-right: 1px solid;" | 20 | style="border-left: 1px solid; border-right: 1px solid;" | 13½ | style="border-left: 1px solid; border-right: 1px solid;" | 450 |- | style="border-left: 1px solid; border-right: 1px solid;" | 7th Ditto | style="border-left: 1px solid; border-right: 1px solid;" | 1610 | style="border-left: 1px solid; border-right: 1px solid;" | 248½ | style="border-left: 1px solid; border-right: 1px solid;" | 241½ | style="border-left: 1px solid; border-right: 1px solid;" | 15 | style="border-left: 1px solid; border-right: 1px solid;" | 15½ | style="border-left: 1px solid; border-right: 1px solid;" | 490 |- | style="border-left: 1px solid; border-right: 1px solid;" | 8th Ditto | style="border-left: 1px solid; border-right: 1px solid;" | 1880 | style="border-left: 1px solid; border-right: 1px solid;" | 314½ | style="border-left: 1px solid; border-right: 1px solid;" | 205½ | style="border-left: 1px solid; border-right: 1px solid;" | 11 | style="border-left: 1px solid; border-right: 1px solid;" | 16½ | style="border-left: 1px solid; border-right: 1px solid;" | 520 |} Ricardo emphasises the illustrative nature of the figures in the table: "It is scarcely necessary to observe, that the data on which this table is constructed are assumed, and are probably very far from the truth. They were fixed on as tending to illustrate the principle,—which would be the same, whether the first profits were fifty per cent. or five,—or, whether an additional capital of ten quarters, or of one hundred, were required to obtain the same produce from the cultivation of new land."([[:s:en:An Essay on the Influence of a Low Price of Corn on the Profits of Stock#11|p. 11]], footnote). The table shows, according to Ricardo that "during the progress of a country the whole produce raised on its land, will increase, and for a certain time that part of the produce which belongs to the profits of stock, as well as that part which belongs to rent will increase; but that at a later period, every accumulation of capital will be attended with an absolute, as well as a proportionate diminution of profits,—though rents will uniformly increase. This is a view of the effects of accumulation which is exceedingly curious, and has, I believe, never before been noticed. It will be seen by the table, that, in a progressive country, rent is not only absolutely increasing, but that it is also increasing in its ratio to the capital employed on the land; thus when four hundred and ten was the whole capital employed, the landlord obtained three and a half per cent.; when one thousand one hundred—thirteen and a quarter per cent.; and when one thousand eight hundred and eighty—sixteen and a half per cent. The landlord not only obtains a greater produce, but a larger share.(p. [[:s:en:An Essay on the Influence of a Low Price of Corn on the Profits of Stock#12|12]]- 13). This is the essence of Ricardo’s theory of ground rent in the context of a growing economy: as the economy grows – that is, as population growth and capital accumulation take place – the rate of profit will fall, whilst the income accruing to landowners will rise both in absolute and relative terms. The bringing into use of inferior land leads, through competition amongst capitalists – who are repeatedly forced to settle for a lower rate of profit – to the continuous absolute and relative enrichment of landowners. == Rent: a 'pure surplus' == {| | | {{pad|700px}} |"Corn is not high because a rente is paid, but a rent is paid because corn is high."<br />({{harvp|Ricardo|1817|p=38}}). |} Having drawn his central conclusion regarding the development of income distribution in his model of economic growth, Ricardo observes: “Rent (...) is in all cases a portion of the profits previously obtained on the land. It is never a new creation of revenue, but always part of a revenue already created.” ({{harvp|Ricardo|1815}}, [[:s:en:An Essay on the Influence of a Low Price of Corn on the Profits of Stock#15|p. 15]]) Various authors emphasise this treatment of rent as a 'pure surplus'.<ref name=pure>See for instance {{harvp|Howard|King|1975|page = 77}} (second edition 1985:{{harvp|Howard|King|1985|page = 76}}); and see {{harvp|Hollander|1973|page=273}} for an explanation of the significance of Ricardo’s analysis of ‘interest as a transfer payment’ in contrast to Malthus’s concept of rent.</ref> For Ricardo, rent is price-determined rather than price-determining: rent does not form part of the value of production, which is, after all, determined by the conditions of production on marginal land. For Adam Smith, however, rent does constitute a constituent part of the price of production. Benetti points out that, in Ricardo's view, rent is certainly not an original surplus: "it is never a newly created revenue". On the contrary: the "surplus product" originally takes the form of profit, but is increasingly ''redistributed'' in favour of landowners in the course of economic development {{sfnp|Benetti|1975|page = 12}}. Consequently, in Ricardo's theory, the formation of rent does not depend on private ownership of land. Its appropriation by landowners, however, does. Benetti draws the conclusion from this that the treatment of rent (in the theory) must in fact follow the treatment of the determination of the rate of profit. A curious phenomenon arises in this context: in Ricardo's approach, good-quality land is initially freely available, so there is no rent, and the entire 'net product' is appropriated by the capitalist farmers. At a certain point, land of poorer quality is brought into use and the 'first settlers' (those who have settled on the best land), as owners of that best land, will receive rent. In this approach, therefore, the first settlers are initially merely entrepreneurs, but at a certain point become both entrepreneurs and landowners.<br />In the view presented by Ricardo, there are initially only the classes of the ‘labouring poor’ and capitalists. It is only when land of inferior quality is brought into use that a class of landowners emerges, who appropriate the rent. It might be said that the (socio-)logical and historical analyses do not correspond at all in this respect.<ref name=Veen82>This observation was made by Robert J. v.d. Veen in his lecture at University of Amsterdam, 1982-10-04-10-82 (lecture notes db, pp. 6–7).</ref> Blaug points out that Ricardo's interpretation of rent as 'pure surplus' implies the assumption that land has no alternative uses other than those for which it is actually being used at any given time. {{sfnp|Blaug|1985|page=82}}<br />"From the notion that 'rent does not enter into price', that rent is not payment for the using up of resources, the classical economists drew the practical corollary that it would make no difference if landlords were thrown to the wolves."{{sfnp|Blaug|1985|page=82}} The expropriation (or taxation) of rent by the state would not affect the scale of production. == "The share of landlords" == {| | | {{pad|700px}} |"The landlord not only obtains a greater produce, but a larger share."<br />({{harvp|Ricardo|1815}}, [[:s:en:An Essay on the Influence of a Low Price of Corn on the Profits of Stock#13|p. 13]]). |} In the second paragraph, above, it was stated that, according to Ricardo, the trends in the figures in the table show: * that total output rises in the course of the process of economic development; * that the share of output accruing to the capitalists as profit also increases for a time, but subsequently falls both in absolute and proportional terms; * that the rate of profit falls continuously; * that the sum of rent rises continuously; * that rent rises in relation to the capital invested: "the landowner receives not only a greater product, but also a greater share". Hollander points out that Ricardo "did not emphasise the behaviour of the rental and profit shares in the net or gross product."{{sfnp|Hollander|1973|p=274}} Indeed, apart from a highly ambiguous passage regarding "a larger share" in the ''Essay,'' there is no statement on the development of the share of rent in total (gross) production. From the context, one might infer that Ricardo has in mind here the ratio to the capital invested. The table does contain a column labelled "Rent per cent. on the whole capital"<ref name=whole>Compare the statement in {{harvp|Hollander|1973|p=274}} that Ricardo "is in fact mistaken" in that "beyond the eighth period the fraction (rent to agricultural capital) actually falls". {{harvp|Eatwell|1975|p=187}} (footnote) replies to this.</ref> (denoted in the next table as: R<sub>t</sub> / K<sub>t</sub>), but no column in which the share of rent in total production is calculated. However, the numerical example has apparently been chosen in such a way that the share of rent in total production rises continuously – as can be seen from the next table in the R<sub>t</sub> / K<sub>t</sub> column. {| style="border-collapse: collapse; border: 1px solid; text-align: center; margin: 1em auto 1em auto;" |+'''''Table, showing the part of rent in total production''''' |- style="border: 1px solid; font-size: 70%; hyphens: auto; text-align: center;" | rowspan=2 style="vertical-align: top; border-collapse: collapse; border-right: 1px solid;" | period | colspan=3 style="align: center; border-collapse: collapse; border-right: 1px solid;" | in quarters wheat | rowspan=2 style="align: center; border-collapse: collapse; border-right: 1px solid;" | "rent<br /> &nbsp; per cent.&nbsp; &nbsp; <br />on the<br />whole<br />capital" | rowspan=2 style="align: center; border-collapse: collapse;" | part<br />of rent<br />in<br />total<br /> &nbsp; production &nbsp; |- style="border: 1px solid; font-size: 70%; hyphens: auto; text-align: center;" | style="border-right: 1px solid;"| total<br /> &nbsp; production &nbsp; | style="border-right: 1px solid;"| "whole<br />capital<br />employed" &nbsp; | style="border-right: 1px solid;"| "whole<br /> &nbsp; amount &nbsp;<br />of rent" |- style="border: 1px solid; font-size: 100%; hyphens: auto; text-align: center;" | style="border-right: 1px solid;" | t | style="border-right: 1px solid;"| X<sub>t</sub> | style="border-right: 1px solid;"| K<sub>t</sub> | style="border-right: 1px solid;"| R<sub>t</sub> | style="border-right: 1px solid;"| R<sub>t</sub> / K<sub>t</sub> | R<sub>t</sub> / X<sub>t</sub> |- | style="border-right: 1px solid;"| 1 | style="border-right: 1px solid;"| 300 | style="border-right: 1px solid;"| 200 | style="border-right: 1px solid;"| {{-}} | style="border-right: 1px solid;"| {{-}} | {{-}} |- | style="border-right: 1px solid;"| 2 | style="border-right: 1px solid;"| 600 | style="border-right: 1px solid;"| 410 | style="border-right: 1px solid;"| 14 | style="border-right: 1px solid;"| 3,41 | 2,33 |- | style="border-right: 1px solid;"| 3 | style="border-right: 1px solid;"| 900 | style="border-right: 1px solid;"| 630 | style="border-right: 1px solid;"| 42 | style="border-right: 1px solid;"| 6,67 | 4,67 |- | style="border-right: 1px solid;"| 4 | style="border-right: 1px solid;"| 1200 | style="border-right: 1px solid;"| 860 | style="border-right: 1px solid;"| 81 | style="border-right: 1px solid;"| 9,42 | 6,75 |- | style="border-right: 1px solid;"| 5 | style="border-right: 1px solid;"| 1500 | style="border-right: 1px solid;"| 1100 | style="border-right: 1px solid;"| 125 | style="border-right: 1px solid;"| 11,36 | 8,33 |- | style="border-right: 1px solid;"| 6 | style="border-right: 1px solid;"| 1800 | style="border-right: 1px solid;"| 1350 | style="border-right: 1px solid;"| 180 | style="border-right: 1px solid;"| 13,33 | 10,00 |- | style="border-right: 1px solid;"| 7 | style="border-right: 1px solid;"| 2100 | style="border-right: 1px solid;"| 1610 | style="border-right: 1px solid;"| 248,5 | style="border-right: 1px solid;"| 15,43 | 11,83 |- | style="border-right: 1px solid;"| 8 | style="border-right: 1px solid;"| 2400 | style="border-right: 1px solid;"| 1880 | style="border-right: 1px solid;"| 314,5 | style="border-right: 1px solid;"| 16,73 | 13,10 |} A number of authors are quite adamant in their view that, according to Ricardo, the share of rent in total output will rise, at the expense of the combined share of wages and profits. However, in general, the various authors do not appear to be particularly careful in citing the source for this view. O’Brien, for example, notes: “Ricardo made very clear predictions regarding the direction of change for all three shares” {{sfnp|O’Brien|1975|p=132}}. He argues that this can be clearly seen in the table in the “Essay”. He then immediately moves on to the question of whether Ricardo was justified in drawing the general conclusion from the table that rent rises as a proportion of output – an issue that will be discussed at length further in this resource. The fact that such a conclusion is not explicitly present in Ricardo’s work–at least in the ''Essay on Profits''–has already been noted above, but is not mentioned by O’Brien. Barkai speaks of "the definite conclusion" which Ricardo draws, "that in proportion to the progress of wealth and the difficulty of producing food, although the value of the labourer's portion will be increased, his real share will be diminished, whilst that of the landlord will not only be raised in value but will also be increased in quantity. The remaining quantity of the produce of the land constitutes the profit of stock . . . as society advances, this proportion of the whole produce will be diminished.<ref name=Barkai1>Barkai here points to {{harvp|Ricardo|1817}}, page 112 (Sraffa ed.).</ref> The proof of this theorem consists of an arithmetical example spread over three chapters of his Principles.”{{sfnp|Barkai|1959|p= 245}} Hollander’s statement at the beginning of this paragraph – that Ricardo places little emphasis on the behaviour of the shares of rent and profit in the net and/or gross product – is, in any case, difficult to refute. It may be concluded here that there is certainly reason to assume that Ricardo believed that the share of rent (as a proportion of total income or as a proportion of ‘the whole capital employed’, i.e. R/X or R/K) would rise in the course of the process of accumulation, and that this is certainly the case in the numerical example he gives in the ''Essay'' (and, incidentally, also in the numerical example in the ''Principles''), but that it is premature to assert this assumption with absolute certainty, and that the various authors who do so generally provide little or no evidence to support it.<br /> It can equally be argued that Ricardo is not very categorical in his conclusions regarding the development of rent, profit and wages as a proportion of total income. What will be set out in the page "on the mathematical interpretation of the ''Essay''," following this resource should be read against this background. A further difficulty that arises in this context – and to which the final paragraphs of the page "on the mathematical interpretation of the ''Essay''" are devoted – is that, if Ricardo believed he could conclude that the share of ground rent would rise in the course of economic development, this conclusion cannot possibly be derived consistently from the assumptions on which Ricardo based his model.<br /> It will become apparent that additional restrictions (particularly regarding the form of the production function used) are necessary to conclude that the share of ground rent will always rise when capital accumulation takes place under conditions of "diminishing returns". It is worth noting that [[:w:en:Edward West (economist and judge)|Edward West]], in his exposition of the "law of diminishing returns," as evidence for this law, argues, amongst other things, that the share of ground rent as a proportion of total agricultural output has fallen over the course of history. Stigler demonstrates that his reasoning in this regard is incorrect.{{sfnp|Stigler|1952|p=197}} == See also: == This resource will be followed by some pages, in which aspects of some issues in the ''Essay on Profits'' will be examined a bit deeper: * [[David Ricardo/An Essay on Profits/Graphical|Graphical representation]] of the Ricardian theory of rent * [[David Ricardo/An Essay on Profits/Mathematical|Mathematical representation]] of the Ricardian theory of rent * [[David Ricardo/An Essay on Profits/Nature|Some remarks]] concerning the role of "nature" in the Ricardian theory == Footnotes == {{references}} == Sources == * {{cite journal | last = Barkai | first = H. | author-link = :w:de:Haim Barkai | date = 1959 | title = Ricardo on Factor Prices and Income Distribution in a Growing Economy | journal = Economica | volume = 26 | issue = 103 | pp = 240–250 | doi = 10.2307/2550483 | url = https://www.jstor.org/stable/2550483 | ref = harv }} {{small|(access via [[File:Wikipedia Library owl.svg|12px|alt=12 px Wikipedia Library Owl]] [[:w:en:Wikipedia:The Wikipedia Library|The Wikipedia Library (TWL)]])}} * {{cite book | last = Benetti | first = C. | date = 1975 | title = Valeur et répartition | lang = fr | location = Grenoble | publisher = Maspero | id = {{OCLC|461493533}} | ref = harv }} (2nd ed.) * {{cite book | last = Blaug | first = Mark | date = 1985 | title = Economic Theory in Retrospect ''(fourth ed.)'' | location = Cambridge etc. | publisher = Cambridge University Press | ISBN = 0 521 30354 0 | ref = harv | url = http://digamo.free.fr/blaug62.pdf }} * {{cite journal | last = Eatwell | first = John | author-link = w:en:John Eatwell | date = 1975 | title = The Interpretation of Ricardo's Essay on Profits | journal = Economica | volume = 42 | issue = 166 | pp = 182–187 | doi = 10.2307/2553593 | url = https://www.jstor.org/stable/2553593 | ref = harv }}<br />{{small|(access via [[File:Wikipedia Library owl.svg|12px|alt=12 px Wikipedia Library Owl]] [[:w:en:Wikipedia:The Wikipedia Library|The Wikipedia Library (TWL)]])}} * {{cite journal | last = Hollander | first = Samuel | author-link = :w:en:Samuel Hollander | date = 1973 | title = Ricardo’s Analysis of the Profit Rate, 1813-15 | journal = Economica | volume = 40 | issue = 159 | pages = 260–282 | doi = 10.2307/2552797 | url = https://www.jstor.org/stable/2552797 | ref = harv }}<br />{{small|(access via [[File:Wikipedia Library owl.svg|12px|alt=12 px Wikipedia Library Owl]] [[:w:en:Wikipedia:The Wikipedia Library|The Wikipedia Library (TWL)]])}} * Hollander 1975 * {{cite book | last1 = Howard | first1 = M.C. | last2 = King | first2 = J.E. | date = 1975 | title = The political economy of Marx | location = London | publisher = Longman | url = https://archive.org/details/politicaleconomy0000howa | url-access = restricted | ref = harv }} * {{cite book | last1 = Howard | first1 = M.C. | last2 = King | first2 = J.E. | date = 1985 | title = The political economy of Marx (2nd ed.) | location = London | publisher = Longman | id = {{OCLC|10483279}} | ref = harv }} * {{cite book | last = Malthus | first = T.R. | author-link = :w:en:Thomas Robert Malthus | date = 1815 | title = An Inquiry Into the Nature and Progress of Rent | location = London | publisher = John Murray | ref = harv | url = https://en.wikisource.org/wiki/An_Inquiry_Into_the_Nature_and_Progress_of_Rent }} * {{cite book | last = O'Brien | first = D.P. | author-link = :w:en:Denis Patrick O'Brien | date = 1975 | title = The Classical Economists | location = Oxford | publisher = Clarendon Press | ISBN = 0 19 877015 4 | url = https://archive.org/details/classicaleconomi0000obri | url-access = restricted | ref = harv }} * {{cite book | last = Ricardo | first = David | author-link = David Ricardo | date = 1815 | title = An Essay on the Influence of a Low Price of Corn on the Profits of Stock | location = London | publisher = John Murray | ref = harv | url = https://en.wikisource.org/wiki/An_Essay_on_the_Influence_of_a_Low_Price_of_Corn_on_the_Profits_of_Stock }} * {{cite book | last = Ricardo | first = David | author-link = David Ricardo | date = 1817 | title = On the Principles of Political Economy; And Taxation | location = London | publisher = John Murray | ref = harv | url = }} (when referring we will use the page numbering of the Sraffa ed.) * {{cite journal | last = Stigler | first = George J. | author-link = :w:en:George Stigler | date = 1952 | title = The Ricardian Theory of Value and Distribution | journal = Journal of Political Economy | volume = 60 | issue = 3 | pp = 187–207 | ref = harv | url = https://www.jstor.org/stable/1826451 }}<br />{{small|(access via [[File:Wikipedia Library owl.svg|12px|alt=12 px Wikipedia Library Owl]] [[:w:en:Wikipedia:The Wikipedia Library|The Wikipedia Library (TWL)]])}} gjdrsxkjpyaketpemg62pta8k24it0b 0 332040 2844481 2834384 2026-11-04T16:16:10Z Calixto Rosa 3111607 /* Projects on Wikiversity */ 2844481 wikitext text/x-wiki [[File:Cortext_logo.svg|center|frameless|350x350px]] <div style="margin: 2em;"> <inputbox> type=fulltext placeholder=Search the Cortext pages on Wikiversity searchbuttonlabel=🔍 break=no searchfilter=subpageof:"Cortext" </inputbox> </div> '''Cortext''' is a research infrastructure specializing in computational methods for the humanities and social sciences. Its goal is to provide researchers with computational analysis tools that are conceptually appropriate and user-friendly, typically to support qualitative research, document analysis, or field surveys, and in particular for processing corpora derived from their projects or related databases. * [https://www.cortext.net/ Website] == Projects on Wikiversity == * [[Cortext/Projects/Refactor Network Mapping|Refactoring Network Mapping]] * [[Cortext/Projects/Computational reproducibility|Computational reproducibility]] * [[Cortext/Projects/Distributed computing|Distributed computing]] * [[Contributing to Research Software]] 6bv7f15mtgebc62tiqu8c039skyp5cv 2844485 2844481 2026-11-04T16:51:08Z Calixto Rosa 3111607 2844485 wikitext text/x-wiki [[File:Cortext_logo.svg|center|frameless|350x350px]] <div style="margin: 2em;"> <inputbox> type=fulltext placeholder=Search the Cortext pages on Wikiversity searchbuttonlabel=🔍 break=no searchfilter=subpageof:"Cortext" </inputbox> </div> '''Cortext''' is a research infrastructure specializing in computational methods for the humanities and social sciences. Its goal is to provide researchers with computational analysis tools that are conceptually appropriate and user-friendly, typically to support qualitative research, document analysis, or field surveys, and in particular for processing corpora derived from their projects or related databases. * [https://www.cortext.net/ Website] == Projects on Wikiversity == * [[Cortext/Projects/Refactor Network Mapping|Refactoring Network Mapping]] * [[Cortext/Projects/Computational reproducibility|Computational reproducibility]] * [[Cortext/Projects/Distributed computing|Distributed computing]] * [[Cortext/Projects/Contributing to Research Software|Contributing to Research Software]] rhxbgm8b23c0hri1c4olei0qpec13ct 0 332412 2844533 2844347 2026-11-04T23:24:09Z Scogdill 1331941 2844533 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 championship boxing match in Bruges ended with cheating on the British side: British Jim Smith was losing, so his supporters entered the ring, took advantage of Smith's holding Australian Frank Slavin and beat him up, using fists, sticks, "knuckle-dusters," "life preservers" and boots. The referee ruled the match 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> A knuckle-duster was "A metal instrument made to cover the knuckles,"<ref>“Knuckle-duster, ''N.''”  ''Oxford English Dictionary'', Oxford UP,  December 2024, https://doi.org/10.1093/OED/6130372797.</ref> like brass knuckles, and a life preserver was "A stick or truncheon weighted at one end" with lead.<ref>“Life-preserver, ''N.''”  ''Oxford English Dictionary'', Oxford UP,  September 2026, https://doi.org/10.1093/OED/1055046410.</ref> One meeting of the boxing-oriented London Pelican Club was held almost immediately. Then a series of meetings following an investigation occurred in the first week of January (the [[Social Victorians/Boxing/Pelican Club Meeting Smith-Slavin match|Pelican Club meetings]] referred to below). Slavin followed this fight with a challenge for American John L. Sullivan, saying he would fight in America or Australia, but not in a private venue in England. Many, many newspapers covered this fight and the reaction to it in the boxing community. Copy-and-paste journalism and the lack of copyright laws means that many of the stories are all or in part identical to others, so no attempt has been made to offer complete coverage. ==Logistics== * 23 December 1889, 8:00 a.m. * Stakes: £1,000 ==Related Events== * [[Social Victorians/Boxing/Pelican Club Meeting Smith-Slavin match|Pelican Club Meetings about the Smith-Slavin match]] * the "[[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 Sportsman The Tuesday, 24 December 1889 issue of ''The Sportsman'' devoted page 3 (7 columns) entirely to this story. This reporter's writing style is very ornate, and he also uses slang and local vocabulary: sabots, Brums, Newspaper Title: The Sportsman 24 December 1889, Tuesday: 3 [of 4], Col. 1a–7c [of 7]. British Newspaper Archive https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001055%2F18891224&page=3&article=057&stringtohighlight=mandeville+viscount. Same print title and p. THE GREAT FIGHT. SMITH & SLAVIN FOR £1000. DISORDERLY PROCEEDINGS. BRUTAL ATTACK ON SLAVIN. SMITH KNOCKED DOWN REPEATEDLY. HE LEAVES THE RING. A DRAW DECLARED AFTER 14 ROUNDS. The long-talked-of fight between Jem Smith, Champion of England, and Frank P. Slavin, Champion of Australia, took place between eight and nine o'clock yesterday morning in the private grounds of a gentleman's residence near Bruges, Belgium. The contest, as everybody knows, was for 1000''l''. Thanks to the most cowardly and disgraceful behaviour of a gang of ruffians in Smith's corner, the referee was forced to declare the fight a draw at the close of the fourteenth round. Full details are given below. THE JOURNEY TO THE CONTINENT. (By our special reporter.) [?] travellers accustomed to make long and frequent journeys do not trouble themselves much about starting. The pangs of parting are easly effaced, and after a few years of "globe-trotting," as the Yankees are pleased to call it, beyond looking after personal comforts the travelling Briton has very little trouble. I have made many notable journeys upon all sorts of errands, but must confess that entering upon my last jaunt brought not a few  qualms — other than those likely to be experienced at sea. The preliminaries of the international fight, which was the reason of the trip from town, had not been carried through without many untoward circumstances arising to spoil the fun. On Saturday, however, there was every prospect of the business being brought to a successful termination, and when word was brought me that on the following morning our party were to leave London at an early hour there was no need for further hesitation, and preparations for the journey were rapidly made. What a day it was! Rain of a most annoying and penetrating kind fell until well into the night, making every one hurry to bring their day's business, or pleasure, to an end, that they might get indoors out of the dismal, depressing weather. Just as I had thought all safe for the [night came] A Rude Shock. [Arriving??] at the office of THE SPORTSMAN I found a well-known sporting journalist, whose expressive features at once conveyed to my practised eye that "something had happened." His right hand convulsively clutched a telegram, and it was quickly patent that this innocent-looking piece of paper was the cause of the trouble. "What do you think of this?" asked he, handing over the document. Only a few words were pencilled thereon, but these were enough to cause us alarm. "Stoppage, but think can get bail. — Slavin." Here was a pretty "go," the mystic words being translated meaning that the Australian had been "pinched" at Margate. Before we had thoroughly grasped the situation a second wire arrived containing the definite information that Slavin had been arrested at the Kentish watering-place and that he had a difficulty in finding bail among strangers. This dire news sent our spirits down to zero. Two acts of the little drama we were enacting had been very depressing, but the third brought sunshine again. Close upon the heels of the second telegram came a third, and it was with trembling fingers that at the request of my friend I opened the missive. A glance was enough — Slavin was free, and on his way to Dover. This sent us home satisfied that all would be well on the morrow now that both the principal actors in the contest were on the spot or nearby. Daylight brought with it A May Morning. Long before the earliest purveyor of the lacteal fluid arrived in the quiet suburban street to wake the sleepers with his hideous yelping, a cross between that of a beaten dog and a Comanche Indian, I had turned out, as there was no time to be lost. It was a delightful morning, the rain clouds having vanished, and being replaced by brilliant sunshine the weather being more akin to early summer than that usually associated with late December. The early promise did not last, as before eight o'clock the south-westerly breeze brought up rain clouds once more, and a fine free sample of "London peculiar" had taken possession of the streets as I wended my way towards Cannon-street Station. Ten o'clock was the hour named, but half an hour before that time several of the party had arrived at the trysting place. First to be noticed was Bob Habbijam, in company with two less-known sports, and this trio had not the dim terminus long to themselves. Jack Lewis, looking "all broke up," put in an appearance a few minutes later, while at the same time three scribes with pockets bulging from a surplusage of note-books dropped in. It was very soon evident that the company would be much larger than intended, as cabs rattled up continually, bringing a further complement. Tom Smith, Bob Kirby, Barney Sheppard, "Pop," and Frank Hinde were the next to join us, and in quick succession came Teddy Bayley, Will Riley, and Joe Vezey, who was to occupy the unthankful office of referee. According to the original conditions, a number of the intending spectators should have proceeded by way of Calais overnight, but by some means or other they managed to miss the train, and it was owing to this fact that so many were present on Sunday morning. There was no disguising the fact that such a crowd would draw attention, and militate against the secrecy necessary to insure us against intervention. There was no help for it, however, and before the Continental express came in from Charing-Cross "The Squire," Ike White, J. Kilby, C. Horton, C. Parsons, and J. Fleming had swelled the ranks of the ''voyageurs''. Down to Dover. It was only by judicious squeezing and packing that room could be found for all in the train, already well freighted with passengers, but eventually everyone managed to find a corner, luggage being conspicuous by its absence. I had a real sportsman for companion, Jack Kilby, of the Ferry Hotel, Cookham. Having missed the train overnight, not to be thwarted of his promised journey he arranged to proceed to town by road in the early morning. Instructing two lads to sit up through the night so as to be awakened in good time, Kilby turned in. At four o'clock he had the horse put in the shafts of a dog-cart, and drove slick away to Cannon-street. It was a particularly quiet trip, broken only by stoppages at Ashford and Folkestone, where some of the irrepressibles saw fit to leave the train and perambulate the platform. This, to say the least of it, was a foolish proceeding, as it fairly "gave the game away," and there was almost an air of expectancy about the officials when, after a run of nearly two hours, the South-Eastern express pulled up on Dover [Col. 1c–2a] Pier. Thoughts of coming trouble had ere this sobered down the playful ones, and it did not add to their joy when, looking seawards, they saw what is known as "dirty" weather in store. On the Briny. The transfer of passengers from the pier to the deck of the Princess Henriette was rapidly effected, and "once on board the lugger" my fellow-travellers disappeared as if by magic, but as rapidly came to light again, having, in view of developments likely to appear later, made arrangements below. There was a general changing of headgear, the bowler being discarded for the more useful tight-fitting cap — and what a variety were donned! As on the appearance of the clown in the pantomime, now the fun commenced. Before noon, when the hawsers were cast off, several shining lights in the boxing world prepared for the worst, the long rollers which broke in against the pier giving them the greatest alarm. Many had never previously ventured upon anything of a more stormy character than the Thames, and but for the fear of reproach would have turned back. As the "Princess" swung off from the pier the two "Bobs," Kirby and Habbijam, betook themselves to the "rail" in expectation of an attack of ''mal de mer''. What a pitiable sight they presented, but in a few minutes they found many brothers in misfortune, Tom Smith caving in. For a time it was a bit rough, and one of the good sailors commenced quoting Byron. "O'er the glad waters of the dark blue sea," sang he. Well, it was not too cerulean, but there was no question as to its gladness. The English Channel is generally built that way, and it was evidently making extra merry on account of the season. Once away from the shore-swell it was smooth enough, and the sickly division came round wonderfully. Frank Hinde, who played his old part of low comedian, amused them, and at length by his wonderful flow of spirits — animal and otherwise — had the worst cases on their legs again. A few exceptions there were, half a dozen or so of the swells having "sought the seclusion which the cabin grants," and there they stayed. Our low-comedy merchant now found a a [sic] serious rival in "Pop," and between them they kept the coterie on the quarter-deck from being dull. Yarns were spun by this pair of drolls, and the merry jest went round until more substantial fare was in request at the table. Both the aforementioned did their share, the way Hinde put away the humble yet satisfying cold meat and pickles causing intense envy in the breasts of the less hardy mariners. A very quick passage was made across the streak, and at half-past three the coast appeared in sight. At the same time rain commenced to fall, and as we steamed past the lighthouse on Ostend breakwater it came pelting down sharply. At a quarter to three the good ship was secured alongside the wharf, and out came the lost sheep looking white and miserable, one passenger, although not of us, who had sat on deck more dead than alive the whole way over, fairly dancing with joy at his release from the purgatory of sea-sickness. Ostend to Bruges. While those unfortunate enough to be hampered with luggage stood by for the purpose of having it searched the rest of us secured seats in the train already drawn up alongside. It was then discovered that we did not start for half an hour, and the interval was turned to account by a  visit to the buffet. ''Cafe-au-lait'' was the general order, and mightily refreshing it proved to be. Everything comes to him who waits, but before the journey was continued sharp to the time mentioned, it was very patent that our descent upon Belgium in such numbers had attracted much attention. One dark-bearded "foreigner" who had watched us closely on the way across, actually grew so confident that he had exploited our movents that he grew bold beyond discretion. Coming forward with all the effrontery imaginable he rather startled us by asking, "Well, when is the 'mill' coming off?" In a twinkling one and all assumed a look of stony indifference, and on the question being repeated he was forcibly yet politely informed that we had nothing to do with any "mills." As we left the station our inquisitive friend was seen gesticulating to a belaced Ostender, evidently proud of his perspicuity. Thirty minutes' run through the flat and thoroughly soaked country brought the gable roofs of ancient Bruges in sight. Night had not yet darkened the streets, but there was mist enough to hide our movements to a great extent. It had been agreed that the brigade should split up as much as possible and spread themselves over the city. This was done in a great measure, Tom Dunning piloting my section across the  broad station-square to the Hotel de l'Univers. On the threshold I met Mick Taylor, who naturally looked somewhat aghast at the advent of so big a muster. It must have surprised him still more when others strolled in, and eventually the Britisher's party reached no less than fifteen. Fortunately our worthy host had plenty of available room, and after climbing up the winding stairs the whole troupe were comfortably housed. Water was at a premium, while soap could only be obtained by borrowing from the one or two lucky ones who had thought to bring that useful saponaceous compound with them. Entering the saloon after an outward and inward refresher, I ran across Josh Cosnett, and close behind the Brum stood, Frank Slavin, looming fine and large. The Australian Champion looked in grand condition, his face bronzed by the sea air and the toughening preparation he uses. Gently puffing a cloud from his Belgian weed, he greeted the arrivals cheerily, and was as skittish as a kitten, his hearty laugh resounding through the establishment. He had a very favourable passage across Channel the previous night, and amused himself by strolling around the city during the day. It was a piece of good fortune that not only could the ''garçon'' speak English fairly well, but at least two of my companions were conversant with the language of the country; negotiations anent dinner were therefore carried out with celerity. Pugilism at the Piano. While awaiting the refection Slavin opened the piano which stood at one end of the ''salle-à-manger'', and dashing his lingers over the keys coon gave evidence that he was no mean performer on that instrument. Waltz, polka, and march all came as one to him. What a staggerer this would have been to the old timers, who never dared to allow a man in training such liberties. There was not the slightest sign of fear for the result of his coming encounter, and, in fact, Slavin appeared the least interested of the crowd. His piano selection whiled away the interval before dinner, at which the Australian presided. Taking his place at the head of the table Slavin entered into the fun, and no one would have conceived for a single moment that he had such a weighty matter on his mind as a fight. The repast was a lingering affair, and Slavin retired half way over the course, and betook himself once more to his favourite piano, which he wrestled with on and off during the remainder of the evening. While we were enjoying the good things under the roof of the Hotel Univers the rain had increased, and a perfect downpour succeeded. This had the effect of keeping the majority confined to the house, but several of the more venturesome spirits proceeded to explore the city, and they at any rate got a good drenching for their pains. The "pelting of the pitiless storm" continued the greater part of the night. Arrival of Jem Smith. The Champion of England, accompanied by Jack Baldock, Jack Harper, Jem Carney, and several other henchmen, came on from Lille at six o'clock, and arrived at Bruges about 8.30, taking up their quarters at the Hotel de Londres. Jem Smith, after leaving London on Friday, had travelled by easy stages to the Belgian frontier, so as not to miss a meal or proper night's rest and exercise between. He looked wonderfully well, having worked off much superfluous adiposity during the last few weeks' training. Unlike his opponent, Smith kept very quiet during the evening, and retired early. A meeting of the representatives of the men took place at a late hour, and in company with the owner of the ground and others interested, the final arnuigements were made. I was present to assist at this necessary proceeding, and so perfectly had all the preliminaries been settled that there was very little left to be done[.] Owing to the presence of a mob of Birmingham men, who by some means or other had made their way to Bruges, there was fear of interference, but every precaution was taken to defeat anything disagreeable occurring. The behaviour of many in the streets during the evening was not calculated to assist those who desired to see a fair and square battle. Strict orders were given to the ''garçon'' to call us at 5.30 sharp, and to make assurance doubly sure many affixed their names on bedroom doors. Punctually to the moment came the rat-tat on the panel, and a hasty toilet was made by the light of the dip. A glance through the window reassured us with respect to the weather, the rain having ceased, but it was yet too dark to get any idea of the future. Even thus early the patter of sabots on the stones outside betokened the fact that Bruges, sleepy as it is said to be, was well awake. When the sounds from the famous belfry announced six o'clock [Col. 2c–3a] the bulk of the dwellers in the Hotel Univers had found their way to the ''salle-à-manger'', or "the manger" as "Pop" expressively named that apartment. Tea, coffee, fillets de bœuf, and chops were discussed, and the early birds had only been fairly started when with a cheery "Good morning, gentlemen," Slavin appeared. He lost no time, and was soon busy with knife and fork. Time flew by rapidly, and there were signs of an early daybreak. This hurried on the final preparations. Bills were paid, creature comforts taken on board, Slavin, with the greatest ''sangfroid'', seeing to his own business as unconcerned as possible. Our pilot put in an appearance shortly after the half-hour had chimed, and informed us that the other sections had already gone on. The March to the Battlefield. Breaking into groups of half a dozen the advance guard, led by an English resident in Bruges, pushed on, but before going two hundred yards there came a check. The railway had to be crossed, and just as the leading division reached the line they found the gates closed. Although opened in a couple of minutes it allowed the party to amalgamate, but they soon filed out agent. Many streets had to be traversed, and in the now increasing light we were scanned by groups of workmen and workwomen who were on the way to the many factories in the neighborhood. Talking except in whispers was tabooed, so that our nationality might not be discovered, and without further incident a walk of twenty minutes or so brought us to the chosen spot. Entering the private grounds by a wicket-gate, there was light enough to see that the others were present standing among the trees. Four moving objects suddenly appeared as we crossed the garden carrying some white posts, which quite gave Joe Vesey a fright. The principals were hurried off to comfortable quarters near by, Smith having the stable to make his toilet in, while Slavin was accommodated in a barn. Proceeding towards a big plantation I found several quietly at work Building the Ring. Between two belts of trees, formed chiefly of firs and beeches, was a clearing forty feet wide, and here the turf had been swept clear of the fallen leaves — in fact, a native, whose gigantic sabots caused much amusement, was still busy with rake and roller[.] Barney Sheppard superintended the pitching of the ring, an operation which took some time. Eight heavy stakes had been provided and plenty of rope, but the ground was very soft after so much rain, and wedges and guy-ropes had to be freely utilised. There being no tape, the distance was measured out by strides, and when eventually the job was finished it was plain that plenty of room had been allowed. The spectators amused themselves in various ways until, when the clock struck eight, all was in readiness. It was now broad daylight, and as several rustics on their way to employment caught sight of the, to them, peculiar proceeding, and stopped to more closely examine, the place did not promise to be free from interruption. All went well, however, and as the business did not concern them they passed on. There was some delay at this point, and it was close upon half-past eight when the men were reported ready for the fray. The following comprised the company round the ring: — Lord Mandeville, Lord Royston, Hon. M. Sandys, Mr Abington, Mr F. Kotchie, Mr J. Kilby, Mr C. Parsons, Mr Ike White, Mr W. Riley, Mr C. Morton, Mr Harris, Mr E. Bayley, Mr J. Lewis (Australia), Mr J. Vesey, Mr F. Hinde, Mr J. Fleming, Mr M. Taylor, Mr J. Calder, Mr G. T. Dunning, "Pop" Joy, R. Habbijam, G. ("Barney") Sheppard, Jem Carney, Tom Smith, Mangie [?] Hyams, Matthews, Summers, Boulton, and several others, including two or three reporters. Smith, accompanied by his seconds, J. Harper and J. Baldock, entered the ring, quietly going over to the south-west corner with hardly any notice. Baldock at once wanted to know why Slavin had not turned up, and was inclined to be nasty, when the Australian walked briskly to the opposite corner, having, without the customary teasing for choice of positions, allowed Smith to take the higher ground. Josh Cosnett and Bob Kirby accompanied the Australian Champion as seconds, while Mick Taylor superintended the arrangements in his corner. A loud cheer greeted their advent. There was a few minutes' delay after the men had taken up positions, owing to Slavin not being provided with a seat, and Baldock again wished to hurry him up, but matters proceeded peacefully. Mr Joe Vesey, who had been appointed to the unthankful post of umpire, took up a position on the lower side midway between the corners, and Mr J. Calder was selected as timekeeper. On either side of the referee Mr J. Fleming, umpired for Smith, and Mr W. Riley for Slavin, these preliminaries being quickly settled. The disposition of the officials made Slavin's corner appear the stronger, and to still further add to this, a large percentage of the spectators selected that side, so as to get their backs to the wind. Another, and, perhaps, a more potent reason was the fact that The Dreaded Birmingham Division had massed themselves behind Smith and along the higher aide. They had long caused considerable fear among the spectators, but were not so noticeable until they stationed themselves in a phalanx near the Englishman. While the corners were being arranged several of the "boys" worked their way round to Slavin's quarters and stood menacingly by. Mick Taylor was too old a bird to be caught, and with the assistance of Prank Hinde, Lord Mandeville, and one or two trusty friends, guard was kept over the water and other articles. Every necessary for the two athletes had been provided by the proprietor of the grounds, so that there was no trouble on this score. Smith was wrapped in a heavy frieze overcoat, while Slavin, who, during his sojourn in the barn had been divested of his heavy moustache — a loss he deeply regretted — was also covered by a big ulster. It was now half an hour beyond the time fixed for the commencement of hostilities and broad daylight, even the belfry of Bruges being plainly visible in the distance, but curiously enough not a soul beyond the pugilistic party was anywhere near the fighting-ground. At last everything was ''en règle'', and in perfect silence Mr Joe Vesey came forward and said, "I hope all those present will keep quiet and help me all you can," at which brief speech there was a chorus of "hear, hear," especially from Slavin's corner. At 8.35 Belgian time, the two antagonists met in the centre of the arena and shook hands, the seconds, as in time-honoured fashion, doing likewise, forming what is known as the Maltese Cross. Smith, who partook of a draught before commencing, was the first to peel, and although he looked fine enough in the face, when his covering had been removed he appeared fat, although nothing like so obese as when he met Peter Jackson. The Englishman wore short white drawers and green socks, while a strengthening plaister adorned his waist. Slavin looked the picture of health and strength, having trained down very hard, his muscles standing out most prominently. He was clothed in long black drawers and blue stockings, and, like his opponent, wore a deep plaister. In addition Slavin used a suspensory belt loaned to him by Charlie Mitchell, who had found it of great service in his fight with John L. Sullivan. Under weather, which for the business in hand was simply perfect, and particularly warm for the time of year, there was, as far as could be seen by the uninitiated, every prospect of a good [Col. 3c–4a] fight. This was doubted by the majority, the presence of the crowd of roughs betokening trouble when once the battle should commence. The Call of "Time" At twenty-five minutes to nine brought everyone to the ropes, and amid perfect silence the pair of pugilists shaped for the encounter. Slavin from the first smiled confidently, while Smith looked serious. Neither appeared in any hurry to begin, and sparring for an opening, Slavin gradually worked Smith back into his corner. Fifty seconds elapsed before a blow was struck, but this had not passed without our opinion being confirmed, that we might expect trouble from the Brums. Smith's seconds set the ball rolling by hurling the vilest of epithets at the Australian, and their example was followed by their supporters. By some oversight the seconds had not been ordered out of the ring, but attention was at once called to the fact, but, constituting himself referee, Baldock refused to obey the order to retire when given by the appointed officials. This boded ill, and before a round had been fought it was pretty plain to all that Slavin would have little chance of victory. Had the objectionable party confined themselves to vituperation this might have been borne, but going from bad to worse we soon witnessed a scene that was not only a blot upon the good name of Englishmen as lovers of fair play but one never equalled even in the worst days of the ring. Slavin was the first to lead, but only tapped his opponent lightly on the face, and he had the best of several exchanges, but did not put forth all his power into his left-handers. Smith's blows were also comparatively harmless, until the last half minute, when they got to close quarters and hammered away sharply amidst a chorus of yells from the Smithites, until they fell together. No sooner were the two men down than Baldock rushed up, and in picking his man up knelt on Slavin. A foul was claimed, but no notice was taken of it, although it was very palpable. Warnings were given to the Australian not to go into Smith's corner, but without avail, and he entered upon the second round full of pluck and confidence. Language — if such filthy expressions as were levelled at Slavin could be so designated — of the violent kind greeted him on facing Smith, and this was not confined to the hired mob. Those from whom at least fair play might reasonably have been expected indulged in blasphemy and rowdyism even more pronounced than that of the others. To still further disconcert Slavin a claim of first blood was made. The moment he walked across the ring Smith worked up close at once, and on rushing in was met with the left on the mouth. A brief struggle ensued before they fell together, Smith while on the ground spiking Slavin in the thigh. A foul was again claimed but the referee was so badgered that he could hardly have known where he was, and no notice was taken of the occurrence. In the third round, again a short one, Smith was Knocked Down By a cross on the jaw, and he came up looking the worse for wear next time, while the blackguards in his corner were more threatening than before. Smith again waited in his corner, and, nothing daunted by their menacing attitude, Slavin boldly went after him. As he advanced he was assailed by the miscreants, who now saw that Slavin was getting the best of it, and when after another round had been fought, in which Smith began to blow freely, the Australian was near the ropes he received a blow from one of their number. Every time Slavin was forced to fetch Smith out of his corner, where at this early stage the ropes had been loosened. In the sixth round Slavin now warmed up, sent home a hot left-hander on the right eye, from which the ruby fluid ran freely. This he followed up by some sharp work, eventually fetching Smith down with a vicious right-hander. This treatment of their "Champion" did not suit the villanious [sic] coterie, and Slavin went across the ring next time at great risk. Smith worked Slavin round to the ropes, and while there the dastards held him for the Englishman — save the mark! — to punish, and there was an indescribable scene of confusion. Still, Slavin would not be denied, and in the eighth round sent the right heavily on the jaw, and fairly knocked Smith off his legs. In the following round Smith received a smash on the mouth, but Slavin was terribly handled by the outsiders, who punched him freely while in holds. Smith up to this had hardly made a mark on his antagonist, and beyond a cut on the lip he showed no signs of punishment, but the cowardly ruffians who assisted the "Champion" wounded him severely. So vile was the conduct that the referee was, like every one else had long been, thoroughly disgusted, and he tried without avail to put a stop to it, calling upon Harper to "be a man." Smith was generally short in his deliveries, and Slavin, paying more attention to the body had doled out some telling work. In the twelfth round Smith was felled in his corner, and this brought a crisis. The latter broke away from Slavin's attempts to get on with the left until he was almost backed up against the corner stake, when the Australian dashed the left on the face. No sooner was he fibbing away than the ruffians crowded up and showered blows on his head. In the midst of the mélée a cry of "Here they are" was raised, and A Stampede Ensued from Smith's side. It was evidently a concerted plan, as when the leaders found that no one stirred from Slavin's corner they actually came back, and pointing to the roadway, yelled, "There they are, there they are." By this time Smith had run a hundred yards away towards the house, and Slavin's party, under Rule 26 [?] claimed the fight, but to no purpose. The confusion ensuing upon this manoeuvre looked like putting an end to further fighting. Slavin, who had shown not only admirable pluck and judgment, but also kept his temper, was quieted down when he thought there was yet a chance of winning. Smith was fetched back again, and time was called. Slavin, before putting his hands up, addressing his remarks to the crowd at the other side of the ring, said, "Are you Englishmen? I am an Englishman, and only want fair play." The fight then proceeded. Seeking to settle matters now that he had Smith weakened, Slavin went right over to Smith, and planted the left sharply on the body twice, and then dashed in with both bands, landing on the jaw. As they closed someone of the brutal mob struck Slavin on the right ear again with a "duster," the referee saying, "I saw that." As Slavin threw Smith and the mob again became aggressive, Vesey, who could stand it no longer, said, "I refuse to act any longer. It is impossible for this man — pointing to Slavin — to get fair play." The referee concluded by giving the decision, "It's a draw." The mob heard this with fiendish delight, and at once cleared off from the ring-side. Slavin could hardly contain himself, shouting "There is not a man round the ring can kill me. l came 16,000 miles for a fair fight, and why don't you give a man a chance." There was no help for it, and the latest fight for the Championship ended under circumstances which brought the greatest discredit upon those who were responsible for [Col. 4c–5a] the presence of the wretches, who were prepared for any villany [sic] to prevent Smith being defeated. Not satisfied with their performance at the ring-side, they followed him to his temporary dressing-room, but were soon persuaded to leave him alone, which was perhaps fortunate for at least some of the party. Jem Smith had a carriage in waiting for him, and was rapidly carried off to the town, where it was intended to stay for some hours. There was a rush to the telegraph-office to send messages to interested friends, but only a few minutes were available, as at least a third of  the party intended Making All Speed Homewards. The Ostend train was waiting in the handsome station, and there was no time to be lost[.] Slavin, who walked the whole of the way back into town, was hurried into a first-class compartment, where he made himself comfortable in a corner with a cigar. There was a crowd of hangers-on in and around the station, who evidently were by this time aware of the nature of the morning's work, but without further incident the train steamed out of Bruges. Arriving at Ostend at 10.15 we found the Comte de Flandres ready to receive passengers at the quay-side. This boat is not to be compared with the Princess Henriette, being much smaller and less seaworthy. Slavin had a deck cabin prepared, and therein held a levée, champagne and other sparking wines being ordered up. The Australian, under the careful hands of Mick Taylor, had the cruel wounds inflicted by the ruffians at the ring-aide attended to, and by the time the boat was clear of the offing Slavin was fixed up in shape and paraded the deck. Everyone talked of the fight and its terrible surroundings, and nothing but the severest condemnation was accorded the Englishman's party. One gentleman, who was wearing Smith's colours, was so disgusted that he threw them overboard, and replaced the kerchief with one bearing Slavin's monogram. Dover was reached in due course, a big crowd awaiting Slavin's arrival on the pier, but, shorn of his moustache, he was hardly recognised. He proceeded thence to Margate with M. Taylor and J. Cosnett, where another ovation awaited him. The other portion of our company journeyed direct to London, and on detraining at Cannon-street there was a crowd collected to gather news from the seat of war. Smith was to leave Bruges during the evening, and all being well will arrive in London early this (Tuesday) morning. THE FIGHT. First Round. Slavin left his corner quickly, with a firm, determined step. Smith was much slower in his movements, and the Australian had almost reached his opponent's corner when the men were face to face. The disparity in the height of the men was very marked, Slavin standing well over the Englishman. Each wore a deep strengthening plaister, whilst Slavin, it may also be added, made use of the suspender worn by Charley Mitchell in his memorable fight with John L. Sullivan. Full of confidence, and evidently in splendid condition, the colonial looked every inch a fighter, his length of reach, and lithe, agile movements contrasting strongly with the lumbering position struck by Smith. Though fairly clean in the face Smith looked very heavy below, and undoubtedly carried too much flesh for the business he had taken in hand. Great caution was exercised on both sides, and it was seen that neither man intended to throw the slightest chance away. Weaving in Slavin feinted with his left, but Smith was quickly out of danger, and in turn Slavin dodged an intended blow from the Englishman's left. In this manner fifty seconds sped on, after which Smith got in lightly on the month. Exchanges of a light character followed, Slavin reaching the body and Smith the face. Subsequently the latter, who seemed considerably puzzled by the Australian's style, several times led out of distance at the head, Slavin on each occasion breaking ground very prettily. At last the men got to close quarters, where some sharp infighting ensued. Slavin made the body his mark, where he put in heavy work with the right, Smith in the meantime visiting his opponent's head. The Australian's blows seemed to take the most effect, and the round ended by Slavin having the worst of a fall. Time, 2min 30sec. Second Round. No sooner had the men been removed to their corners than Jack Harper called out in a very loud voice, "First blood," a claim which was of course allowed, the Colonial showing a slight trickle from the lips. On "Time" being called the pair were quick to scratch, that is to say, Slavic walked right across to Smith's corner, the Englishman refusing to come out. A long opening spar took place, but at length blows were exchanged, nothing very serious being done, and closing, the pair had a slight struggle, both falling together. Whilst on the ground a good deal of kicking and struggimg was indulged in, and whether intentionally or no Smith spiked his opponent on the inside of the left thigh, tearing his drawers, and causing the blood to flow considerably. Uproar followed, the round lasting 20sec. Third Round. Though up to this time neither man had sustained much punishment, the exchanges not having been very severe, Smith was always carried to his corner, Slavin, on the other hand, walking beck to his. At the beginning of the third round the referee ordered the seconds out of the ring, but they declined to go, saying that official had no power to order them to do so. On starting Smith forced the fighting, and leading with the left, hit Slavin upon the body. This he followed up rapidly with two more in the same place, all light hits, and some rapid exchanges followed, the Colonial using the right with effect upon the side of the head. Smith paid principal attention to the body, but at length Slavin, drawing back the right, shot it out straight from the shoulder, catching the Englishman fair on the point, and sending him down all of a heap. This magnificent delivery put the Slavinites in excellent spirits, the Colonial smiling all over his face, while the Smithites looked serious. First knock down blow was claimed, and, of course, allowed for Slavin. Time of round, 24sec. Fourth Round. Smith rose from his seat looking very serious indeed, Slavin, on the other hand, being in much better spirits. At the same time a suspicious movement was noticeable among the brigade of Birmingham ruffians, and one of them came round to the Colonial's corner with no good intentions. Had it not been that Mick Taylor and Frank Hinde kept zealous guard over all the water provided for the use of the combatants, there is no knowing what fiendish trick might not have been perpetrated. The remainder of that party stayed in Smith's [Col. 5c–6a] corner. As for the seconds, all remained in the ring. The opponents sparred for the first twenty-five seconds, and at length Slavin, getting an opening, led out of distance at the body. The Englishman seemed glad to have a blow, but Slavin weaving in delivered lightly with the left upon the cheek. Again the pair faced each other in the centre of the ring, and Slavin went for his man, and feinting with the left, just failed to bring off the right, Smith skipping out of the way. Again the Antipodean feinted with the left, and following up the charge, this time successfully caught Smith flush on the mouth. Then quick as lightning getting to close quarters, Slavin delivered heavily right and left on the body, Smith being very slow in responding to this vigorous attack. He had, however, other assistants, for a vile ruffian from over the ropes deliberately struck Slavin a blow on the head. A shout was at once raised, but when charged with it the rascal denied that he had done anything. A scuffle ensued between the men, and to escape from the difficulty Smith cross-buttocked his opponent, who fell, the seconds, rushing to the centre of the ring, falling over them. Time, lmin 15sec. Fifth Round. "Don't go into his corner," urged Taylor and others of Slavin's party to their man, but nothing could restrain him when time was called, and, springing up with a bound, he advanced towards the opposite comer, where Smith was standing up. Once or twice the Englishman seemed inclined to go into the middle of the ring, but was always prevented by his yelling seconds, who stood on either side of him. On toeing the mark Slavin appeared with a suspicious-looking redness of the left eye, caused by the miscreant above referred to. Though appealed to the referee declined to give any decision on the subject, and the fight proceeded amidst some of the most horrible language it has ever been our misfortune to listen to. This proceeded entirely from Smith's corner, who, seeing that their man was already beginning to blow and look distressed, prepared to stop at nothing. Slavin, after wasting a second or two, followed Smith into his corner. One or two exchanges, quickly delivered, of a light nature took place, and the Colonial got in on the body with the right, following up with the same on the head. This Smith considered ample justification for sinking gracefully upon the ground, the round having lasted only sixteen seconds. Sixth Round. Smith came up blowing Iike a grampus, whilst on the other hand, Slavin, seeing his advantage, was eager to begin. Of course, it was necessary to go right over into Smith's corner, that worthy not moving two feet from his chair, whilst the shouts, jeers, and din which greeted the Australian were of an inexpressibly disgusting nature. Still, nothing daunted, the Antipodean followed his man up, Smith leading at the head. This Slavin easily turned aside by a lithe movement, and in turn was stopped by Smith. Resuming, Slavin banged in the left on the mouth, Smith countering on the body. A smart counter ensued, and at this juncture the referee entered the ring, making a fruitless appeal to Smith's corner to behave themselves. Nothing, however, could silence the foul language which proceeded from that quarter. Once more facing each other, the Australian shot in the left fist heavily on Smith's left eye, opening a severe cut, which bled freely. Some vicious infighting followed, and Smith twice got the right in upon the head. Slavin countered with a terrific right-handed blow on the eye, knocking Smith down. Time, 1min 20sec. Seventh Round. In vain did Slavin's friends implore him not to go into Smith's very "warm corner." It was plain that if he did not do so there would be no fight, Smith never coming many feet away from the rascals who had been hired to protect him. Getting to close quarters Smith delivered the left lightly upon the face and banged in directly afterwards the right on the body. Slavin did not see the Englishman having matters all his own way, and a desperate rally took place, during which Smith fought Slavin on to the ropes, and there held him. Here was a grand opportunity for the miscreants, who did not fail to make use of it. One of them kicked Slavin repeatedly, and others beat him about the head, whilst, naturally, Harper and Baldock were yelling at the top of their voices. The confusion wars something terrible, and the ring looked very much like being invaded. The ropes in Smith's comer were pulled right down, so that an entrance might be effected at any moment, and it was a relief when the pair wrestled and came down in a dog-fall together, the round having lasted one minute. Eighth Round. Crossing over at once to the only place where Smith was to be found (''i.e.'', in his own corner), Slavin worked his opponent round to the side of the ring, and there let him have the left lightly upon the face. Standing at this moment close up to the ring-side, the Colonial received a couple of smart blows from a stick which a ruffian coward carried.  An immediate cry went up, and the individual finding himself under very close observation immediately slunk behind. Taking not the smallest notice of these interruptions Slavin pursued his legitimate opponent, both Smith's following him about and jeering in his face. Again the referee entered the ring, and cautioned the seconds to move away. This had very little effect, and in the midst of half-arm fighting of a very severe kind, Smith, who was receiver-general, judged it advisable to seek mother earth.  Time of round, 1min 30sec. Ninth Round. This was altogether a very curious round. When time was called Slavin got up to find his opponent, who barely moved a foot from his chair, and this fact  should have been sufficient warning that something awkward was intended. In the corner the two sparred for about fifteen seconds amidst a horrible din. Finding that the Englishman was in no hurry to begin, Slavin made the first movement, and landed with the left lightly on the head. Following up this, he brought the right round heavily on the dial, making Smith wince, but a second blow in the same quarter was avoided. Still in the corner, the pair got to half-arm fighting, once more bringing them close to the ropes. The Birmingham mob here set about Slavin with great brutality, kicking him and punching him on all parts of the body. The referee, though appealed to again and again, took no notice, though admitting that he saw what was going on. Getting away from the ropes Slavin let go the right, and, reaching Smith's mouth, caused the blood to spring forth profusely. This last blow sent Smith to grass, the round occupying 50sec. [Col. 6c–7a] Tenth Round. The Champion of England by no mean relished the treatment received in the last round, and at the expiration of thirty seconds was puffing and blowing freely. He attempted to spar for wind, but the Australian seeing his advantage bored in when he could find an opening. Delivering right and left, and receiving only slight returns from Smith, the Australian pulled his man over, and the round terminated in 40sec. Eleventh Round. Neither was quite so quick to time, and Harper, noticing that, said, "This is not a two-round job, Patsey." Something like fifteen seconds was spent in sparring. Following Smith up the Australian failed to get an opportunity for some time. At length he put his hands down and invited Smith to "Come on and have a round." This was evidently not a particularly pleasing invitation to the Champion, as he remained in his own corner. Not to be denied, Slavin went after his man, but a left-hander meant for the head was put aside by Smith. At the end of a minute Slavin landed heavily on the Englishman's left cheek, a great red blotch immediately appearing. A rapid exchange followed, and Slavin was again home on the body. A heavy blow on the mouth from the Antipodean sent his opponent to the turf, the round having lasted 1min 35sec. Twelfth Round. A great change was noticeable in the appearance of the rivals on meeting each other for this round. Smith was showing signs of punishment on the face to a very large extent, his left eye being cut, his lip split, and there being a number of marks of Slavin's handiwork on his face. He was very distressed, and blowing heavily. On the other hand, Slavin, beyond the bad left eye (not done by Smith, but by one of the gang of hirelings), had apparently nothing the matter with him. Springing up brightly to his work, the Australian roamed about the ring until he found Jem Smith secreted in his own corner, only just away from his chair. The Australian reached the mouth lightly, and endeavoured by this means to draw his opponent out, but failed to achieve his object, Smith being strenuously warned not to move away from his seconds. The corner was encroached upon by Smith's crew, Harper and Baldock this time being almost between the men. The tumult was very great, but it seemed impossible to daunt the Sydney man, who followed Smith up, though little harm was done. Sparring was the order of the day until Slavin worked in with the left, delivering heavily. With great rapidity he drove the right with terrific force right on to Jem's chin, knocking him clean off his pins, and ending the round. Time, 1min 30sec. Thirteenth Round. Smith came up thoroughly dazed, the terrible blow in the last round having taken a lot of the steam out of him. The Birmingham gang, seeing that this was the case, got all of a heap in Smith's corner, and though Slavin was most earnestly advised to keep away, he would not do so. Smith stood just by ''his'' chair, and going right up to him, Slavin let out with the left, and bringing round the right. Both of these visitations Smith missed, but not so in the case  of a second smack from the right, which made Jem wink. However, turning round, the Englishman forced his opponent up against the ropes, and here occurred one of the most disgraceful scenes ever witnessed at a pugilistic encounter. Some villain drove his knuckle-dustered fist right on to Slavin's ear, cutting it almost in two and causing the Australian to exclaim, "Oh!" Other of these demons kicked him, a scene of indescribable confusion ensuing, in the midst of which the men continued fighting until the pair came down together. Time. 1min 28sec. No sooner was the round finished than cries were raised of "Here they come: the coppers, the coppers; and all Smith's gang, including his seconds, rushed away from the ring, at the same time calling upon Slavin to do so likewise. As for the Champion of England, he headed the runaway party, though nobody could detect the shadow of a "blue." The Australian never left the ring, much to the chagrin of the Smith party, who, seeing that the referee was being called upon for a decision, returned rather sheepishly. Baldock was in a dreadful fright, thinking the battle has been lost, but when the words "fight on" came he and his friends were very delighted. Fourteenth and Last Round. When "Time" was called Slavin folded his arms and waited for Smith to "come on," but Mr Smith didn't want to come on, and remained in his corner. Seeing it was impossible to make him stir Slavin once more crossed over, and a spar of nearly a minute took place. Then Slavin drove home the right on the body heavily, and the pair closed in Smith's corner. This gave another opportunity to a miscreant to strike Slavin on the head. Others in the gang were not slow to follow, and again the round ended in horrible confusion. Time. 2min 15sec. No sooner was this round over than the referee called out. "I refuse to act any longer. It is impossible for Slavin to get fair play. It's a draw." Then, going up to Slavin he said, "I will not stay to see you killed." The Antipodean stood up and exclaimed, "If you are Englishmen why don't you give me fair play? I have not had fair play." After this came a stampede. Remarks. Judging by the disgraceful proceedings recorded above, we are surely near the end of what are known as "fights in the old  style." That such a scene as that witnessed yesterday could possibly be enacted by any man with a drop of Christian blood in his veins seems almost incredible. Take the picture. An athlete, one of our stock, journeys 16,000 miles to measure blades with the best we can produce. Confident he will receive fair play in a land which boasts of nothing foul, he, after trying in vain to ratify a match to meet England's Champion, to be decided before an impartial and unbiassed crowd, puts down his hard-earned hundreds and meets our man in the style the latter insists upon. And with what result? Battling under unheard-of difficulties, not fighting one man but nearly a dozen, he narrowly escapes being murdered, and in the end is robbed of a victory he had won not once but in almost every round contested. If the nail driven hard into the coffin of pugilism yesterday is not the last, it is indeed ten thousand pities. Nothing can flourish, however, without support, and, providing the remarks passed by the majority of the spectators at the ring-side at San Michael are of any account, it will be many a long day before a knuckle-fight receives the patronage of the "upper crust." Slavin, we say, narrowly escaped murder, a fact proved by the words the referee, who, as well as others present, stood in fear of his Iife. One gentleman, an old ring-goer, affirmed he had never witnessed such disgraceful proceedings, and when the ring was finally cleared a feeling of relief was manifest on all sides. Unfortunately we are called upon to say that the riot and murderous intent were all in Smith's corner. On the other side not a man was present against whom a breath of suspicion could be raised. Opposed to Slavin was a hired gang of desperadoes ready to commit any act necessary to serve the ends of those by whom they were engaged. Terms of imprisonment had been worked out by each, and, prepared with knuckle-dusters, revolvers, and knives, they appeared ready to butcher the company at the raising of a finger. In face of all this the Australian fought his way to a moral victory. Almost from the outset Smith was outmatched, and beaten at the game considered his own. Sticking to his corner, of course by order, he showed none of his boasted bull-dog pluck and courage. His dreadful body-blow was easily put aside by Slavin, whilst instead of being "bashed" on the ground and battered out of shape, the Australian fairly held his own in wrestling. We are not blaming Smith for the doings of the miscreants present, and we also know that he was acting under orders by staying in his corner and affording them every opportunity to kick, strike, and prod the Australian, but it was altogether un-English to hold Slavin on the ropes whilst the mob belaboured him with "dusters" and sticks. Smith was, besides being in poor condition, slow and ponderous. After the third round his chance was hopeless, and his supporters were surprised at the form shown by Slavin. Contrary to general expectation, the Cornstalk proved himself far and away a better knuckle-fighter than a "glover." With his long reach he did great execution by left-handed deliveries, and the way he toppled Smith's fourteen stone over with the terrible right was a sight [p. 3, Col. 7c – p. 4, Col. 1a] Page Number: 4 that would have gladdened the heart of old Bill Richardson. In a word Slavin was entirely a different man from the Slavin that fought Bill Goode. Brimful of confidence he faced dangers that few would have encountered, and had the battle been conducted on fair lines he would have been Champion of England and Australia in less than half an hour. In all his battles he has never known defeat nor met with a knock-down blow. Yesterday it was not once found necessary to carry him to his corner, and as all the fighting was done in Smith's corner he must have walked a matter of eighteen yards each round. Slavin has a method of his own in pugilism. Possessed of staying powers, or "bottom" of a most remarkable character, he tires his man and then fights him. Yesterday, providing he had met with no punishment from outsiders, he would have left the ring almost without a scratch, and, as he himself said at the finish, "he was just getting warm." Remarkably clever on his feet, Slavin was in and out with the activity of a feather-weight, and feinting with the left, which he brought forward, he sent his right under the ribs with crushing effect. For some time past it has been an open secret that if Smith did not win he would surely not lose, and it is a blot on the proverbial love of English fair play to think that a mob should have been organised, and such a cowardly attempt made to rob a stranger amongst us of a victory so hardly earned and so clearly deserved. From the first the rules were broken by allowing the seconds to be in the ring whilst the fight was in progress, and although Smith's men threw everything in the way of the Australian, Cosnett and Kirby never transgressed the rules of fair play. Why the referee did not award the fight to Slavin when Smith left the ring in the thirteenth round can only be explained by his fear of the consequences that might follow.  Bearing on this subject, Rule 26 says: — "That if a man leaves the ring, either to escape punishment or for any other purpose, without the permission of the referee, unless he is involuntarily forced out, he shall forfeit the battle." Rule 24 is also plain enough in all conscience, "That any man voluntarily quitting the ring previous to the deliberate judgment of the referee being obtained shall be deemed to have lost the fight." Undoubtedly Mr Vesey lacked firmness, and perceiving this at the start, the "boys" and Smith's seconds, not to mention another couple who used their hands and feet at every opportunity, saw the game was in their own hands, and simply did as they liked. From first to last the proceedings were of a most disgraceful character, and the result showed plainly the reason why Smith's side objected to all the referees nominated by Slavin. Several gentlemen present, in fact all disinterested spectators, expressed their disgust at the treatment meted out to Slavin, and it was suggested on the return journey that a subscription be opened to provide funds for a testimonial to the plucky but unfortunate Colonial, particulars of which appear elsewhere. Even after the fight Slavin's assailants could not let him rest. Not content with robbing him of victory, they followed him to his place of dressing, but a determined front shown by Mr Taylor, Cosnett, and Kirby kept the blackguards back. These are well-known, and will, we believe, shortly meet with the punishment they so richly deserve. One rough made a rush for Lord Mandeville just before the close, but "Kim" soon showed he could hold his own with the "boy," who beat a hasty retreat at the "sight" of six-inch "Jack." We now take our leave of this unsavoury affair, which will probably be the last of its kind — "a Championship knuckle fight." The Englishman's boasted love of fair dealing appears lost to the professional fighter, and the sooner "merry mills in the old style" die out, and the hanger-on has to earn his living in some other way, the better. Slavin will visit London today. He leaves Margate by the 12.15 train, and will arrive at T<small>HE</small> S<small>PORTSMAN</small> office about three o'clock. Smith had a nasty lump on the jaw and left cheek, whilst his right eye was cut open deeply. In addition to this the punishment he sustained on the body was of a severe character. Slavin, on the other hand, only received a cut lip until after the fourth round, when the rowdies joined forces with Smith and smashed his eye during a mélée. After all their kicking and the free use of "dusters" the Australian only suffered from a badly-cut ear and the before-mentioned damaged eye. It may be stated that not a single bet was made at the ring side. SLAVIN'S OPINION OF HIS TREATENT. His Future Plans. On the return journey to England yesterday Slavin was the centre of attraction. At Ostend, where the tale of the fight had somehow or another been told, he was met by an eager crowd, and on the boat he held quite a ''levée''. Though at first somewhat disappointed at being robbed of the title of Champion and 500''l''. so well within his grasp, the Australian, surrounded by old friends, and new ones gained by reason of his straightforward conduct, rapidly regained his spirits. With the inner man refreshed by a steak, washed down with some of Heidsiek's [?] best, Slavin, a Havana between his lips, quickly got on good terms with himself and the rest of the  party. Much sympathy was expressed on all sides, and in the course of conversation with a S<small>PORTSMAN</small> representative Slavin said:— Of course, I feel a bit cut up, because I had my man beaten[.] Did you expect any trouble at the ring side? Slavin: Yes, I was prepared for that, as I was constantly receiving letters of a threatening nature, and though these things as a rule are of no account, I had reason to believe there was some truth this time. Besides, I had not forgotten the affair at Astley's Theatre. Do you feel any effects of the fighting? Slavin: The fighting? I feel nothing of what I got from anyone in the ring; it's what those beauties gave me outside. The only time Smith hit me in anything like form was on the jaw. Why did you persist in going to his corner? Slavin: I wanted to get the affair over and enjoy my Christmas dinner, and I knew if the fighting was left to Smith I should want it served in the ring. Of course you heard a mob was got together for Smith? Slavin: Yes, I knew all about that, but I relied on British fair play, and expected the referee to do his duty. Directly I got in the ring I saw what the game was, and knew my only chance was to knock Smith out. What do you think of Smith? — Well, I think so much of him that I'll bet him 200''l.'' I stop him in six rounds, but we must have no more private shows. I've had enough of that business. I suppose you have a poor opinion of English fair play? Slavin: I have been treated like a prince generally, but I can see the fighting men are easily led astray, and I expected trouble directly Smith objected to a referee. If we served an Englishman in the same manner in Australia we should get drummed out of the town, and rightly too. What are your plans for the future? Well, I shall stay in England some time and challenge Sullivan, and you might as well cable it over to America for me. With the gloves? Slavin: No, I went to fight him with the knuckles. I think I should get fair play out there. You prefer the old style to the new? Slavin: I am fond of boxing, but the other is my game, and though not a boaster I fear no one — that is if I can have fair play. Shall you give any exhibitions? Slavin: I may perhaps make a tour round the country, and very likely visit Ireland. You put on weight the last week? Slavin: Yes, I gained eight pounds and felt grand; in fact, just up to dick when Smith bolted. What do you think of our seconds? Slavin: Well, I'd like them to have a job in Australia. They'd never come back again. When do you propose visiting London? Slavin: I'll come up to-morrow and call at T<small>HE</small> S<small>PORTSMAN</small> Office about three. A stroll was now taken on deck. [sic] and reaching Dover Slavin was met by some friends from Margate, who carried him off to the train in triumph. A Testimonial for Slavin. So impressed were some noblemen and gentlemen with the pluck and ability shown yesterday by Slavin, and at the same time thoroughly disgusted at the treatment he had received, that it was decided whilst crossing the Channel on the way home to open a subscription on his behalf. The donations are limited to 2gs, and the following amounts have been already promised Mr G. T, [sic] [Col. 1c–2a] Dunning, T<small>HE</small> S<small>PORTSMAN</small> Office, Fleet-street, who has been requested to act as honorary treasurer to the fund: [names set as unordered list to save space, last 3 column headers: £ ''s.'' ''d.''] * Lord Mandeville 2 2 0 * Lord Royston 2 2 0 * Hon. Michael Sandys 2 2 0 * Mr Fred Ketchie 2 2 0 * Mr Ike White 2 2 0 * Mr W. Riley 2 2 0 * Mr C. Parsons 2 2 0 * Mr J. Kilby 2 2 0 * Mr Frank Hinde 2 2 0 * Mr J. Calder 2 2 0 Any contributions sent to the hon. treasurer will be duly acknowledged in T<small>HE</small> S<small>PORTSMAN</small>, which in a couple of days will contain the names of a committee now in course of formation. In the London clubs yesterday the opinion was very general that Slavin ought to be compensated for the loss he had sustained in consequence of the ruffianly tactics pursued in Smith's corner. The utmost indignation was expressed in reference to the treatment the Australian had received at the hands of so-called Englishmen. We have received a number of letters on the subject. The following is a sample of them: To the Editor of T<small>HE</small> S<small>PORTSMAN</small> Sia, — I have no doubt that many Englishmen will have read with regret the disgracefully unfair treatment that Slavin met with in his encounter with the  "Champion of England" to-day. I trust that a subscription will be started to show him in a substantial manner that the sympathy of most Englishmen who take an interest in boxing is entirely with him. If you are prepared to receive subscriptions I shall have much pleasure in subscribing a guinea, and in collecting those of many friends who will be glad to subscribe. — I am, Sir, yours faithfully, C. B. C. Royal Thames Yacht Club, December 23. [We refer our correspondent to the paragraph above. — Ed. T<small>HE</small> S<small>PORTSMAN</small>. ] [sic bracketed interpolation] Eyewitnesses Interviewed. A representative of T<small>HE</small> S<small>PORTSMAN</small> was at Cannon-street station yesterday afternoon to meet the Continental express, in which a goodly number of those who had witnessed the fight made their return to town. The train was somewhat late, and did not steam into the station until a quarter to six, when our reporter singled out several well-known "sports" (including more then one scion of a noble house) for a few minutes' chat. Asked as to how affairs happened at the outset, one and all agreed that it was plain a row was meant, the demeanour of a certain section of the party, which  did not number more than forty, being particularly noticeable. "How did the combatants' seconds behave?" — "Well, after the first round, it was evident that Smith would soon be at the mercy of his powerful opponent, and his seconds very soon perceived this state of things. In the initial round Harper came down with some force on the Australian's chest, while in the next bout Slavin came in for some rough treatment at the hands of Baldock, the Champion evidently suffering with one of his eyes." "What form was the Englishman displaying?" — "From the outset he made a miserable stand, and was plainly afraid of his rival. He fought as a coward, and by the tactics adopted by several of those at the ring side Slavin was fighting against great odds.  Every time he got into Smith's corner he was met with remarks of a most uncalled-for character, and had to be very sharp to keep at arm's length of several rowdies, who, as the fight proceeded, became extremely noisy." "How do you account for such a noisy corner at the back of Smith?" — "Although they did not number more than eight, they were some of the worst  characters known, and had undoubtedly been commissioned to play the part they did. This was most disgraceful, isasmuch as they not only made Slavin a mark for the knuckledusters and other weapons they were armed with, but likewise made tracks for individual spectators who ventured to express their indignation at the course of proceedings. One rough, more venturesome than the others, "framed" up to a familiar figure at boxing tournaments, but on the latter producing a long bowie knife, accompanied with a few words of a somewhat emphatic nature, the villain rejoined his  party." Are these men known at all?" — "Yes, they are mostly from Birmingham, and are of the lowest order. It was reported that more than one had been kept at the country's expense, one for a very lengthy period." "Was not this sort of trouble expected?" — "It had been given out by one influential patron of the English Champion that there would be an absence of rowdyism, but it was also stated on good authority that certain roughs had been specially engaged, so that trouble was naturally anticipated." In reply to questions as to how the contest proceeded, it was the opinion that the thirteenth round would be the last, as Smith's party made a determined effort to bring about an undesirable finish. On the two fighters getting to close quarters more rough work ensued, and on some pretence Smith left the ring. Slavin, however, stood his ground manfully. "How did the crowd behave towards the finish?" — "There was a lot of excitement on all sides, but the only fear was that the rough element would resort to extreme measures in endeavouring to save their Champion. Others were ready to insult the Australian by using most filthy language towards him, and altogether affairs began to look serious." "Is it true that attempts were made to tamper with the water used by Slavin's seconds?" — "Yes! a man was noticed to be hovering about in that quarter, but, fortunately, his efforts to drug the fluid were of no avail." "Did Smith deal out much punishment?" — "His blows lacked the force of his rival, and when the referee gave his verdict as a draw the Australian looked very little the worse for his exertions, and could have fought far two hours more. He could have beaten Smith and both his seconds into the  bargain, given fair play." "Was Slavin injured during the fight?" — "No; not from the blows of his rival, as he heeded them very little. He, however, received a nasty blow from a knuckle-duster by one of the cowardly ruffians, which caused his ear to bleed." "As to the referee's decision?" — "That official was undoubtedly afraid of his life, and thought it best to stop the proceedings before any serious harm happened." "What becamiof Smith after the fight." — "He returned to his hotel, and will no doubt arrive by a later train. His friends were disgusted with the form he displayed, and it will be a long time before he recovers his former prestige, if ever he does." "How did the Cornstalk behave in the ring?" — "He fought in the most manly style throughout, and gained the highest opinion from all respectable people present. A testimonial to Slavin has already been decided upon, and no efforts will be spared by those concerned in making it worthy of the occasion." Mr Ike White, a well-known sportsman in the city, and one who, as a rule, does not mind going to considerable trouble to witness a "mill," was very severe in his comments on the gang who were so lost to all sense of decency and fair play. Mr White said: — "The story of the fight does not need much telling. Slavin had his man beaten at every point, and was winning easily. Slavin is a very brave man. He bad to go into Smith's corner every time, and when there had to fight not one but six men; the ruffianism was simply awful." "Do you think it was for the best that the contest was stopped as it was?" "Yes, certainly. Goodness knows what would have happened before long. Probably Slavin would have been killed, as the villains that had congregated in Smith's earner seemed determined, at all hazards, to prevent him from losing. More than once the Australian had Baldock on one side and Harper on the other, both jeering at him, whilst Smith was in front and the 'boys' were making all sorts of attempts to injure him, but he seemed to pay no attention to anyone but Smith. Slavin would not have had a mark from Smith, such as he does now having been inflicted entirely by the mob[.]" "What do you think of Slavin's abilities?" "He is a very great natural fighter, and as for his gameness a pluckier man never pulled off a shirt." On the subject of who was responsible for the presence of the rowdies that caused the bother Mr White was reticent, but not so when the subject of a testimonial to the Australian was mooted. "He is coming to town to-morrow," said the speaker, "and shall have a great reception. Some good sportsmen  have promised to leave no stone unturned to give Slavin a testimonial to be proud of, for he richly deserves it." "What seemed the general opinion on the ground?" "Why every ''man'' (with great emphasis on this word) was sick and disgusted: all but the mob gradually forsook Smith's side of the ring, and the Englishman will lose many friends in consequence of the turn affairs took." Reception of the News in Birmingham. Considering the ''penchant'' of the natives of Birmingham for fighting, and their interest in all matters concerning the ring, a stranger in the city yesterday knowing this would have been rather taken by sur prise [sic] at the indifference manifested in the result. At the various local sporting resorts, however, the cause was plain enough to those in the know. Early in the week it was an open secret that a warm corner was being provided for the English Champion, consequently betting was at almost a standstill. In what little speculation took place the odds were in favour of Smith, as the ''cognoscenti'' were confident that the Australian would not be allowed to win. Amongst the better class of sportsmen in the city the whole of yesterday's disgraceful proceeding were viewed with the greatest disgust. There are still many staunch supporters of the ring in the district who would gladly hail any signs of its regaining its past glory, but they look upon the fight of yesterday as the death-blow to the institution. By the few that thought the affair might possibly be allowed to be decided on its merits Slavin was expected to win, but his only chance was considered to be in his forcing the fighting at the start [Col. 2c–3a]  and knocking his man out early. George Prober hoped to hear that the fight had been decided fairly, and he regretted the unsatisfactory ending. How the Match Was Made On the evening of Saturday, August 10, the P. and O. steamship Carthage cast anchor in the Albert Dock, Limehouse. On bard was an individual, anxiously expected by the British public — namely, Frank P. Slavin, the Champion at Australia, who was accompanied by his manager, John Lewis. Without loss of time, Slavin made his way to T<small>HE</small> S<small>PORTSMAN</small> Office, and on the succeeding Monday the following interesting particulars respecting the visitor appeared in the columns of this paper:— "He has a long record, including victories with both gloves and knuckles, and has never yet known defeat. His bearing is that of a thorough gentleman. He is a fluent talker, and can converse on any sporting subject. Devoid of conceit, he yet has a full comprehension of his own powers, and speaks with confidence as to the result of any encounter in which he may take part. Slavin is a splendid representative of the nation which is growing up across the seas. The contour of his features is distinctly English, square cut and firm. Beneath an intellectual forehead are two deep-set piercing blue eyes, and his features are bronzed with the burning suns of the south. From this it will be gleaned that the Antipodean is a formidable-looking customer, especially when we add that he stands no less than 6ft 1in, and in condition fights about 13st 4lb. Long before his arrival 100''l.'' had been posted at T<small>HE</small> S<small>PORTSMAN</small> office for Slavin to box any man in England, but the money was waiting nearly four months before any of our big guns would cover it. Jem Smith, the Champion, turned a deaf ear to the straightforward offer of the Colonists, and Mitchell  was called away to Yankeeland to lend assistance to his friend Kikrain [?]. For some time it looked as though our visitor would have to return as he came, without giving anyone a hiding or receiving one himself, until a gallant little English middle-weight stepped into the breach to box for the honour of the old country, and agreed to face the stalwart Cornstalk. This was none other than William Goode, who in consequence of his politeness, urbanity, and elegence of dress was christened by his friends 'Chesterfield.' Goode had previously lowered the colours of another Australian, Tom [Less?], a much bigger man than himself, and, bearing this in mind, his friends freely plunked down 200''l.'' These were Messrs Robert Abel and Harry Hart, two well-known figures in the world of sport, articles after little delay being signed at T<small>HE</small> S<small>PORTSMAN</small> office, whereby Slavin and Goode agreed to box the bout of twelve rounds, under the usual rules, on October 17, for the sum stated. Many old ring-goers and patrons of the games shook their heads when the articles were published, and not a few blamed 'Chesterfield' for his rashness in tackling at catch-weights a man so much his superior in size. Goode, however, comes of a fighting stock, is full of pluck and stamina, and belongs to the order which never say die. Excepting in a competition, his maiden attempt, 'Chesterfield' had never known defeat, and the making a match with such overwhelming odds against him his bravery received due acknowledgment on all hands. Slavin held an unbroken record of nineteen successes, and throughout his career he had never known the effects of a knock-down blow. The articles stipulated that the men should box in a 24ft ring, and at the time of signing Mr B. J. [Angle?] was agreed to by both sides as referee. A very necessary clause was inserted, giving the referee power to award the stakes in case of what he might deem an organised interruption to the man who, in his opinion, had the best of the boxing at the time, and if all agreements contain such a clause, it will have a deterrent effect on those who invariably go on the sin, tie, and wrangle system. Slavin placed himself under the charge of that painstaking trainer and old-time boxer, Jack Start. The pair made Mr J. Kilby's, the Ferry Hotel, Cookham-on-Thames, their head-quarters, but things went very badly at the start. During the second week of his training the Australian caught a severe cold, and rheumatism setting in, the services of a doctor were required. The illness was, for obvious reasons, kept a profound secret, but so bad did Slavin get that the first man of medicine deemed it necessary to call in a second. Ten days were spent in the house before Slavin pulled round, and though all went well for some time, he contracted another bad cold. Start, however, wa unremitting in his attention to his charge, and Slavin came to the scratch in fairly good shape. The result of the encounter, a win for Slavin in the fifth round, despite a grand stand by Goode, will be fresh in the memory of our readers." [sic: quotation begins at the top of the paragraph] Smith's failure to make a match with the Australian caused general surprise, but the Englishman would have nothing to do with the glove business, despite the repeated offers of Slavin. The latter's friends, on the other hand, saw little but trouble attached to a knuckle fight, and though Slavin himself preferred the fists to the mittens, there appeared small chance of a match between the pair until Mr Frank Hinde stepped forward and put up a deposit of 50''l.'' at T<small>HE</small> S<small>PORTSMAN</small> office for the Australian to fight Smith for the championship and 200''l.'' a-side. The old question of stakeholder then cropped up, and, after a Iot of bickering, the Colonial gave way, and agreed that the money should be deposited with private gentleman, who for obvious reasons shall be nameless. A meeting took place, and eventually the following contract was drawn up by a representative of T<small>HE</small> S<small>PORTSMAN</small> and signed: Articles of Agreement entered into this day (Monday, September 16, 1889), between Frank Slavin, of Melbourne, Australia, and James Smith, of London, England. The said Frank Slavin and James Smith hereby agree to fight a fair stand-up fight according to the new rules of the London Prize Ring, by which the said Frank Slavin and the said James Smith hereby mutually agree to be bound. The said fight shall be for 200''l.'' a-side, open for 1000''l.'', lor the Championship of England and Australia, and shall take place in 1889 out of England. The two representatives to select the battle-ground issue all invitations (not exceeding fifty persons a-side), and carry out the arrangements for bringing off the fight. The backers of the said Frank Slavin and the said Jem Smith this day (Monday, September 16) deposit 200''l.'' with the stakeholder. The representative of each man to supply the other with a list containing names of spectators, and to have mutual power to strike out a name they object to. The said James Smith and the said Frank Slavin each to have power to appoint an umpire. The referee to be mutually agreed upon by the duly authorised umpires of the men at least seven days before the day fixed for the fight, but in case they cannot agree, the final stakeholder to have power to appoint a referee. The men to be in the ring between the hours of six a.m. and twelve noon, or the man absent to forfeit the battle money. The expense of the ropes and stakes shall be borne by the two contestants equally. In case of magisterial interference the referee, if appointed, or the stakeholder if not, shall (if possible on the same day) name the next time and place of meeting, and either party failing to appear at the time and place specified to lose the battle money. The stakes not to be given up unless by mutual consent, or until fairly lost by a fight, and due notice shall be given to both parties of the time and place of giving up the money. In the event of any question arising which may not be provided for in these articles the referee to have full power and authority to decide such question, his decision to be final, conclusive, and subject to no appeal in any court of law. The stakeholder shall in any and every case be exonerated from all responsibility upon obeying the direction of the referee. In pursuance of this agreement we hereunto subscribe our names. Frank Hinde (for Frank Slavin). John Fleming (for James Smith). Witness for Smith, F. Montague. Witness for Stavin, John Lewes. Subsequently Slavin, as stated, defeated Goode, following which he fulfilled an engagement at the Royal Aquarium extending over thirteen nights. The "Cornstalk," though suffering from an injury to his left hand, offered 20''l.'' each evening to any man whom he could not step in four thee-minute rounds, and though several tried to earn the money Slavin always accomplished his self-imposed task, and brought his engagement to a close in a blaze of triuumph by stopping two men the last evening in four rounds. Slavin had now come into funds. Through the columns of this paper he expressed a wish to put some of his own money down, and, confident in his ability to uphold the honour of Australia, he offered to increase the stakes to 500''l.'' a-side. Smith, with a millionaire at his back, had little difficulty in getting the extra 300''l.'', and accordingly the money was made 1000''l.'', the sum fought for by Sullivan and Mitchell. Smith in the meantime, though ignoring Slavin's challenge to box under Queensberry rules, strange to say, entered into an engagement with Jackson under the conditions he so stoutly objected to when put forward by the white man[,] His ignominious show with the coloured Champion may be passed over without comment, and it will suffice to say that after he and his conqueror had boxed together for ten nights at the Aquarium the Englishman began to prepare himself for the affair under notice. The Men in Training Slavin, when he entered upon his training at the Nayland Rock Hotel, Margate, was 12st 9lb in weight, while fourteen days later, in the buff, he scaled 13st 6lb, and was as hard as nails. The increased weight was accounted for by the splendid development of the muscles in the arms, shoulders, and back. His daily programme while at Margate was as follows:— Up at seven sharp, and twenty to twenty-five minutes' strong exercise with 4lb dumb-bells, followed by a gentle stroll of a couple of miles before breakfast, at 8.30, which consisted of porridge, 1lb of chops or steak, dry toast, and a cup of tea. After the perusal of the morning papers he indulges in a ten-mile walk at a smart pace, averaging five miles per hour. Back again to the hotel, into a bath, and then a rub down with whisky, which was completed in time to give him a good rest before dinner, timed for 1.30, and composed of either roast beef, mutton, or chicken, with vegetables, light pudding, and a pint of seven-year-old bitter. The meal being digested over a game of billiards, he started at three o'clock on a six-mile stroll, varied at intervals by sprints of one hundred yards. On his return to his quarters he punched the ball for half an hour, and then sparred lightly with an Australian gentleman who was at the time on a visit to this country, and holds the Amateur Championship of Australia, he having kindly consented to assist Slavin. This was followed by a sponge and rub down and a good slapping of the muscles before a tea, at 6.30 p.m„ [sic] of cold meat, half a dozen eggs, dry toast, and tea. The wind-up of the day was a quiet stroll to the pier and back, and a game of billiards before retiring to rest, at ten o'clock sharp. Smith's programme was something of the same description, excepting the ball punching, the Englishman preferring the sack. He made the Greyhound,  Newmarket his head-quarters, most of his exercise being taken on the Heath, though, as visitors were forbidden and interviewing tabooed, little was gained [Col. 3c–4a] as to his doings. Under the care of Jem Howes, his old trainer, he was, however, reported to be doing well, and much finer than when he fought Jackson. Finding the Battle-Ground. (By one of the Explorers.) At a meeting held on Saturday, November 16, Mr John Fleming and Mr G. T. Dunning were deputed to act on behalf of Smith and Slavin respectively, their duties being to select the battle-ground, issue invitations, and carry out the arrangements for bringing off the fight. This was rather a strong order, and one of a most trying nature, as anyone conversant with the peculiarities of those attached to the P.R. may know. Wednesday, December 4, was a cold, raw day, and with the thermometer down to freezing-point travelling at night was anything but a luxury. Reaching Paddington soon after 9 p.m., Mr Fleming and I, thanks to an energetic guard, soon found ourselves comfortable in a first-class "smoker" attached to the Weymouth express. At 9.15 to the tick we steamed away from the Great Western terminus, and in due time reached Reading. The hour had passed quickly, pleasantly whiled away by interesting tales of old-time bouts told by the manager of the Pelican boxing. Such men as Punch Dowsett and Bill Charlton graduated in Mr Fleming's school, and the way these celebrities used to set about each other for as many shillings as the boxer of to-day gets pounds is nothing less than remarkable. A ten minutes' halt at Swindon was utilised to the fullest extent, and then bowling along we reached the end of our journey, so far as land was concerned, soon after 2 a.m. on Thursday. When we left town the air was crisp and frosty, with hardly a breath of wind stirring. We were, therefore, greatly surprised when stepping on beard the Gazelle we were informed that a "choppy sea" was running outside. The fact of a passenger's hat going over the side proved the truth of the assertion, and then this same passenger, a likely-looking fellow, bemoaned his fate and bad luck. "How confoundedly annoying,"  he said. "This is the sixth time I have been across during the year, and not once has it been calm." "Better have a sleeping berth, sir," said a sprightly youth at the top of the stairs. "Sleeping berth be hanged. I'll remain on deck," said he, and here for a time we left him inquiring if his horses had gone across in the morning. Guernsey, dear reader, was our destination, and with the idea of making ourselves comfortable we made for below. I felt a bit "peckish" and cast a longing eye on the good things set out in the dining saloon. This was, however, as far as I went in the line, and diving a floor lower doffed my boots and courted the god of sleep. Morpheus, however, scorned my attentions, thanks to the clatter on deck and the noise of the machinery, to say nothing of the "magging" [?] of our friend of the horses. This gentleman, after three or four futile attempts to drag me into a controversy about traveling, "struck ile [?]" with an elderly party, occupying the berth behind me, and for nearly an hour I had a pretty lively time of it. The "elderly party" had a voice greatly resembling the dividing of tin hoops with a file, and in addition displayed a shocking habit of making horrid and atrocious puns. Just as I thought of giving up sleep for a bad job a gale sprang up. Though the Geselle is a record breaker she soon felt the effects of the breeze and began to roll heavily. The increase in the motion of the steamer was followed by a decrease in the conversation around me, and by degrees the punster went from had to worse. The "[hussey??]" [one?] also dropped into a walk, and after monosyllables had been exchanged the two came to a dead stop. Offering up a prayer of thanksgiving for this I closed my eyes again, and being a good sailor was quickly sleeping the peace of, I hope, the just. The concussion of the Gazelle against the Guernsey Pier, accompanied by a prod from Fleming's stick, at length aroused me. Among the Channel Islands. The time by "Joe Vickers" was 7 a.m., and half-awake, I took to my boots. "Need not lace them; we shall be there in a minute," said Fleming, and in a drowsy, submissive, take-me-anywhere-if-only-to-bed sort of way l followed him on deck. A Babel of voices and a clutch at my bad and rug thoroughly aroused me, and jumping into a trap the Pelican and I were in about three minutes set down at Gardner's Royal Hotel. A venerable-looking elder, whom Fleming voted a "[sheenie?]" (goodness knows what he meant, though), made us welcome, and agitating the communicator he handed us over to the care of a dark-eyed chambermaid. I forget her name, and as Fleming is not here I cannot give it. I will say no more than that after a cup of tea diluted with delicious cream, we took a well-earned and mush needed rest. Up and doing soon after tea, a delicate task was before us. Fleming had a friend on the land, a gallant captain, who knew of a spot where we could fight for a month. Guernsey, dear to the "holiday maker" and the "honeymooner," is, as everybody knows, or should know, situate in the English Channel near the coasts of Brittany and Normandy. It is 68 [?] miles from Weymouth, 7 from Sark, 19 from Aldersey [?], and 25 from Jersey. I mention these distances, for we explored the whole group. Encompassed with rocks, almost hidden in high water, navigation is exceedingly dangerous at all times near the island, the knowledge of which quickly knocked one of our plans on the head. Standing at the door of Gardner's Royal Hotel on Thursday morning Castle Coronet came in view, while in the distance, almost hidden by a heavy sea mist, could just be discerned the islands of Herm [?], Jethon, and Sark. Guernsey, it should be mentioned, though under the protection of England, boasts a sort of Home Rule government, rather irksome to the stranger within its gates, especially if that stranger is on mischief bent and gets caught. Making, in a cautious manner, inquiries at the breakfast table, we found a friend in "Mike." Said I, "He's a Frenchman."  "Get out," replied Fleming, "he's a 'bark.'"  "Drinks he isn't," "Done; drinks he is." "Mike" had not at that time spoken, but had noiselessly passed to and fro with a military step and a hand behind him. In reply to a call from Fleming he approached, and a "Yes, yer honour," told me I should have to "part." "Mike" was a trump, and I loved him from the start. For eleven years he had been in the service of a colonel at 1''l.'' a week, and three suit of clothes in the year. Hotel pay was not so good, and though he wade "a bit" in the summer, the young ones were generally there when the bills were paid, if not when the work was on. "Mike" told us lots of things in what many consider the "fascinating brogue."' He informed us — and how our hearts beat at the news — that the island, which had a population of over 32,000, employed but six policemen. "There's not a beggar in the land," said Mike, "and but one Hansom cab, and," pointing  to the window, "be jabers, there it goes. " So far so good, but after a walk round the son of Erin gave us a shock by saying, in reply to a query from  "They've a way of thdr own on this  and, sad mealy ivory ether men L .1  ee  3.4.11  nsertessoholy we bad Pat  nowt,table." W  and, •••••= ar easik Fleming, "They've a way of their own on this island, and nearly ivery [sic] other man is a special constable." Fortunately we had just finished our repast, and, supporting each other, we tottered faintly to the door, partially regaining strength with a breath of the pure sea breeze. Beads of perspiration stood on the brow of the Pelican, whilst I was suffering from a "trickling" down the back of anything but a pleasing nature. Visions of a dungeon-cell under Castle Coronet flashed through my brain as a voice and a step were heand behind, and then I was somewhat reassured. "When would you like a chair fell softly on mine ear. Afraid that we had been thought of as bit week on the leg, I turned to the speaker and asked what he meant. Explanations ensuing, we found the "chair" was a sort of Bath chair on wheels used throughout the island. Declining the offer, we took a stroll, and putting this and that together, came to the conclusion that our task was of a very delicate nature. The first thing was to interview the captain, who, in reply to a note, courteously waited upon us. He gave us the geography of the island and an introduction to Mr., a friend, who we unfortunately found away from home. Taking a "chair" we then started on a voyage of exploration, making, as advised, our way to L'Ancresse [?] Common. Leaving St. Peter's Port and passing through St. Sampson's — the Vale — we came across acres of vineyards, in one instance extending nearly a mile along the road-side. Half an hour's drive brought us to the common mentioned, and dismounting we tried in vain to find a level piece of turf. Windmills abounded on all sides, and plodding on we reached L'Ancresse Bay, a lovely little spot, where the sea rippled softly over a bed of almost snow-white sand. Nowhere could be found a suitable spot whereon to pitch the "magic square," and as the shades of night were falling fast we rattled back to the "Royal." A wet night offered no inducement for further travel, and it was a case of early to bed and early to rise, the next day finding Fleming and myself at the door of Mr M. Acquainting this gentleman with the Nature of the business in hand he, like a good sportsman, fell into the scheme and at once volunteered to drive us to "the best spot in the world." "Bessie" was harnessed to the village cart, and away we went at the rate of fully fourteen miles an hour in the direction of the Forest. I have sat behind many a good goer, but must for her size award the palm to Bessie a beauty barely twelve hands high. Throwing the reins in the bottom of the vehicle, Mr M. '''let the pony have her head''', and the way she went uphill, down dale, and past other traps was truly astonishing. Great St. Peter's was passed, and lovely scenery encountered, the gardens being full of flowers, geraniums and roses blooming on all sides. An enjoyable drive of forty-five minutes found us at Boequaire [?] Bay, and directly in front stood the island of Lihou. As many of our readers may know, Lihou is a small hill of land that at low water can be reached on foot from Guernsey. As a fact, a paved roadway is laid in the bed of the ocean, across which both foot passengers and carts easily make their way. With the tide up communication is effected by boats, but this being low at the time we had the novel experience of Driving Through the Sea. Gaining the other side we "explored" and as the piece of land is only some three hundred yards by a hundred and fifty, we were not long about it. Two excellent patches of level turf were found, but just as we were making for the trap a gaunt, Robinson Crusoe-looking individual scaled one of the small stone walls and barred the way. Of course, he wanted to know our business, and, after a rather rough opening, quickly "listened to reason." In the course of conversation we found that the population of the island was limited to three, the old "josser" and his two daughters, with a "moke" thrown in. Pointing a long skinny finger, which might have belonged to the Witch of Endor, towards the quadruped mentioned, the old man said, "Us two keeps the family. He draws the seaweed and I burn it."  "And why do you burn it?" "Well, to make [Col. 4c–5a] ashes, and from the ashes iodine is manufactured." "And how much do yen earn a week by that?" said  I. "Oh, about 4''s.''; not 5''s.'' any week." Great Scott! and this was all the old man, his two daughters,  and the "moke" had to live on. Handing him a franc each, which nearly stupefied the old chappie, Fleming and myself returned to Mr M., who drove us rapidly back to St. Peter's Port. The next day we crossed to Sark and Jersey, but found nothing so suitable as the ground at Lihou, and fairly well satisfied with the result of our visit, a day later boarded the Lynx for the return journey. During our visit the weather had been of a fairly favourable description, but the voyage home was made under very depressing circumstances. Rain fell heavily and a thick mist hung over the water, which after going a few miles was alive with porpoise. Some of these animals came within a few yards of the Lynx, but the speed of the boat proved too good for the finney ones, who were eventually left behind. With a breeze springing up sail was hoisted, and dashing through a heavy sea in splendid form the Lynx made some sixteen knots in the hour. In this way half the journey was completed, when all of a sudden we were caught in a squall. In an instant the wind chopped round, and before you could say "Jack Robinson" the good vessel was turned broadside to. The order to take in sail was promptly obeyed, but the clatter on deck brought many an anxious passenger rapidly from below. A very heavy sea was now running, and though the Lynx kept her course bravely the decks were swept from stem to stern. A huge East India-man now hove in sight, and, as one of the men informed me, she had the audacity to cross our bows instead of passing under our stern. This breach of ocean etiquette ruffled the skipper of the Lynx, who chased the trader, and eventually made her alter her course. This little "diversion" caused much amusement amongst our passengers, and after a cold, rough passage we landed safely at Weymouth. Chilled to the bone, we made for the Victoria Hotel, where a chop and a glass of old October revived the almost stagnant circulation. London was reached about 9.30, and, tired and worn out, I was glad to lay my head on a pillow I had not seen for four days. From London to Ostend. Unforseen [sic] difficulties having cropped up, the Channel Island scheme had to be abandoned; consequently we were once more called upon to leave our native land. This time Belgium was our destination, and having carefully studied the map of that country, we took our departure one fine starlight evening from Charing-cross by the 8.15 Continental express. In the previous week we had the luxury of a compartment to ourselves, but on the occasion under notice we were not so fortunate. Two individuals, one with a strong Midland accent, and the other a Frenchman speaking bad English, shared the carriage with us, and at such a rate did the former's tongue go directly he entered that I gave up reading as a bad job, with a sigh put away my unfinished book of the previous trip. Fleming fared better, and having a knack of ignoring all around him, once more tackled "Blood Money," and almost read himself off to sleep.  In the meantime our friends in the opposite corner made "hay," and I soon found out they were both in the lace trade. With a tongue that went a couple of hundred to the minute, the Midlander almost monopolised the conversation, from which I gleaned the two were bound for Calais, and thanked goodness we should part at Dover. Reaching this port about ten we boarded the Ostend boat, which, as regards passengers, was very poorly patronised. Hardly a breath of wind was stirring, and with the sea calm as a mill-pond we had a pleasant passage over, and landed safely at Ostend soon after 3 a.m. A cup of coffee was hastily swallowed, and a half-hour's run in the Brussels express found us At Bruges. Leaving the train, we had considerable difficulty in finding our way out of the station. For a moment we caught sight of a drowsy old porter, but the last carriage had not passed away from the platform ere he was lost to view. Seeing an opening on other side we made for it, only to have our passage barred by a half-dressed Belgian who, whilst displaying a total ignorance of English, Irish, and Scotch, directed us to return the way we came. For some time we tried in vain to find an outlet, and just as I had made up my mind to walk the platform until daylight Fleming found an iron gate ajar. Without loss of time we were in the street, and crossing the round, knocked loudly at the door of the nearest hotel. All was in darkness, and with no response to our fourth appeal for admittance, a return to the station or a "tramp round" appeared to be our only alternative. A slight noise inside followed by a shuffling of feet, again raised hope within us, and when, after some trouble, the door swung back and brough to view a shock-headed Belgian our relief was great. No time, I can assure you, was lost in getting between the sheets, and rising betimes we again went on a voyage of discovery. Fleming had a chum in the town, and to his house of call our steps were directed. Again disappointment was in store for our wanderers, and a feeling of disappointment came over me when we found Mr G. was away. There was, under the circumstances, nothing left but to wait, and Fleming, who puts on hair rapidly, decided to place himself under the care of the town barber, or, I should, properly speaking, my ''coiffeur''. For this piece of indiscretion he had to come away minus a slice of his lower lip, which, fortunately, was not large enough to seriously reduce his avordupois [sic?], and heaping blessings (?) [sic] on mankind generally, not in any way forgetting the "Bruges Barber, we roamed over the old town, noted for its magnificent cathedral and many seats of learning. The hours sped on pleasantly enough, and in due time our friend was unearthed. Being pledged to secrecy I am not going from my word by giving full particulars of The Battle Ground placed at our disposal. This is in close proximity to the town — in fact, not half an hour's walk, and on reaching the spot we were shown a lovely piece of turn. A clump of trees on each side partly hid the house from view, and when I say that the ring was to be pitched on a lawn tennis ground, it will be seen that neither side could claim an advantage as regards standing ground, &c., the lawn being as level as a billiard table. Though our efforts so far had been crowned with success, we did not get permission to use the "pictch" without much trouble. The matter had to be seriously considered, and Mr G. would not give a decided answer without waiting another day. Consequently our stay had to be extended and the following morning, after repeated promises that the affair should be conducted in an orderly manner with not a rough at the ring-side, we got the answer, "Yes." "But," said Mr G., "if there is any row or crowd I can assure you you will have a journey for nothing; therefore if trouble is at all probable keep away and save your money and time." As long as faith was not broken by either side we saw no reason why the matter should not be settled in a proper manner, and making all the necessary arrangements and laying down plans of a most complete description, we returned to town well satisfied with the result of a visit to Bruges. TROUBLE ABOUT THE REFEREE.  ppo~Oaa to town we received satisfactory ren  ef dr week iime by satih el the eentesbale.  SIAM% limbed sfler byJesit Oseitett, the Birmingham  beavymeigist, and bs Mal lit MUM Taylor, of  Australia, though rather MAI, mes in good MAL  At one tines he melnd ink Weit Mb. but • rims  =f a r melt immalit bin balk te aid.  tbe imperbinee o f Me berieses in head,  sad ib bearing ea Mum sw.ts, IBMs bed Kass  cease for enxiety. In Maim to this be was  for 3001. of bin own  semay, b eszi en =m bas tl i i:=  to tie Memel *Miss to tie  The ant queliea was Ohs sidoetica of ssessass  and millier to II wall-isegerlewl pest of  capable and  Amain to this setialsa is was Mt toles  slakolsoldar le sprat the liked ansed  tin usposmatedoes failed be agrmid gi  ia " 4l 4ll a  Sim goon had le bo adopted, arise to tha  objeslisms bawd bi Smith. Ths podiums  =l"llkeshdo side was Mr A*, when  prassedly **OM to. Nod witsssita  souse sI sambseme (a radium Mai  the haisshia ladirsaa Opp a C ff u it l l p c a  mdpa  sow was) with no better sesellt. Mr  Bet Morph, strange te say, mead ass sail, sad Men  in " No" was said le Me means ef  Messrs G. H. Vise, A. T. sea Z.  Bayley. AI these gerillatem, It mast be 11.1.14,  were proposed by Skil. and objected to by IBM*  who in the meantime had aimed a member el the  Pdiesa Club, a gewillawa of limited nerselmee in  lbw matters, and the landlord ef Me hem hem  which he was training, Mr W. lifikst, Neinmeheit.  This brought about a diatloelt, awl the stabs of  affairs getting wind, ugly rumen were map  afloat, and even money, and in may mum  odes, were betted that there  11111111 d be li n t O  fight." All this while Stubs was Midas sad  fuming at the delay, whilst at the mate thee his  representative and Smith's were at week almost night  and day endeavouring to bring shout a settkesent of  the difficulty. For • long time it Ignited as though  the matter would end in smoke, as after the services  of the stakehcdder bad been call ed ups great difficulty  wu esperienoed in getting anymm te aseept the office  of referee. So late as Friday Mama nething bed tern  done, and it was not until the mein et that day that  Mr Joe Veiny, consenting to stand, was nominated  by the stake holder. === Dundee ''Evening Telegraph'' === This Wednesday, 25 December 1889 article describes Slavin's reception in London after the fight. Viscount Mandeville was there, suggesting that he was a Slavin supporter:<blockquote>There was (says a sporting correspondent) great excitement in London yesterday over the fight of Monday, and all classes of the community expressed their abhorrence of the proceedings adopted by the Smith party. It was known early that Slavin would journey up from his quarters at Margate about mid-day, and by two o'clock there was a large gathering in and about the Holborn Viaduct Station. By three o'clock the crowd had increased enormously. The train arrived behind time. As soon as Slavin and Mr Taylor were recognised, a loud cheer was raised. So great was the crush that it was only with the greatest difficulty that the Australians could make their way to the street. The reception accorded the champion, although very gratifying, was somewhat rough, Slavin and his party being forced across the road before they could take refuge in a hansom. Among those present were John Lewis, Slavin's manager; [[Social Victorians/People/Mandeville|Lord Mandeville]], E. Bayley, and other well-known sportsmen. From the station Slavin went, in answer to an urgent request, to the Stock Exchange, where another enthusiastic reception waited him. A move was then made to the office of the ''Sportsman'' in Fleet Street. Among those who were present to offer their sympathy and congratulations to Slavin on what all considered to be a victory were [[Social Victorians/People/Mandeville|Lord Mandeville]], Ike White, E. Bayley, George Knight, W. Riley, C. Mitchell, J. Goldberg, J. Kilby, F. Symes, F. Hinde, and a score of well-known boxers. Shouts for Slavin were kept up outside, and the crowd could not be silenced until Slavin went to the window. He bowed his acknowledgments, and after much difficulty obtained a hearing. He thanked the multitude for his great reception, at which three cheers wore raised for him, and a corresponding number of groans for Smith. Later Slavin dined at the Albert Club where a Testimonial Committee was formed. At seven o'clock Slavin departed for Margate. Smith remained in Bruges until one o'clock on Monday, when, in company with Baldock and Harper he left for London ''via'' Calais. They arrived in London at eleven o'clock, Smith going home in company with a friend. Smith has a wound over the right eye, and his face is much swollen.<ref>"London Gives the Colonist a Great Reception." ''Dundee Evening Telegraph'' 25 December 1889, Wednesday: 3 [of 4], Col. 6b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000453%2F18891225&page=3&article=052&stringtohighlight=mandeville+viscount. Print title: ''TheEvening Telegraph'', n.p.</ref></blockquote> === ''Reynolds's Newspaper'' === The Sunday, 29 December 1899, of ''Reynolds's Newspaper'' published a detailed account of the fight.<blockquote>The fight between Slavin and Smith took place on Monday morning upon private property in the outskirts of Bruges. Various spots, both at home and abroad, were from time to time named as the possible battle-field, but finally it became known to a certain select few that an English gentleman, Mr. Atkinson-Grimshaw, residing near Bruges, had placed his grounds at the disposal of the combatants. The host is a well-known breeder of fowls; and he certainly did not expect the quality of fighting cock which responded to his invitation. The ground was encircled by large trees, which secured the utmost privacy. About forty persons assembled at the ring-side, including Viscount Royston, Viscount Mandeville, Mr. Abington, Mr. Harris, Mr. W. Riley, Mr. P. Kotchie, Mr. Ike White, the Hon. M. Sandys, Mr E. Bayley, Mr. C. Parsons, Mr. J. Kilby, Mr. C. Morton, and other well-known "sports." Smith was dressed first, and had to wait a few minutes for Slavin. The Englishman appeared in white flannel drawers, white socks, and lace-up boots; while the Australian wore black drawers, blue socks, and lace-up boots. Harper and Baldock were seconds, Howes bottle holder, and Mr. Fleming umpire for Smith; Cosnett and Bob Kirby, Mr. Taylor, and Mr. W. Riley acting in similar capacities for Slavin. Mr. Joe Vesey officiated as referee and Mr. Calder as time-keeper. The men were in the ring at 8.25, and ten minutes later the fight began. After the first two rounds everything went in favour of Slavin, who in the third round knocked Smith clean off his legs with a terrific right-hander. He repeated this operation four or five times subsequently and surprised many of the spectators by the determined way in which he fought. In the thirteenth round he doubled Smith up with a fearful blow under the left ribs, and the Englishman's friends, seeing that Smith was almost beaten, raised a cry of "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. Prior to this Slavin had been repeatedly kicked on the legs and hit on the body and head with sticks and knuckle-dusters. These gross and cowardly outrages were committed in Smith's corner, where all the fighting took place from start to finish. The next round, the fourteenth, was the last. Slavin fought in a most determined fashion, and when he was punishing Smith heavily, the latter caught the Australian round the waist and held him on to the ropes, whilst the ruffians standing around beat and kicked him in a most cowardly fashion. Amid cries of foul, the pair fell to the ground, and, on the men taking their corners, the referee said he would act no longer, as it was perfectly clear Slavin was not to have fair play. This created immense excitement, during which the referee declared the light a draw, and, crossing to Slavin, said, "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. Smith and his friends were left behind at Bruges. Round 1. — Slavin stood over Smith with a longer reach. Both sparred cantiously at the opening, Slavin at length leading off and landing heavily on Smith's chest, jumping back out of danger. Sparring again Slavin landed full on Smith's mouth. Heavy in-fighting followed, the men closing and falling together. When Slavin rose it was seen that he was bleeding from the mouth. Time, 1min. 30sec. Round 2. — Slavin fought Smith right into his own corner. There was a disgraceful scrimmage there among Smith's friends, Slavin being badly used. Both men fell together. Time, 15sec. Smith's party shouted out that their man had been spiked by Slavin's boots. The referee, thus appealed to, declared that the men had spiked each other. Round 3. — Smith showed to better advantage in this round, getting home both right and left twice in quick succession on the Australian's body. Sharp close in-fighting followed, Smith going down before a terrific right-hander. (First knock-down blow to Slavin.) Round 4. — Both men were blowing when they came up for this round. The fighting was again fast and furious, Smith at length cross-buttocking Slavin. The Englishman, however, did not fall on him. Round 5. — A light round, Smith going down before a light blow. Time, 11sec. Round 6. — Slavin came up very determined, and forced the fighting at a great pace. He drove Smith all over the ring, landing with his right a heavy blow over the right eye. Smith went down bleeding freely from a gash on the forehead. Round 7. — Smith came up stronger and did some of the forcing work, beating Slavin for a moment on to the ropes. While in this position one of the crowd of roughs deliberately kicked the Australian, who, however, took no notice of the cowardly attack. Both the combatants got away from the ropes and fought to the middle of the ring, where Smith slipped, and both were taken to their corners. Round 8. — Slavin did the opening work in this round, driving Smith round the ring and finally into his corner, where, as usual, his friends were ready to protect him. Smith came out again, when Slavin landed heavily with right and left, and Smith sought the ground. Round — 9 There were heavy exchanges from the outset, Slavin giving his man no rest, and driving him back into his corner. While there one of the mob struck Slavin on the head a violent blow. Time, 1min. Round 10. — This was a far more sportsmanlike round. Smith stood up well in the centre of the ring, and fought it out without the assistance of his friends. There was hard fighting from first to last of a close order, both getting home some terrific body blows. Both fell together. Round 11. — Smith looked tired and led off feebly while Slavin returned with right and left on the head, cutting open the Englishman's cheek on the left side. Smith landed his left again, Smith countering heavily. Smith then rushed upon the Australian, who met him with a severe blow on the mouth, and then sent him down with a terrific right-hander. Round 12. — The disturbances at the ring had now become so unbearable that the referee cautioned the seconds to keep to their corners, and not to interfere with the principals during the progress of the round. The round was of the cautious order, Smith finally being sent to grass again from a heavy right-hand blow on the jaw. Time, 1min. 30sec. Round 13. — Slavin again did all the leading, fighting the Englishman round the ring, and forcing him into his corner. There were some heavy exehanges, Smith's friends again assisting their man by hitting freely at Slavin. Round 14 and last. — Just a second before "Time" was called, Slavin, remarking on the treatment he had received, said, "Are you Englishmen, boys?" This was received with derisive shouts from Smith's party, and then, in an instant, a cry of "Police" was raised. The whole of the Smith party instantly decamped, hurrying their man along with them. Slavin, however, declined to stir, and his friends shouted that it was only a ruse to stop the fight. After a time the Smith  party returned, and Slavin then said to Baldock, the most prominent of Jem Smith's seconds, "Why don't you give me fair play?" Slavin's seconds claimed the fight, but the referee ordered them to fight on. The tactics of the Englishman's supporters were now very evident, but Slavin, who was only anxious to have the fight out to an end, advanced into the  centre of the ring, and again beat the Englishman back into his own corner. A scrimmage ensued, both men landing heavily, and the mob in Smith's corner then closed upon Slavin, who was again kicked and struck. The Australian, however, paid no attention to anyone except Smith, and he finished up the round by throwing him heavily. The referee then said, "I shall refuse to act any longer. This man" — pointing  to Slavin — "can't have fair play. I declare it a draw." Smith seemed well at the finish, and Slavin was certainly full of fight and disgusted at having been robbed of victory by the cowardly attacks of roughs. The Australian had fought in the fairest and most plucky manner throughout, forcing the Englishman back into his corner in nearly every round, and keeping him there. Here the fight came to an end, the referee exclaiming, "I shall refuse to act any longer. I shall declare it a draw." There was considerable surprise among Slavin and his supporters, and the Australian said, "I came 16,000 miles to win, and why don't you give a man a chance?" After some little delay the company dispersed. The time at the end of the fight was eight minutes to nine. Mr. Cecil Raleigh, hon. secretary of the Pelican Club, 21, Denman-street W., writes as follows:— "As an impression prevails that the Smith and Slavin fight was arranged under the patronage of this club, I should esteem it a favour if you would allow me to state emphatically that the Pelican Club gave no countenance to the affair in any way whatever, and had nothing to do with it." Directly the news became known in London the greatest indignation prevailed in all the clubs at the dastardly way in which Slavin had been treated. Subsequently some forty or fifty gentlemen proceeded to the Pelican to demand the instant and public repudiation on the part of the committee with any official cognizance of the arrangements for the fight, and also for the summary punishment of those who had dragged prize fighting down to the lowest depth of degradation and blackguardism. In accordance with these demands, and in full sympathy with them, those members present, conjointly with the proprietor, immediately sent out telegrams summoning an extraordinary meeting of the committee. That Slavin has now a right to the title (empty though it be) of "champion" is clearly supported by the 24th rule of the Prize Ring, which runs as follows:— "Any pugilist voluntarily quitting the ring previous to the deliberate judgment of the referee being obtained shall be deemed to have lost the fight." It seemed a pity that Mr. Vesey should not have had sufficient support to enforce what was uadoubtedly the judgment of all impartial spectators of round thirteen. Arrival of Slavin at Margate. Slavin arrived at Margate at half-past seven on Monday evening, and met with a flattering reception. During his training in the town he had become a general favourite with those who took an interest in the event. Sympathy was almost wholly on his side, and during the day news was anxiously awaited. At the station he was met by a four-in-hand and the town band, and proceeded to the Elephant Hotel, the band playing, "See the conquering hero comes." Along the Parade, on which was assembled a large crowd of people, he was loudly cheered. At the Elephant Hotel, there was a large crowd of people, and the reception was most enthusiastic. Slavin briefly addressed the gathering, thanking them for the reception he had received. His intention, he said, when [Col. 4c–5a] he left England was to win the championship or lose it fairly, and he was sorry that the fight had ended in such a fiasco. Inside the hotel the scene was very exciting, and Slavin received the hearty congratulations of his friends at the slight punishment he had received. He then left for the Nayland Rock Hotel, his quarters, and on the road was loudly cheered. Upon arriving at the hotel, Slavin retired, accompanied by his second, Joss Cosnett, after which the correspondent of the Press Association was introduced to him. The first question put was, "Well, Slavin, how have you fared?" "Very badly," responded the Australian. "As far as the roughs were concerned, I never witnessed anything more disgraceful in all my experience. My treatment at their hands has been most cruel, and nothing I could say would sufficiently emphasize what I mean. I am very pleased indeed at  the reception I received on my arrival at Margate. Nothing could have been more enthusiastic. You see what punishment I have received. It is not from Smith so much as it is from the outsiders." — Joss Cosnett: Smith did not bit him any more than I am hitting you. The conduct of the roughs round the ring was shameful. Slavin: The treatment I have received from them was most unmanly. I am sure the roughs never meant it to be fought out. They intended to have a draw if they found their man losing. Several of Smith's backers, including two well-known noblemen, were disgusted, and threw his colours away and put mine on, as a protest, I suppose, against the treatment I had received. To speak the truth, I would not like to see a dog treated as I was. All I am anxious for now is to meet Sullivan for the championship of the world for £500 to £1,000, or anyone else. I am willing and ready to meet and fight anybody for the championship before I go back to Australia. The Pelican Club and the Fight. A meeting was held at the Pelican Club at one o'clock on Tuesday afternoon. At the conclusion the hon. secretary, Mr. Cecil Raleigh, furnished the following report:— "In view of the reports of what had occurred at the Smith and Slavin fight a special meeting of the committee of the Pelican Club was instantly convened by telegraph for this afternoon. There was a large attendance of the committee, among those present being the Marquis of Queensbury, Mr. John Corlett, and other well-known sportsmen. It was unanimously determined to address a letter of sympathy to Frank Slavin, expressing the appreciation of the committee of the courage he exhibited, in face of the most  blackguardly conduct which ever disgraced the ring side. The committee added that they fully recognised that he had indisputably established his claim to the championship of England, and they further informed him of their intention to present to him a belt and purse in recognition of his pluck. The committee then determined that Mr. John Fleming, late manager to Jem Smith, who had hitherto acted as master of the ceremonies at the Pelican Club, should be suspended from his office until such time as he should afford an explanation of his connection with the Smith and Slavin fight, which explanation would be laid before a general meeting of the members of the club on Tuesday, January 6th. The indignation of the members of the club knows no bounds, and every effort will be made to discover who were the real instigators of the disgraceful business." It is most likely that the belt and purse will be presented to Slavin on January 14th, when the club move into their new premises. The following letter has been addressed by the pugilist, F. P. Slavin, to the Pelican Club:— "In reply to your very kind and encouraging letter, accept my warmest thanks for the high compliment you are about to pay me in presenting me with a belt, called the Champion of England belt. You may rely it will be my duty, as it has been all through my career, to do my best to win any contest I may enter into in a fair and manly manner. — Again thanking you, I am, sir, yours truly, "F. P. SLAVIN. "Nayland Rock Hotel, Margate." Mr. Fleming's Statement. Mr. Fleming states that the fight was arranged before the boxing match with Jackson. He went over to Belgium and selected the ground. As regards the rough element present, he did not know one of the men. No doubt they were engaged by some friends of Smith. The only persons who went across with them were four or five gentlemen. All connection between Smith and him had ceased, as he felt that he could not act in any way for Smith after what had occurred. Every fight that he had previously had the management of had always been brought off without the intrusion of any of the rough element at the ring side. He took no part in the proceedings, being merely there as a spectator, and he had not seen Smith since the fight. He did not think that he should have any difficulty in proving that he had no connection with the disgraceful proceedings at the ring side. The Preliminary Arrangements. The preliminaries of the match will, perhaps, be interesting. When, in 1885, Jem Smith, who had made a name among English heavy-weight boxers, met and defeated Jack Davis, of Birmingham, there was a sign of a revival of pugilism in this country. This was increased when Smith fought Alf Greenfield, and, although the battle ended in a draw, it was not long before the "lost art" was freely discussed, and there was every appearance of the revival of the prize-ring. Smith, as will be well remembered, fought a memorable battle with Jake Kilrain, who came from America to wrest the championship honour from the Englishman. John L. Sullivan, the Boston Boy, soon followed, and, although he failed to beat Charles Mitchell, his claim to the title of champion of the world was made good last July, when be met and defeated Kilrain. Then came new stars in the firmament of boxers, but glove-fighting had almost extinguished the "old style." Prominent among the late arrivals were Peter Jackson, a coloured man, and Frank P. Slavin, both hailing from Australia. Their records were of the highest calibre, Slavin having been successful in all the nineteen contests in which he had engaged, either with gloves or with bare hands, while Jackson's was but little less noticeable. The latter, however, carried all before him in America, and then sought fresh fields in England. Only a few weeks ago he vanquished Jem Smith in a couple of rounds. Slavin arrived in London a week prior to Jackson, and, after a lengthy wait, he was accommodated with a match. No one but W. Goode cared to meet him with the gloves, and Goode was easily disposed of. After signing articles to fight Jem Smith under prize ring rules, both men went into training, Smith at Newmarket, under the care of Jem Howes, and Slavin at Margate, with Fred Symes as mentor. The most difficult part of the arrangements had yet to be settled, and it was not without great trouble that a battle ground could be selected which afforded facilities for the accommodation of a party of sportsmen to witness the contest. At length a point was pitched upon near the Belgian capital, easy of access and fairly safe from interference. Another difficulty now arose over the appointment of a referee, and no less than five good men were named, all of whom were in turn objected to by Smith. The possibility of this having been foreseen it had been stated in the articles that the two parties failing to agree, the stakeholder, Mr. C. W. Blacklock, should appoint that official. This he did with satisfaction to both parties, naming Mr. J. Vesey, a well-known sporting man, to act. The stakes had then been increased to £500 a side. There is a great difference in the physique of the two men. Slavin stands 6ft. 1in., and scales in condition thirteen stone, is a very muscular athlete, and particularly quick on his legs; while Smith is of the sturdy order, and is only 5ft. 8<sup>1</sup>/<sub>2</sub>in. high. His weight when fit is 12st. 10lbs., and although his muscles are very massively developed, giving proof of great strength, he has not the quick hitting power of the Australian. Performances of the Men. Frank P. Slavin, the champion of Australia, was born on January 27, 1863, at Maitland, N.S.W. He stands 6ft. 1in. in his stockings, and in condition scales 13st. 5lb. or 13st. 6lb. Shavin served his time or apprenticeship to a coachbuilder and blacksmith, and first came out as a pedestrian. After winning several competitions as an amateur, Slavin's first fight was with knuckles against M. Power for £50 a-side, and he won after seven rounds, lasting thirteen minutes. He then beat Jem Burke with two-ounce gloves for £200 a-side and the championship of Queensland, winning in three rounds. This was in December, 1885, and in 1886 he went to Sydney and entered into an engagement with Larry Foley. Here Slavin met Taylor, a pupil of Jackson's, for a purse with small gloves, and stopped him in two rounds. He was then matched with a big follow named Bligh for £10 a-side, and beat him in one round. After a rest of two months articles were signed for him and Mike Dooley to box for £100 a-side, and Slavin won in nine rounds. He next met Mike Costello, an American, a good man, and after a big battle in Sydney, lasting two hours and a half, a draw was the result. Another match was made for £400, and £50 a-side was posted, but Costello forfeited this. Afterwards Slavin's backers matched him for £25 a-side to stop Costello in six rounds, but the latter is very clever and tricky, and lasted the time. Subsequently Slavin journeyed to Sydney, and finding Mike there, challenged him again. The result of this was a match for £100 a-side with skin gloves, which Slavin won in twenty minutes. Slavin at this time had any amount of backing, and issued a challenge to fight Peter Jackson for £250 a-side and the championship. Jackson's party wanted a match with gloves for a purse. They could not, however, come to terms. In January last year the Australian made a trip to New Zealand. The first man put up was Smith, who was backed to stand four rounds, but he had had enough at the end of the first. In Dunedin Slavin beat a big fellow in four minutes, and journeyed a long distance to fight Laing, the champion. The pair were matched for  £200 a-side, and fought with two-ounce gloves at Wanganui, Laing failing to come up at the end of the fifth round. After this Slavin returned to Melbourne, and met Jack Burke, but after Slavin had had all the best of it, the contest was declared a draw. Mike Dooley was now in Sydney, and issued a challenge to him. This led to a match for £200 with small gloves, which ended in Mike being knocked out in 2min. 40sec. Slavin returned to Melbourne, and fought Burke, Slavin undertaking to stop him in six rounds; £200 a-side was put up and the two fought with six-ounce gloves, and agreed to divide the gate, and Burke was beaten in the middle of the third round. Previous to this Billy Farnan, the conqueror of Jackson, and Slavin, fought for £200, with the result that the latter added another winning bracket to his name, Farnan failing to answer to the call of time in the third round. He then received forfeit from Woolf Bendoff. An easy victory over Bill Buck in two rounds followed. He then came to England, and his late victory in his glove fight with the plucky young Englishman, W. Goode, must still be fresh in the memory of our readers. Jem Smith was born in Whitecross-street on January 21, 1863. He stands 5ft. 8<sup>1</sup>/<sub>2</sub>in. high, and weighs in condition 12st. 10lb. Round the chest he measures 43<sup>1</sup>/<sub>2</sub>in.; waist, 36<sup>1</sup>/<sub>2</sub>in.; hips, 41in.; thigh, 24<sup>1</sup>/<sub>2</sub>in.; calf, 16<sup>3</sup>/<sub>4</sub>in.; biceps, 15<sup>1</sup>/<sub>2</sub>in. For some years he was employed in one of the metropolitan timber yards, but developing a taste for boxing after one or two rough-and-tumbles with his fellow-workmen, he was taken in hand by Jack Knifton ("eighty-one tonner") and later by Mr. F. Grimm, of the Central Club. His first success was in a competition at the Griffin, Shoreditch. Later on he beat Snowey, of Holloway, and was then matched to box Harry Arnold with gloves for a £10 purse. The affair was looked upon as a good thing [Col. 5c–6a] for Smith, but it took him nearly an hour to polish off this last opponent, though he had slightly the best of the encounter from start to finish. Smith's next engagement was with Woolf Bendoff to box to a finish with gloves for £25 a-side, which came off at an East-end club. In this contest Smith injured his right arm, but, nevertheless, proved successful at the end of twelve  rounds. Smith also won Tom Symonds's All-comers' Competition on February 26, 1885, at Bill Richardson's, defeating Sugar Goodson, Jack Wannop (the wrestler), and Tom Longer. Smith's next battle was with Jack Davis, for £100 a side and the championship, on the turf, and came off on the borders of Surrey and Sussex on Wednesday, December 17. In all six rounds were fought, occupying 7min. 56sec., Davis being knocked out of time with a swinging right-hander. He subsequently met Alf Greenfield, of Birmingham, for £300, at Maisons-Laffite, near Paris, on Tuesday, February 16, 1886. Thirteen rounds were fought, mostly in Smith's favour, and the men fell, whereupon a cry of foul was raised by Greenfield's party, and a crowd of roughs broke into the ring, stopping the fight. The referee (Jem Mace), at the request of the principal backers of the men, agreed to decide the battle a draw. Soon after Smith was matched to fight Knifton, the "eighty-one-tonner" as he is called. This proved a fiasco. The memorable struggle between Smith and Kilrain, brought off on an island in the Seine, came next, darkness putting an end to the affair, which also resulted in a draw, after the men had faced each other for 106 rounds, on December 19th, 1887. This brings us to his ten-round glove contest with Jack Wannop, Smith laying £250 to £200, and winning easily. After which, as all the pugilistic world knows, Smith was easily defeated by the coloured champion, Peter Jackson, in a glove fight at the Pelican Club, the last named winning on a foul by Smith, who was at the same time well beaten.<ref>"The Great Prize Fight. £1,000 in Stakes. Brutal Scenes in the Ring." ''Reynolds's Newspaper'' 29 December 1889, Sunday: 1 [of 8], Col. 4a–6a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000101%2F18891229&page=1&article=006&stringtohighlight=mandeville+viscount. Same print title, n.p.</ref> </blockquote> === The ''People'' === On Sunday, 29 December 1899, the ''People''<blockquote>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 the special meeting of the committee of the Pelican Club it was unanimously determined to address a letter of sympathy to Slavin, expressing the appreciation of the committee of "the courage he exhibited in face of the most blackguardly conduct which ever disgraced the ring side.["] The committee added that they fully recognised that he had indisputably established his claim to the championship of England, and they further informed him of their intention to present to him a belt and purse in recognition of his pluck[.] The committee determined that Mr. John Fleming, late manager to Jem Smith, who had hitherto acted as manager of the ceremonies at the Pelican Club, should be suspended from his office until such time as he should afford an explanation of his connection with the Smith and Slavin fight, which explanation would be laid before a general meeting of the members of the Pelican Club on Tuesday, January 7th. It is most likely that the belt and purse will be presented to Slavin on January 14th, when the club will move into their new premises. While Salvin was at the Stock Exchange on Tuesday, a subscription in his interest was started, and £110 was quickly subscribed. Slavin Challenges J. L. Sullivan. Frank Slavin, the champion of Australia, is prepared to meet John L. Sullivan, under London prize ring rules, for £500 or £1,000 a side and the championship of the world. Slavin has lodged a sum of £100, and will contest the title with Sullivan in either America or Australia in six months from signing articles. We have cabled this challenge to Mr. R. K. Fox, of New York, and the money lodged with him or the editor of the ''Sportsman'' will lead to a match.<ref>"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.</ref></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. {{paragraph break}} 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 {{paragraph break}} The subscriptions towards the above now amount to over 400''l.'', and it having been decided to close the list on the 20th, the committee will be obliged if those who have promised donations will forward the same to the hon. sec. and treasurer, Mr. G. T. Dunning, T<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}} 9t00sh2x8njtj1bksh9faf9ed41jtyt 0 332551 2844596 2842803 2026-11-05T10:44:25Z Numismaticus 3112922 add one 2844596 wikitext text/x-wiki Unillustrated lot project: a permanent record of coins which were offered in old auction sales or sold in private treaty, but without illustrations, to preserve documentary evidence which may otherwise be lost in trade or by institutions. {| class="wikitable" |+ !Item ID !Territory !Reference !Coin photograph !Document photograph !Notes |- |1 |Kingdom of Macedon |Price 316 |[[File:Alexander_III_AE_as_Price_316.png|300x300px]] |[[File:Ticket_re._Price_316.png|thumb|Macedonien, Alexander IV. od. III. 323-311 bronze Herakles kopf r. Rs. Kende n. Koecher 15,-]] |6.39 g; Ticket: Germany circa. 1960s or earlier. |- |2 |Seleucid |SC 340 |[[File:Unil2.jpg|300x300px]] |[[File:Unil2 ticket.jpg|thumb|Heinrich Pilartz VII, [19]64, no. 93]] |1.39 g. ex-Heinrich Pilartz VII, 1-3 October 1964, (Cologne, Germany), lot 93 |- |3 |Byzantine |Sear 518 |[[File:Byzantine, Maurice Tiberius follis, rengal year 6, Kyzikos, first officina.png|300x300px]] | |11.19 g ex-Adalbert Balvin collection, Dorotheum sale (September 2026), lot 928 pt. |- |4 |Roman Provincial |Dattari 1465 |[[File:Alexandria tetradrachm as Dattari 1465.jpg|300x300px]] |[[File:Pilartz VII lot 343 envelope.jpg|227x227px]] |24 mm, 13.16 g Pilartz catalogues as"Antioch" in error. ex-Heinrich Pilartz VII, 1-3 October 1964, lot 343 |} nq1qdw4e3zqfxd4h0j1sye8yj6kiw0q 0 332682 2844511 2843039 2026-11-04T19:49:45Z Dick Bos 24466 cont 2844511 wikitext text/x-wiki This subpage is dedicated to the graphical representation of the theory of David Ricardo, as developed in his ''[[David Ricardo/An Essay on Profits|Essay on Profits]].'' Several scholars have tried to represent the essential elements of the Ricardian theory of rent in the ''Essay'' in a graphical way.<br /> Here some of these representations will be shown in some detail. An older one, by [[:w:en:Nicholas Kaldor|Nicholas Kaldor]], dates back to 1955. The second graphical representation we will show is by O'Brien (1975). == Graphical representation of the ''Essay'' by Kaldor (1955) == In his essay "Alternative Theories of Distribution" (1955){{sfnp|Kaldor|1955}}, [[:w:en:Nicholas Kaldor|Kaldor]] provides a graphical representation of Ricardo’s theory of rent, which has been widely cited since. His (concise) treatment of Ricardo’s theory will be summarised here:<br /> Ricardo’s theory is based on two principles: the "marginal principle" and the "surplus principle".<ref name=surplus>The term "surplus principle", as used here in line with Kaldor, may be somewhat confusing in the light of what was discussed in [[David Ricardo/An Essay on Profits#Rent: a 'pure surplus'|section 3]] of the previous page in this resource. There, rent was in fact described as 'pure surplus'.</ref> The "marginal principle" explains the share of rent; the "surplus principle" explains the distribution of the residual between wages and profits. The economy is divided into two sectors: agriculture and industry. The forces at work in agriculture determine the distribution across the entire economy. Kaldor illustrates "the agricultural side of the picture" using the diagram in Figure 1. He assumes that there are fixed technical coefficients for the ratio between fixed capital and labour, and that the coefficients for the land–labour ratio vary. The existence of the two distinct curves representing marginal and average labour productivity is a consequence of the downward slope of the average product curve, that is to say, of the assumption of "diminishing returns". [[Image:KaldorRicardo.png|center|550px|Graphical representation of Ricardo's approach in ''An Essay on Profits''|alt=Graphical representation of Ricardo's approach in ''An Essay on Profits'']] Given a certain size of the labour force (OM), total output is OCDM.<br /> "Rent is the difference between the product of labour on 'marginal' land and the product on average land, or (allowing for the intensive, as well as the extensive, margin) the difference between average and marginal labour productivity which depends on the elasticity of the p-Ap curve, i.e., the extent to which diminishing returns operate."{{sfnp|Kaldor|1955|p=84}}<br /> According to Ricardo’s theory, the marginal product of labour is divided between wages and profits. The wage rate is determined exogenously and is assumed to be constant in terms of grain. The demand for labour is determined by capital accumulation, and more specifically by the available wage fund. As capital accumulation continues, the wage fund grows, and consequently the quantity of labour employed increases, assuming the wage rate OW remains constant. Ultimately, profits are then a residual. The rate of profit is therefore entirely determined by the conditions of production on marginal land, which (in a closed economy and given a certain level of technical knowledge) in turn reflect the extent to which capital has been accumulated. Kaldor quotes a statement by [[James Mill]], according to whom "the diminishing fertility of the soil is the great and ultimately only necessary cause of a fall in profit."<ref name=Mill>James Mill (1773-1836), quoted in {{harvp|Kaldor|1955|p=86}}</ref> Kaldor also points out that ongoing technical progress causes the p-Ap and p-Mp curves to shift to the right, thereby suspending the operation of the law of diminishing returns.<br /> Conversely, taxation and agricultural protection (which lead to an increase in the wage bill) will reinforce the trend towards "that ultimate state of gloom, the Stationary State, where accumulation ceases."{{sfnp|Kaldor|1955|p=215}} Profits will become so low that they will no longer provide sufficient compensation for the effort the capitalist must expend and the risk he runs when he employs his capital productively (cf. {{harvp|Ricardo|1817|page=122}}). The essence of what has been set out above can be summarised in the following diagram: [[Image:DiagramRicardoDistribution.png|center|550px|Diagram concerning Ricardo's theory of distribution|alt=Diagram concerning Ricardo's theory of distribution]] == Graphical representation of the ''Essay'' by O'Brien (1975) == In O’Brien’s exposition of Ricardo’s theory of ground rent,{{sfnp|O'Brien|1975|p= 37–41}} several new elements are highlighted in comparison with the approach of Kaldor:<br /> Because land is not homogeneous, the application of increasing amounts of capital and labour to land already in use results in diminishing physical returns; whilst the cultivation of new land, which is less fertile than the land already under cultivation, results in diminishing returns to scale. Both the average and marginal products of labour and capital decline as a result of these effects. The demand for grain, and through this, the extent of the cultivated area, are functions of population size. Population size itself is determined in accordance with a (specifically Ricardian) interpretation of Malthus’s theory of population.<ref name=popul>See {{harvp|O’Brien|1975|p=56f}}; {{harvp|Stigler|1952|p=187–195}}</ref> The theory of (ground) rent was first developed by [[James Anderson]] in 1777 and was published, both by Malthus and Edward West, in 1815.<ref name=MalthusWest>{{cite book | last = Malthus | first = T.R. | author-link = :w:en:Thomas Robert Malthus | date = 1815 | title = An Inquiry Into the Nature and Progress of Rent | location = London | publisher = John Murray | ref = harv | url = https://en.wikisource.org/wiki/An_Inquiry_Into_the_Nature_and_Progress_of_Rent }}; {{cite book | last = [West] | first = [Edward] | date = 1815 | title = Essay on the application of capital to land, with observations shewing the impolicy of any great restriction of the importation of corn, and that the bounty of 1688 did not lower the price of it. By a fellow of University College, Oxford | location = London | publisher = T. Underwood | url = https://books.google.nl/books?id=-i0ZA8rzAP0C }}</ref> Ricardo independently developed his own theory of rent.<ref name=Ricrent>{{harvp|Stigler|1952|p=195f.}}; {{harvp|Sraffa|1951}}, [https://archive.org/details/workscorresponde0004rica/page/4/mode/1up p. 5f.]</ref> Ricardo treats rent as an intra-marginal surplus: the price of the product is determined by the supply price at the "margin of cultivation", and the total rent is the sum of the differences between the price and the costs of production on all intra-marginal types of land. O’Brien illustrates this in a diagram: see Figure 2a. O’Brien also presents a ‘linear version’ of Kaldor’s graph: see Figure 2b.<ref name=Fig2b>See also {{harvp|Blaug|1968|p=92}}</ref> [[Image:O'BrienRicardoA.png|center|550px|Graphical representation of rent according to O'Brien 1975, p. 39 (Image 2a) |alt=Graphical representation of rent according to O'Brien 1975, p. 39 (Image 2a)]] [[Image:O'BrienRicardoB.png|center|550px|Graphical interpretation of the representation of Kaldor by O'Brien 1975, p. 39 (Image 2b) |alt=Graphical interpretation of the representation of Kaldor by O'Brien 1975, p. 39 (Image 2b)]] According to O’Brien, this figure shows how Ricardo used his model to derive predictions about the development of income shares over time (none of which, according to O’Brien, were in fact confirmed by the course of events). The total product ORNQ is divided between ground rent, wages and profits. Ground rent is the difference between the average and marginal products at the output PRNT determined by the size of the population.<br /> O’Brien notes that this share is equal to the sum of the differences between the marginal products prior to the most recent expansion and the marginal product of the most recent expansion of the cultivated area – the area of the triangle PBT in Figure 2b;<ref name=area>As will be explained in our page…. (old: paragraph 8 of Part III) this line of reasoning only holds if we assume that the yield from the land, when no labour (or capital) is employed, is zero – in the terms of the relevant paragraph: if we assume that ƒ(0) = 0.</ref> a similar line of reasoning can also be applied in the case of non-linear curves, such as those in Kaldor’s model. The appropriation of ground rent by landowners is ensured by competition amongst tenants/entrepreneurs, who will be satisfied with the marginal rate of profit. Wages are set at a subsistence level: OWW'Q. Profits (WPTW') constitute the residual.<br /> If these profits exceed what entrepreneurs need to "save" – that is, to refrain from consumption – they will begin to compete with one another for the available labour, thereby driving wages above the subsistence level. Then, through the population mechanism, the population will rise and wages will fall back to the subsistence level. If, thereafter, the marginal product is still sufficient to generate profits greater than what entrepreneurs consider the minimum necessary, the process outlined above will repeat itself. This will continue until the marginal product has fallen to a level that no longer provides the incentive for continued accumulation. This will not so much be a state of no profits as a state in which profits have fallen to "some minimum" ({{harvp|O’Brien|1975|p=41}}).<ref name=minimum>See also {{harvp|Ricardo|1817|p=120–121}}.</ref> In any case, the population will now cease to grow, wages will stabilise at a subsistence level, and the economy will have reached a stationary state. According to O’Brien, the above exposition forms the core of Ricardo’s system (as set out in the ''Principles''): a mechanical model that demonstrates the tendency towards a steady state in the not-too-distant future. == Footnotes == {{references}} == Sources == * {{cite book | last = Blaug | first = Mark | date = 1985 | title = Economic Theory in Retrospect ''(fourth ed.)'' | location = Cambridge etc. | publisher = Cambridge University Press | ISBN = 0 521 30354 0 | ref = harv | url = http://digamo.free.fr/blaug62.pdf }} * {{cite book | last = O'Brien | first = D.P. | author-link = :w:en:Denis Patrick O'Brien | date = 1975 | title = The Classical Economists | location = Oxford | publisher = Clarendon Press | ISBN = 0 19 877015 4 | url = https://archive.org/details/classicaleconomi0000obri | url-access = restricted | ref = harv }} * {{cite journal | last = Kaldor | first = Nicholas | date = 1955 | title = Alternative Theories of Distribution | journal = The Review of Economic Studies | volume = 23 | issue = 2 | pages = 83–100 | doi = 10.2307/2296292 | url = https://www.jstor.org/stable/2296292 | ref = harv }}<br />{{small|(access via [[File:Wikipedia Library owl.svg|12px|alt=12 px Wikipedia Library Owl]] [[:w:en:Wikipedia:The Wikipedia Library|The Wikipedia Library (TWL)]])}} * {{cite book | last = Ricardo | first = David | author-link = David Ricardo | date = 1815 | title = An Essay on the Influence of a Low Price of Corn on the Profits of Stock | location = London | publisher = John Murray | ref = harv | url = https://en.wikisource.org/wiki/An_Essay_on_the_Influence_of_a_Low_Price_of_Corn_on_the_Profits_of_Stock }} * {{cite book | last = Ricardo | first = David | author-link = David Ricardo | date = 1817 | title = On the Principles of Political Economy; And Taxation | location = London | publisher = John Murray | ref = harv | url = }} (when referring we will use the page numbering of the Sraffa ed.) * {{cite book | last = Sraffa | first = Piero | author-link = :w:en:Piero Sraffa | date = 1951 | chapter = Note on "Essay on Profits" | title = The Works and correspondence of David Ricardo (vol. IV) | location = Cambridge | publisher = Cambridge University Press | url = https://archive.org/details/workscorresponde0004rica | ref = harv }} * {{cite journal | last = Stigler | first = George J. | author-link = :w:en:George Stigler | date = 1952 | title = The Ricardian Theory of Value and Distribution | journal = Journal of Political Economy | volume = 60 | issue = 3 | pp = 187–207 | ref = harv | url = https://www.jstor.org/stable/1826451 }}<br />{{small|(access via [[File:Wikipedia Library owl.svg|12px|alt=12 px Wikipedia Library Owl]] [[:w:en:Wikipedia:The Wikipedia Library|The Wikipedia Library (TWL)]])}} eouol6bmaxir0oeg9b8gsthodleu8o3 2844594 2844511 2026-11-05T10:41:36Z Dick Bos 24466 H&K 2844594 wikitext text/x-wiki This subpage is dedicated to the graphical representation of the theory of David Ricardo, as developed in his ''[[David Ricardo/An Essay on Profits|Essay on Profits]].'' Several scholars have tried to represent the essential elements of the Ricardian theory of rent in the ''Essay'' in a graphical way.<br /> Here some of these representations will be shown in some detail. An older one, by [[:w:en:Nicholas Kaldor|Nicholas Kaldor]], dates back to 1955. The second graphical representation we will show is by O'Brien (1975). A third contribution we will discuss here is by Howard & King (1975). == Graphical representation of the ''Essay'' by Kaldor (1955) == In his essay "Alternative Theories of Distribution" (1955){{sfnp|Kaldor|1955}}, [[:w:en:Nicholas Kaldor|Kaldor]] provides a graphical representation of Ricardo’s theory of rent, which has been widely cited since. His (concise) treatment of Ricardo’s theory will be summarised here:<br /> Ricardo’s theory is based on two principles: the "marginal principle" and the "surplus principle".<ref name=surplus>The term "surplus principle", as used here in line with Kaldor, may be somewhat confusing in the light of what was discussed in [[David Ricardo/An Essay on Profits#Rent: a 'pure surplus'|section 3]] of the previous page in this resource. There, rent was in fact described as 'pure surplus'.</ref> The "marginal principle" explains the share of rent; the "surplus principle" explains the distribution of the residual between wages and profits. The economy is divided into two sectors: agriculture and industry. The forces at work in agriculture determine the distribution across the entire economy. Kaldor illustrates "the agricultural side of the picture" using the diagram in Figure 1. He assumes that there are fixed technical coefficients for the ratio between fixed capital and labour, and that the coefficients for the land–labour ratio vary. The existence of the two distinct curves representing marginal and average labour productivity is a consequence of the downward slope of the average product curve, that is to say, of the assumption of "diminishing returns". [[Image:KaldorRicardo.png|center|550px|Graphical representation of Ricardo's approach in ''An Essay on Profits''|alt=Graphical representation of Ricardo's approach in ''An Essay on Profits'']] Given a certain size of the labour force (OM), total output is OCDM.<br /> "Rent is the difference between the product of labour on 'marginal' land and the product on average land, or (allowing for the intensive, as well as the extensive, margin) the difference between average and marginal labour productivity which depends on the elasticity of the p-Ap curve, i.e., the extent to which diminishing returns operate."{{sfnp|Kaldor|1955|p=84}}<br /> According to Ricardo’s theory, the marginal product of labour is divided between wages and profits. The wage rate is determined exogenously and is assumed to be constant in terms of grain. The demand for labour is determined by capital accumulation, and more specifically by the available wage fund. As capital accumulation continues, the wage fund grows, and consequently the quantity of labour employed increases, assuming the wage rate OW remains constant. Ultimately, profits are then a residual. The rate of profit is therefore entirely determined by the conditions of production on marginal land, which (in a closed economy and given a certain level of technical knowledge) in turn reflect the extent to which capital has been accumulated. Kaldor quotes a statement by [[James Mill]], according to whom "the diminishing fertility of the soil is the great and ultimately only necessary cause of a fall in profit."<ref name=Mill>James Mill (1773-1836), quoted in {{harvp|Kaldor|1955|p=86}}</ref> Kaldor also points out that ongoing technical progress causes the p-Ap and p-Mp curves to shift to the right, thereby suspending the operation of the law of diminishing returns.<br /> Conversely, taxation and agricultural protection (which lead to an increase in the wage bill) will reinforce the trend towards "that ultimate state of gloom, the Stationary State, where accumulation ceases."{{sfnp|Kaldor|1955|p=215}} Profits will become so low that they will no longer provide sufficient compensation for the effort the capitalist must expend and the risk he runs when he employs his capital productively (cf. {{harvp|Ricardo|1817|page=122}}). The essence of what has been set out above can be summarised in the following diagram: [[Image:DiagramRicardoDistribution.png|center|550px|Diagram concerning Ricardo's theory of distribution|alt=Diagram concerning Ricardo's theory of distribution]] == Graphical representation of the ''Essay'' by O'Brien (1975) == In O’Brien’s exposition of Ricardo’s theory of ground rent,{{sfnp|O'Brien|1975|p= 37–41}} several new elements are highlighted in comparison with the approach of Kaldor:<br /> Because land is not homogeneous, the application of increasing amounts of capital and labour to land already in use results in diminishing physical returns; whilst the cultivation of new land, which is less fertile than the land already under cultivation, results in diminishing returns to scale. Both the average and marginal products of labour and capital decline as a result of these effects. The demand for grain, and through this, the extent of the cultivated area, are functions of population size. Population size itself is determined in accordance with a (specifically Ricardian) interpretation of Malthus’s theory of population.<ref name=popul>See {{harvp|O’Brien|1975|p=56f}}; {{harvp|Stigler|1952|p=187–195}}</ref> The theory of (ground) rent was first developed by [[James Anderson]] in 1777 and was published, both by Malthus and Edward West, in 1815.<ref name=MalthusWest>{{cite book | last = Malthus | first = T.R. | author-link = :w:en:Thomas Robert Malthus | date = 1815 | title = An Inquiry Into the Nature and Progress of Rent | location = London | publisher = John Murray | ref = harv | url = https://en.wikisource.org/wiki/An_Inquiry_Into_the_Nature_and_Progress_of_Rent }}; {{cite book | last = [West] | first = [Edward] | date = 1815 | title = Essay on the application of capital to land, with observations shewing the impolicy of any great restriction of the importation of corn, and that the bounty of 1688 did not lower the price of it. By a fellow of University College, Oxford | location = London | publisher = T. Underwood | url = https://books.google.nl/books?id=-i0ZA8rzAP0C }}</ref> Ricardo independently developed his own theory of rent.<ref name=Ricrent>{{harvp|Stigler|1952|p=195f.}}; {{harvp|Sraffa|1951}}, [https://archive.org/details/workscorresponde0004rica/page/4/mode/1up p. 5f.]</ref> Ricardo treats rent as an intra-marginal surplus: the price of the product is determined by the supply price at the "margin of cultivation", and the total rent is the sum of the differences between the price and the costs of production on all intra-marginal types of land. O’Brien illustrates this in a diagram: see Figure 2a. O’Brien also presents a ‘linear version’ of Kaldor’s graph: see Figure 2b.<ref name=Fig2b>See also {{harvp|Blaug|1968|p=92}}</ref> [[Image:O'BrienRicardoA.png|center|550px|Graphical representation of rent according to O'Brien 1975, p. 39 (Figure 2a) |alt=Graphical representation of rent according to O'Brien 1975, p. 39 (Figure 2a)]] [[Image:O'BrienRicardoB.png|center|550px|Graphical interpretation of the representation of Kaldor by O'Brien 1975, p. 39 (Figure 2b) |alt=Graphical interpretation of the representation of Kaldor by O'Brien 1975, p. 39 (Figure 2b)]] According to O’Brien, this figure shows how Ricardo used his model to derive predictions about the development of income shares over time (none of which, according to O’Brien, were in fact confirmed by the course of events). The total product ORNQ is divided between ground rent, wages and profits. Ground rent is the difference between the average and marginal products at the output PRNT determined by the size of the population.<br /> O’Brien notes that this share is equal to the sum of the differences between the marginal products prior to the most recent expansion and the marginal product of the most recent expansion of the cultivated area – the area of the triangle PBT in Figure 2b;<ref name=area>As will be explained in our page…. (old: paragraph 8 of Part III) this line of reasoning only holds if we assume that the yield from the land, when no labour (or capital) is employed, is zero – in the terms of the relevant paragraph: if we assume that ƒ(0) = 0.</ref> a similar line of reasoning can also be applied in the case of non-linear curves, such as those in Kaldor’s model. The appropriation of ground rent by landowners is ensured by competition amongst tenants/entrepreneurs, who will be satisfied with the marginal rate of profit. Wages are set at a subsistence level: OWW'Q. Profits (WPTW') constitute the residual.<br /> If these profits exceed what entrepreneurs need to "save" – that is, to refrain from consumption – they will begin to compete with one another for the available labour, thereby driving wages above the subsistence level. Then, through the population mechanism, the population will rise and wages will fall back to the subsistence level. If, thereafter, the marginal product is still sufficient to generate profits greater than what entrepreneurs consider the minimum necessary, the process outlined above will repeat itself. This will continue until the marginal product has fallen to a level that no longer provides the incentive for continued accumulation. This will not so much be a state of no profits as a state in which profits have fallen to "some minimum" ({{harvp|O’Brien|1975|p=41}}).<ref name=minimum>See also {{harvp|Ricardo|1817|p=120–121}}.</ref> In any case, the population will now cease to grow, wages will stabilise at a subsistence level, and the economy will have reached a stationary state. According to O’Brien, the above exposition forms the core of Ricardo’s system (as set out in the ''Principles''): a mechanical model that demonstrates the tendency towards a steady state in the not-too-distant future. == Contribution by Howard & King (1975) == Howard & King point out that, for a given form of the production function – and thus of the average and marginal productivity curves – it is possible that, whilst the rate of profit may fall continuously, a steady state may nevertheless not be reached. [[Image:Howard&King.png|center|550px|A falling rate of profit, but not a steady state, after Howard & King 1975, p. 86 (Figure 3) |alt=A falling rate of profit, but not a steady state, after Howard & King 1975, p. 86 (Figure 3)]] Suppose the production function is such that the average and marginal productivity curves follow the pattern shown in Figure 3: the marginal productivity curve approaches asymptotically a horizontal line that lies at the same level as, or higher than, the wage curve WW': "accumulation can continue indefinitely, with a positive, though always declining, rate of profit".{{sfnp|Howard|King|1975|p=86}} == Footnotes == {{references}} == Sources == * {{cite book | last = Blaug | first = Mark | date = 1985 | title = Economic Theory in Retrospect ''(fourth ed.)'' | location = Cambridge etc. | publisher = Cambridge University Press | ISBN = 0 521 30354 0 | ref = harv | url = http://digamo.free.fr/blaug62.pdf }} * {{cite book | last1 = Howard | first1 = M.C. | last2 = King | first2 = J.E. | date = 1975 | title = The political economy of Marx | location = London | publisher = Longman | url = https://archive.org/details/politicaleconomy0000howa | url-access = restricted | ref = harv }} ** see also: {{cite book | last1 = Howard | first1 = M.C. | last2 = King | first2 = J.E. | date = 1985 | title = The political economy of Marx ''(2nd ed.)'' | location = London | publisher = Longman | id = {{OCLC|10483279}} | ref = harv }} * {{cite journal | last = Kaldor | first = Nicholas | date = 1955 | title = Alternative Theories of Distribution | journal = The Review of Economic Studies | volume = 23 | issue = 2 | pages = 83–100 | doi = 10.2307/2296292 | url = https://www.jstor.org/stable/2296292 | ref = harv }}<br />{{small|(access via [[File:Wikipedia Library owl.svg|12px|alt=12 px Wikipedia Library Owl]] [[:w:en:Wikipedia:The Wikipedia Library|The Wikipedia Library (TWL)]])}} * {{cite book | last = O'Brien | first = D.P. | author-link = :w:en:Denis Patrick O'Brien | date = 1975 | title = The Classical Economists | location = Oxford | publisher = Clarendon Press | ISBN = 0 19 877015 4 | url = https://archive.org/details/classicaleconomi0000obri | url-access = restricted | ref = harv }} * {{cite book | last = Ricardo | first = David | author-link = David Ricardo | date = 1815 | title = An Essay on the Influence of a Low Price of Corn on the Profits of Stock | location = London | publisher = John Murray | ref = harv | url = https://en.wikisource.org/wiki/An_Essay_on_the_Influence_of_a_Low_Price_of_Corn_on_the_Profits_of_Stock }} * {{cite book | last = Ricardo | first = David | author-link = David Ricardo | date = 1817 | title = On the Principles of Political Economy; And Taxation | location = London | publisher = John Murray | ref = harv | url = }} (when referring we will use the page numbering of the Sraffa ed.) * {{cite book | last = Sraffa | first = Piero | author-link = :w:en:Piero Sraffa | date = 1951 | chapter = Note on "Essay on Profits" | title = The Works and correspondence of David Ricardo (vol. IV) | location = Cambridge | publisher = Cambridge University Press | url = https://archive.org/details/workscorresponde0004rica | ref = harv }} * {{cite journal | last = Stigler | first = George J. | author-link = :w:en:George Stigler | date = 1952 | title = The Ricardian Theory of Value and Distribution | journal = Journal of Political Economy | volume = 60 | issue = 3 | pp = 187–207 | ref = harv | url = https://www.jstor.org/stable/1826451 }}<br />{{small|(access via [[File:Wikipedia Library owl.svg|12px|alt=12 px Wikipedia Library Owl]] [[:w:en:Wikipedia:The Wikipedia Library|The Wikipedia Library (TWL)]])}} le6whftukvstobx8yesse0f5u6s6cl3 2 332696 2844454 2844373 2026-11-04T12:34:42Z Atcovi 276019 +! 2844454 wikitext text/x-wiki {{displaytitle|The Sayings of the Scholars & the Salaf}} '''Aaqib Azeez''' <big>بِسْمِ اللَّهِ الرَّحْمَٰنِ الرَّحِيمِ</big><br> ''In the Name of Allah, the Beneficent, the Merciful'' <big>إنَّ الْحَمْدَ لِلَّهِ نَحْمَدُهُ وَنَسْتَعِينُهُ مَنْ يَهْدِهِ اللَّهُ فَلاَ مُضِلَّ لَهُ وَمَنْ يُضْلِلِ اللَّهُ فَلاَ هَادِيَ لَهُ وَأَشْهَدُ أَنْ لاَ إِلَهَ إِلاَّ اللَّهُ وَحْدَهُ لاَ شَرِيكَ لَهُ وَأَشْهَدُ أَنَّ مُحَمَّدًا عَبْدُهُ وَرَسُولُهُ أَمَّا بَعْدُ</big><br> ''Praise be to Allah, we seek His help. Whomsoever Allah guides will never be led astray, and whomsoever Allah leaves astray, no one can guide. I bear witness that there is none worthy of worship but Allah (alone with no partners) and I bear witness that Muhammad is His slave and Messenger. To proceed:'' Allaah, the Exalted, says: * {''O you who have believed, fear Allaah as He should be feared and do not die except as Muslims (in submission to Him).''} [[https://quran.com/ali-imran/102 Quran 3:102]] * {''O mankind, fear your Lord, Who created you from one soul and created from it its mate and dispersed from both of them many men and women. And fear Allaah, through whom you ask one another, and the wombs. Indeed, Allaah is ever, over you, an Observer.''} [[https://quran.com/an-nisa/1 Quran 4:1]] * {''O you who have believed, fear Allaah and speak words of appropriate justice. He will (then) amend for you your deeds and forgive you your sins. And whoever obeys Allaah and His Messenger has certainly attained a great attainment.''} [[https://quran.com/al-ahzab/70-71 Quran 33:70-71]] This page is intended to compile quotes and beneficial knowledge of Islam through the understanding of the righteous scholars and the Salaf. == Abdulaziz ibn Abdullah Al Baz == * '''Swearing by anything other than Allah is not permissible.''' Rather the oath should be by Allah Alone, may He be glorified and exalted, because it is proven that the Messenger of Allah (blessings and peace of Allah be upon him) said: “Whoever swears an oath, let him not swear by anything except by Allah, or else let him remain silent.” '''And he said: “Whoever swears by anything other than Allah has associated something with Him (shirk).” Swearing by anything other than Allah is one of the haram actions that constitute kufr (disbelief), but it comes under the heading of minor shirk, unless he meant that this by which he swore was as great as Allah, or that it has control over the universe, or that it deserves to be called upon instead of Allah, in which case it becomes major kufr (al-kufr al-akbar) – Allah forbid.''' So if he said “By the life of So and So” or “By the life of the Messenger” or “By the life of Musa” or “By the life of ‘Isa” or “By my father’s grave”, or he swore by honesty or by the Ka`bah and so on, all of that is swearing by something other than Allah, and all of that is not permissible and is an evil action. What should be done is to not swear by anything except Allah, may He be glorified and exalted, or by one of His attributes, or by one of His names. The Quran is the Word of Allah, and the Quran is one of the attributes of Allah. So if a person says “By the Quran” or “By the life of the Qur’an”, there is nothing wrong with that, because the Quran is the Word of Allah, may He be glorified and exalted.” (Fatawa Nur ‘ala ad-Darb by Ibn Baz, p. 236-237) === Refuting Terrorism === *"From that which is known to everyone who has the slightest bit of common sense is that '''hijacking airplanes and kidnapping children and the like are extremely great crimes''', the world over. Their evil effects are far and wide, as is the great harm and inconvenience caused to the innocent; the total effect of which none can comprehend except Allaah. Likewise, from that which is known is that these crimes are not specific to any particular country over and above another country, nor any specific group over and above another group, rather it encompasses the whole world. There is no doubt about the effect of these crimes; '''so it is obligatory upon the governments and those responsible from amongst the scholars and other than them to afford these issues great concern, and to exert themselves as much as possible in ending this evil.'''" **SOURCE: [https://abdurrahman.org/wp-content/uploads/2014/10/hijackings-kidnappings-and-bombings-shaykh-ibn-baaz-authentic-translations-com.pdf 1] == [https://www.emaanlibrary.com/wp-content/uploads/2022/01/The-Autobiography-of-the-noble-Shaykh-the-Muhaddith-Abu-%E2%80%98Abd-al-Rahman-Muqbil-b.-Hadi-al-Wadii.pdf Muqbil al-Wadi'i] == * "'''To me, reading the ''Sahihayn'' (of al-Bukhari and Muslim) is the sweetest pleasure in the world'''. When I open ''Bukhari'' and read: "'Abd Allah b. Yusuf narrated to us: Malik narrated to us..." or I read ''Imam'' Muslim say: "Yahya b. Yahya narrated to us...." I forget all the concerns of the ''dunya'' (world) and its problems" ** ''Al-Sahih al-Musnad'', SOURCE: [https://www.emaanlibrary.com/wp-content/uploads/2022/01/The-Autobiography-of-the-noble-Shaykh-the-Muhaddith-Abu-%E2%80%98Abd-al-Rahman-Muqbil-b.-Hadi-al-Wadii.pdf 1] === Refuting Osama bin Laden === * The scholar of Yemen, Shaykh Muqbil Ibn Hādī al-Wādi’ī, said in an interview with the Kuwaiti newspaper, “al-Ra’yu al-'Ām” (no. 11503, 12/19/1998), “'''Here, before Allāh, I completely reject and declare myself free of Bin Laden, for he is an evil threat and tribulation to the (Islamic) nation, and his actions are heinous'''.” And in the same interview, the interviewer said, “It seems Muslims are beginning to face difficulties in western countries because of the bombings that have taken place in almost every place in the world.” Shaykh Muqbil replied: I know. Some brothers from Britain even called me, complaining about the hardships they now face there, asking if it is permissible to publicize their rejection of and disassociation from Osama Bin Laden. '''We told them that we have already rejected and disassociated ourselves from him and his crimes a long time ago. And this current state is evidence that Muslims in the West suffer difficulties because of movements like those of ‘[https://simplysalafiyyah.com/refutations-r.php?id=51&sayyid-qutub%27s-deviations-and-its-insignias-on-the-muslim-brotherhood---part-one The Muslim Brotherhood]’ and others. And from Allāh we ask help.''' ** SOURCE: [https://abdurrahman.org/wp-content/uploads/2014/10/the-position-of-the-scholars-regarding-osama-bin-laden-various-scholars-authentic-translations-com.pdf 1] ==Salih al-Fawzan== *Reflect upon why you were created, and reflect upon where you are heading. For the intelligent person contemplates his present condition and what lies ahead, and prepares for what is before him. This worldly life is a place of passage, and the Hereafter is the place of permanent residence. The worldly life is the abode of action, and the Hereafter is the abode of recompense. Know your reality - may Allah have mercy upon you. '''Let each person reflect within himself: what has he prepared for meeting Allah, the Most High?! So whoever hopes for the meeting of his Lord - then let him do righteous deeds, and not associate anyone in the worship of his Lord.''' **SOURCE: [https://www.youtube.com/shorts/5lfgD0oUKSc 1] == Muhammad ibn Sāliḥ al-Uthaymin == == Muhammad ibn Abd al-Wahhab == * Abu Hurairah asked the Messenger (ﷺ): "Who will be the most fortunate and be granted your intercession?" The Messenger said: "Whoever professes with sincerity in his heart and declares: 'There is no deity worthy of worship but Allah.'" This initial intercession is only granted by the grace of Allah to the believers. Undoubtedly, those who associate Allah with others will have no share of it. The truth is that intercession is a manifestation of Allah's glory and grace upon the sincere believers. Whoever Allah permits to intercede by good prayer, Allah forgives them their sins. The sole purpose of which is to honor their faith and raise them to a praiseworthy position. However under various circumstances the Qur'an denies intercession that includes any form of shirk. The Messenger ﷺ declared that it is restricted to the people of Tawheed and the believers in the oneness of Allah. ** SOURCE: ''Book of Monotheism'', page 27. == Ibn Qayyim al-Jawziyya == * The deputies of Iblis on earth are those who discourage people from seeking knowledge and understanding the religion. These individuals are more harmful to people than the devils among the jinn, for they stand between hearts and the guidance of Allah and His path. ** Miftah Dar al-Sa'adah (1/160) [Al-Baseerah Islamic Center in Plainfield, New Jersey] * I heard Shaykh al-Islam ibn Taymiyyah (may Allah have mercy upon him) say: "'''Part of the perfection of ettiquette in salah is for the worshipper to stand before his Lord with his head bowed, lowering his gaze toward the ground and not raising his eyes upward'''." ** Madarij al-Salikin (2/364) [Al-Baseerah Islamic Center in Plainfield, New Jersey] *A scholar said, "To have patience means that one's common sense and religious motives are stronger than one's whims and desires." It is natural for people to have an inclination towards their desires, but common sense and the religious motive should limit that inclination. The two forces at war: sometimes reason and religion win, and sometimes whims and desires prevail. '''The battlefield is the heart of man'''. **''Patience and Gratitude | An abridged translation of 'Uddat as-sabirin wa dhakhirat ash-shakirin by Ibn Qayyim al-Jawziyyah (translated by Nasiruddin al-Khattab)'', page 14-15. == Ibn Muflih al-Hanbali == * How astonishing it is that so many people wail, mourn and complain bitterly the destruction of their homes, the scarcity of provisions, the harshness of the times, and the rising of prices; '''yet they have never once expressed intense grief over the strangeness of the religion, the disappearance of the Sunan, or the spread of innovations'''. '''Nor have they wept over their own shortcomings. This is due to the weakness of their ''iman'' and the greatness of worldly life in their eyes.''' ** Al-Adab al-Shar'iyyah (3/240) == Muhammad Taqi al-Din al-Hilali == * The intellect alone cannot establish Tawhid without being illuminated by the light of revelation. Therefore, Allah did not make the intellect alone a proof against the people. Allah, the Most High, said what means: "And We would never punish until We had sent a Messenger" [17:15]. ** Hashiyah 'ala Kitab al-Tawhid (pg. 350), [Al-Baseerah Islamic Center in Plainfield, New Jersey] == Abdul-Rahman al-Sa'di == * This religion has not left any good except that it called to it, nor any benefit except that it encouraged it. And it has not left any path that leads to the rectification of religious and worldly affairs except that it urged toward it. Likewise, it has not left any corruption, evil, or harm except that it warned against it. ** Majmu Mu'allafaith (6/166), [Al-Baseerah Islamic Center in Plainfield, New Jersey] == Sufyan al-Thawri == * I have not seen anything that takes away from religion, diminishes manliness, wastes pleasure, or preoccupies the heart more than enmity. ** Mukhtasar Tarikh Dimashq (25/104), [Al-Baseerah Islamic Center in Plainfield, New Jersey] [[Category:Islamic Studies]] [[Category:Atcovi's Work]] gdg020vk3f6fwm13oabdt9t495mqs3z 2844534 2844454 2026-11-04T23:36:31Z Atcovi 276019 +ibn Taymiyya 2844534 wikitext text/x-wiki {{displaytitle|The Sayings of the Scholars & the Salaf}} '''Aaqib Azeez''' <big>بِسْمِ اللَّهِ الرَّحْمَٰنِ الرَّحِيمِ</big><br> ''In the Name of Allah, the Beneficent, the Merciful'' <big>إنَّ الْحَمْدَ لِلَّهِ نَحْمَدُهُ وَنَسْتَعِينُهُ مَنْ يَهْدِهِ اللَّهُ فَلاَ مُضِلَّ لَهُ وَمَنْ يُضْلِلِ اللَّهُ فَلاَ هَادِيَ لَهُ وَأَشْهَدُ أَنْ لاَ إِلَهَ إِلاَّ اللَّهُ وَحْدَهُ لاَ شَرِيكَ لَهُ وَأَشْهَدُ أَنَّ مُحَمَّدًا عَبْدُهُ وَرَسُولُهُ أَمَّا بَعْدُ</big><br> ''Praise be to Allah, we seek His help. Whomsoever Allah guides will never be led astray, and whomsoever Allah leaves astray, no one can guide. I bear witness that there is none worthy of worship but Allah (alone with no partners) and I bear witness that Muhammad is His slave and Messenger. To proceed:'' Allaah, the Exalted, says: * {''O you who have believed, fear Allaah as He should be feared and do not die except as Muslims (in submission to Him).''} [[https://quran.com/ali-imran/102 Quran 3:102]] * {''O mankind, fear your Lord, Who created you from one soul and created from it its mate and dispersed from both of them many men and women. And fear Allaah, through whom you ask one another, and the wombs. Indeed, Allaah is ever, over you, an Observer.''} [[https://quran.com/an-nisa/1 Quran 4:1]] * {''O you who have believed, fear Allaah and speak words of appropriate justice. He will (then) amend for you your deeds and forgive you your sins. And whoever obeys Allaah and His Messenger has certainly attained a great attainment.''} [[https://quran.com/al-ahzab/70-71 Quran 33:70-71]] This page is intended to compile quotes and beneficial knowledge of Islam through the understanding of the righteous scholars and the Salaf. == Abdulaziz ibn Abdullah Al Baz == * '''Swearing by anything other than Allah is not permissible.''' Rather the oath should be by Allah Alone, may He be glorified and exalted, because it is proven that the Messenger of Allah (blessings and peace of Allah be upon him) said: “Whoever swears an oath, let him not swear by anything except by Allah, or else let him remain silent.” '''And he said: “Whoever swears by anything other than Allah has associated something with Him (shirk).” Swearing by anything other than Allah is one of the haram actions that constitute kufr (disbelief), but it comes under the heading of minor shirk, unless he meant that this by which he swore was as great as Allah, or that it has control over the universe, or that it deserves to be called upon instead of Allah, in which case it becomes major kufr (al-kufr al-akbar) – Allah forbid.''' So if he said “By the life of So and So” or “By the life of the Messenger” or “By the life of Musa” or “By the life of ‘Isa” or “By my father’s grave”, or he swore by honesty or by the Ka`bah and so on, all of that is swearing by something other than Allah, and all of that is not permissible and is an evil action. What should be done is to not swear by anything except Allah, may He be glorified and exalted, or by one of His attributes, or by one of His names. The Quran is the Word of Allah, and the Quran is one of the attributes of Allah. So if a person says “By the Quran” or “By the life of the Qur’an”, there is nothing wrong with that, because the Quran is the Word of Allah, may He be glorified and exalted.” (Fatawa Nur ‘ala ad-Darb by Ibn Baz, p. 236-237) === Refuting Terrorism === *"From that which is known to everyone who has the slightest bit of common sense is that '''hijacking airplanes and kidnapping children and the like are extremely great crimes''', the world over. Their evil effects are far and wide, as is the great harm and inconvenience caused to the innocent; the total effect of which none can comprehend except Allaah. Likewise, from that which is known is that these crimes are not specific to any particular country over and above another country, nor any specific group over and above another group, rather it encompasses the whole world. There is no doubt about the effect of these crimes; '''so it is obligatory upon the governments and those responsible from amongst the scholars and other than them to afford these issues great concern, and to exert themselves as much as possible in ending this evil.'''" **SOURCE: [https://abdurrahman.org/wp-content/uploads/2014/10/hijackings-kidnappings-and-bombings-shaykh-ibn-baaz-authentic-translations-com.pdf 1] == [https://www.emaanlibrary.com/wp-content/uploads/2022/01/The-Autobiography-of-the-noble-Shaykh-the-Muhaddith-Abu-%E2%80%98Abd-al-Rahman-Muqbil-b.-Hadi-al-Wadii.pdf Muqbil al-Wadi'i] == * "'''To me, reading the ''Sahihayn'' (of al-Bukhari and Muslim) is the sweetest pleasure in the world'''. When I open ''Bukhari'' and read: "'Abd Allah b. Yusuf narrated to us: Malik narrated to us..." or I read ''Imam'' Muslim say: "Yahya b. Yahya narrated to us...." I forget all the concerns of the ''dunya'' (world) and its problems" ** ''Al-Sahih al-Musnad'', SOURCE: [https://www.emaanlibrary.com/wp-content/uploads/2022/01/The-Autobiography-of-the-noble-Shaykh-the-Muhaddith-Abu-%E2%80%98Abd-al-Rahman-Muqbil-b.-Hadi-al-Wadii.pdf 1] === Refuting Osama bin Laden === * The scholar of Yemen, Shaykh Muqbil Ibn Hādī al-Wādi’ī, said in an interview with the Kuwaiti newspaper, “al-Ra’yu al-'Ām” (no. 11503, 12/19/1998), “'''Here, before Allāh, I completely reject and declare myself free of Bin Laden, for he is an evil threat and tribulation to the (Islamic) nation, and his actions are heinous'''.” And in the same interview, the interviewer said, “It seems Muslims are beginning to face difficulties in western countries because of the bombings that have taken place in almost every place in the world.” Shaykh Muqbil replied: I know. Some brothers from Britain even called me, complaining about the hardships they now face there, asking if it is permissible to publicize their rejection of and disassociation from Osama Bin Laden. '''We told them that we have already rejected and disassociated ourselves from him and his crimes a long time ago. And this current state is evidence that Muslims in the West suffer difficulties because of movements like those of ‘[https://simplysalafiyyah.com/refutations-r.php?id=51&sayyid-qutub%27s-deviations-and-its-insignias-on-the-muslim-brotherhood---part-one The Muslim Brotherhood]’ and others. And from Allāh we ask help.''' ** SOURCE: [https://abdurrahman.org/wp-content/uploads/2014/10/the-position-of-the-scholars-regarding-osama-bin-laden-various-scholars-authentic-translations-com.pdf 1] ==Salih al-Fawzan== *Reflect upon why you were created, and reflect upon where you are heading. For the intelligent person contemplates his present condition and what lies ahead, and prepares for what is before him. This worldly life is a place of passage, and the Hereafter is the place of permanent residence. The worldly life is the abode of action, and the Hereafter is the abode of recompense. Know your reality - may Allah have mercy upon you. '''Let each person reflect within himself: what has he prepared for meeting Allah, the Most High?! So whoever hopes for the meeting of his Lord - then let him do righteous deeds, and not associate anyone in the worship of his Lord.''' **SOURCE: [https://www.youtube.com/shorts/5lfgD0oUKSc 1] == Muhammad ibn Sāliḥ al-Uthaymin == == Muhammad ibn Abd al-Wahhab == * Abu Hurairah asked the Messenger (ﷺ): "Who will be the most fortunate and be granted your intercession?" The Messenger said: "Whoever professes with sincerity in his heart and declares: 'There is no deity worthy of worship but Allah.'" This initial intercession is only granted by the grace of Allah to the believers. Undoubtedly, those who associate Allah with others will have no share of it. The truth is that intercession is a manifestation of Allah's glory and grace upon the sincere believers. Whoever Allah permits to intercede by good prayer, Allah forgives them their sins. The sole purpose of which is to honor their faith and raise them to a praiseworthy position. However under various circumstances the Qur'an denies intercession that includes any form of shirk. The Messenger ﷺ declared that it is restricted to the people of Tawheed and the believers in the oneness of Allah. ** SOURCE: ''Book of Monotheism'', page 27. == Ibn Qayyim al-Jawziyya == * The deputies of Iblis on earth are those who discourage people from seeking knowledge and understanding the religion. These individuals are more harmful to people than the devils among the jinn, for they stand between hearts and the guidance of Allah and His path. ** Miftah Dar al-Sa'adah (1/160) [Al-Baseerah Islamic Center in Plainfield, New Jersey] * I heard Shaykh al-Islam ibn Taymiyyah (may Allah have mercy upon him) say: "'''Part of the perfection of ettiquette in salah is for the worshipper to stand before his Lord with his head bowed, lowering his gaze toward the ground and not raising his eyes upward'''." ** Madarij al-Salikin (2/364) [Al-Baseerah Islamic Center in Plainfield, New Jersey] *A scholar said, "To have patience means that one's common sense and religious motives are stronger than one's whims and desires." It is natural for people to have an inclination towards their desires, but common sense and the religious motive should limit that inclination. The two forces at war: sometimes reason and religion win, and sometimes whims and desires prevail. '''The battlefield is the heart of man'''. **''Patience and Gratitude | An abridged translation of 'Uddat as-sabirin wa dhakhirat ash-shakirin by Ibn Qayyim al-Jawziyyah (translated by Nasiruddin al-Khattab)'', page 14-15. == Ibn Muflih al-Hanbali == * How astonishing it is that so many people wail, mourn and complain bitterly the destruction of their homes, the scarcity of provisions, the harshness of the times, and the rising of prices; '''yet they have never once expressed intense grief over the strangeness of the religion, the disappearance of the Sunan, or the spread of innovations'''. '''Nor have they wept over their own shortcomings. This is due to the weakness of their ''iman'' and the greatness of worldly life in their eyes.''' ** Al-Adab al-Shar'iyyah (3/240) == Muhammad Taqi al-Din al-Hilali == * The intellect alone cannot establish Tawhid without being illuminated by the light of revelation. Therefore, Allah did not make the intellect alone a proof against the people. Allah, the Most High, said what means: "And We would never punish until We had sent a Messenger" [17:15]. ** Hashiyah 'ala Kitab al-Tawhid (pg. 350), [Al-Baseerah Islamic Center in Plainfield, New Jersey] == Abdul-Rahman al-Sa'di == * This religion has not left any good except that it called to it, nor any benefit except that it encouraged it. And it has not left any path that leads to the rectification of religious and worldly affairs except that it urged toward it. Likewise, it has not left any corruption, evil, or harm except that it warned against it. ** Majmu Mu'allafaith (6/166), [Al-Baseerah Islamic Center in Plainfield, New Jersey] == Sufyan al-Thawri == * I have not seen anything that takes away from religion, diminishes manliness, wastes pleasure, or preoccupies the heart more than enmity. ** Mukhtasar Tarikh Dimashq (25/104), [Al-Baseerah Islamic Center in Plainfield, New Jersey] == Ibn Taymiyya == * Whoever says or does that which is kufr, then he becomes a kafir by that itself, even if he did not intend to become a disbeliever, for no one intends kufr except whom Allah wills. ** As-Sarim al-Maslul, 2/339 [[Category:Islamic Studies]] [[Category:Atcovi's Work]] opnyx9j2dc5j8tkyrlaqpsjkhvyqtiz 2844535 2844534 2026-11-04T23:36:57Z Atcovi 276019 /* Ibn Taymiyya */ refuting the Murji'ah 2844535 wikitext text/x-wiki {{displaytitle|The Sayings of the Scholars & the Salaf}} '''Aaqib Azeez''' <big>بِسْمِ اللَّهِ الرَّحْمَٰنِ الرَّحِيمِ</big><br> ''In the Name of Allah, the Beneficent, the Merciful'' <big>إنَّ الْحَمْدَ لِلَّهِ نَحْمَدُهُ وَنَسْتَعِينُهُ مَنْ يَهْدِهِ اللَّهُ فَلاَ مُضِلَّ لَهُ وَمَنْ يُضْلِلِ اللَّهُ فَلاَ هَادِيَ لَهُ وَأَشْهَدُ أَنْ لاَ إِلَهَ إِلاَّ اللَّهُ وَحْدَهُ لاَ شَرِيكَ لَهُ وَأَشْهَدُ أَنَّ مُحَمَّدًا عَبْدُهُ وَرَسُولُهُ أَمَّا بَعْدُ</big><br> ''Praise be to Allah, we seek His help. Whomsoever Allah guides will never be led astray, and whomsoever Allah leaves astray, no one can guide. I bear witness that there is none worthy of worship but Allah (alone with no partners) and I bear witness that Muhammad is His slave and Messenger. To proceed:'' Allaah, the Exalted, says: * {''O you who have believed, fear Allaah as He should be feared and do not die except as Muslims (in submission to Him).''} [[https://quran.com/ali-imran/102 Quran 3:102]] * {''O mankind, fear your Lord, Who created you from one soul and created from it its mate and dispersed from both of them many men and women. And fear Allaah, through whom you ask one another, and the wombs. Indeed, Allaah is ever, over you, an Observer.''} [[https://quran.com/an-nisa/1 Quran 4:1]] * {''O you who have believed, fear Allaah and speak words of appropriate justice. He will (then) amend for you your deeds and forgive you your sins. And whoever obeys Allaah and His Messenger has certainly attained a great attainment.''} [[https://quran.com/al-ahzab/70-71 Quran 33:70-71]] This page is intended to compile quotes and beneficial knowledge of Islam through the understanding of the righteous scholars and the Salaf. == Abdulaziz ibn Abdullah Al Baz == * '''Swearing by anything other than Allah is not permissible.''' Rather the oath should be by Allah Alone, may He be glorified and exalted, because it is proven that the Messenger of Allah (blessings and peace of Allah be upon him) said: “Whoever swears an oath, let him not swear by anything except by Allah, or else let him remain silent.” '''And he said: “Whoever swears by anything other than Allah has associated something with Him (shirk).” Swearing by anything other than Allah is one of the haram actions that constitute kufr (disbelief), but it comes under the heading of minor shirk, unless he meant that this by which he swore was as great as Allah, or that it has control over the universe, or that it deserves to be called upon instead of Allah, in which case it becomes major kufr (al-kufr al-akbar) – Allah forbid.''' So if he said “By the life of So and So” or “By the life of the Messenger” or “By the life of Musa” or “By the life of ‘Isa” or “By my father’s grave”, or he swore by honesty or by the Ka`bah and so on, all of that is swearing by something other than Allah, and all of that is not permissible and is an evil action. What should be done is to not swear by anything except Allah, may He be glorified and exalted, or by one of His attributes, or by one of His names. The Quran is the Word of Allah, and the Quran is one of the attributes of Allah. So if a person says “By the Quran” or “By the life of the Qur’an”, there is nothing wrong with that, because the Quran is the Word of Allah, may He be glorified and exalted.” (Fatawa Nur ‘ala ad-Darb by Ibn Baz, p. 236-237) === Refuting Terrorism === *"From that which is known to everyone who has the slightest bit of common sense is that '''hijacking airplanes and kidnapping children and the like are extremely great crimes''', the world over. Their evil effects are far and wide, as is the great harm and inconvenience caused to the innocent; the total effect of which none can comprehend except Allaah. Likewise, from that which is known is that these crimes are not specific to any particular country over and above another country, nor any specific group over and above another group, rather it encompasses the whole world. There is no doubt about the effect of these crimes; '''so it is obligatory upon the governments and those responsible from amongst the scholars and other than them to afford these issues great concern, and to exert themselves as much as possible in ending this evil.'''" **SOURCE: [https://abdurrahman.org/wp-content/uploads/2014/10/hijackings-kidnappings-and-bombings-shaykh-ibn-baaz-authentic-translations-com.pdf 1] == [https://www.emaanlibrary.com/wp-content/uploads/2022/01/The-Autobiography-of-the-noble-Shaykh-the-Muhaddith-Abu-%E2%80%98Abd-al-Rahman-Muqbil-b.-Hadi-al-Wadii.pdf Muqbil al-Wadi'i] == * "'''To me, reading the ''Sahihayn'' (of al-Bukhari and Muslim) is the sweetest pleasure in the world'''. When I open ''Bukhari'' and read: "'Abd Allah b. Yusuf narrated to us: Malik narrated to us..." or I read ''Imam'' Muslim say: "Yahya b. Yahya narrated to us...." I forget all the concerns of the ''dunya'' (world) and its problems" ** ''Al-Sahih al-Musnad'', SOURCE: [https://www.emaanlibrary.com/wp-content/uploads/2022/01/The-Autobiography-of-the-noble-Shaykh-the-Muhaddith-Abu-%E2%80%98Abd-al-Rahman-Muqbil-b.-Hadi-al-Wadii.pdf 1] === Refuting Osama bin Laden === * The scholar of Yemen, Shaykh Muqbil Ibn Hādī al-Wādi’ī, said in an interview with the Kuwaiti newspaper, “al-Ra’yu al-'Ām” (no. 11503, 12/19/1998), “'''Here, before Allāh, I completely reject and declare myself free of Bin Laden, for he is an evil threat and tribulation to the (Islamic) nation, and his actions are heinous'''.” And in the same interview, the interviewer said, “It seems Muslims are beginning to face difficulties in western countries because of the bombings that have taken place in almost every place in the world.” Shaykh Muqbil replied: I know. Some brothers from Britain even called me, complaining about the hardships they now face there, asking if it is permissible to publicize their rejection of and disassociation from Osama Bin Laden. '''We told them that we have already rejected and disassociated ourselves from him and his crimes a long time ago. And this current state is evidence that Muslims in the West suffer difficulties because of movements like those of ‘[https://simplysalafiyyah.com/refutations-r.php?id=51&sayyid-qutub%27s-deviations-and-its-insignias-on-the-muslim-brotherhood---part-one The Muslim Brotherhood]’ and others. And from Allāh we ask help.''' ** SOURCE: [https://abdurrahman.org/wp-content/uploads/2014/10/the-position-of-the-scholars-regarding-osama-bin-laden-various-scholars-authentic-translations-com.pdf 1] ==Salih al-Fawzan== *Reflect upon why you were created, and reflect upon where you are heading. For the intelligent person contemplates his present condition and what lies ahead, and prepares for what is before him. This worldly life is a place of passage, and the Hereafter is the place of permanent residence. The worldly life is the abode of action, and the Hereafter is the abode of recompense. Know your reality - may Allah have mercy upon you. '''Let each person reflect within himself: what has he prepared for meeting Allah, the Most High?! So whoever hopes for the meeting of his Lord - then let him do righteous deeds, and not associate anyone in the worship of his Lord.''' **SOURCE: [https://www.youtube.com/shorts/5lfgD0oUKSc 1] == Muhammad ibn Sāliḥ al-Uthaymin == == Muhammad ibn Abd al-Wahhab == * Abu Hurairah asked the Messenger (ﷺ): "Who will be the most fortunate and be granted your intercession?" The Messenger said: "Whoever professes with sincerity in his heart and declares: 'There is no deity worthy of worship but Allah.'" This initial intercession is only granted by the grace of Allah to the believers. Undoubtedly, those who associate Allah with others will have no share of it. The truth is that intercession is a manifestation of Allah's glory and grace upon the sincere believers. Whoever Allah permits to intercede by good prayer, Allah forgives them their sins. The sole purpose of which is to honor their faith and raise them to a praiseworthy position. However under various circumstances the Qur'an denies intercession that includes any form of shirk. The Messenger ﷺ declared that it is restricted to the people of Tawheed and the believers in the oneness of Allah. ** SOURCE: ''Book of Monotheism'', page 27. == Ibn Qayyim al-Jawziyya == * The deputies of Iblis on earth are those who discourage people from seeking knowledge and understanding the religion. These individuals are more harmful to people than the devils among the jinn, for they stand between hearts and the guidance of Allah and His path. ** Miftah Dar al-Sa'adah (1/160) [Al-Baseerah Islamic Center in Plainfield, New Jersey] * I heard Shaykh al-Islam ibn Taymiyyah (may Allah have mercy upon him) say: "'''Part of the perfection of ettiquette in salah is for the worshipper to stand before his Lord with his head bowed, lowering his gaze toward the ground and not raising his eyes upward'''." ** Madarij al-Salikin (2/364) [Al-Baseerah Islamic Center in Plainfield, New Jersey] *A scholar said, "To have patience means that one's common sense and religious motives are stronger than one's whims and desires." It is natural for people to have an inclination towards their desires, but common sense and the religious motive should limit that inclination. The two forces at war: sometimes reason and religion win, and sometimes whims and desires prevail. '''The battlefield is the heart of man'''. **''Patience and Gratitude | An abridged translation of 'Uddat as-sabirin wa dhakhirat ash-shakirin by Ibn Qayyim al-Jawziyyah (translated by Nasiruddin al-Khattab)'', page 14-15. == Ibn Muflih al-Hanbali == * How astonishing it is that so many people wail, mourn and complain bitterly the destruction of their homes, the scarcity of provisions, the harshness of the times, and the rising of prices; '''yet they have never once expressed intense grief over the strangeness of the religion, the disappearance of the Sunan, or the spread of innovations'''. '''Nor have they wept over their own shortcomings. This is due to the weakness of their ''iman'' and the greatness of worldly life in their eyes.''' ** Al-Adab al-Shar'iyyah (3/240) == Muhammad Taqi al-Din al-Hilali == * The intellect alone cannot establish Tawhid without being illuminated by the light of revelation. Therefore, Allah did not make the intellect alone a proof against the people. Allah, the Most High, said what means: "And We would never punish until We had sent a Messenger" [17:15]. ** Hashiyah 'ala Kitab al-Tawhid (pg. 350), [Al-Baseerah Islamic Center in Plainfield, New Jersey] == Abdul-Rahman al-Sa'di == * This religion has not left any good except that it called to it, nor any benefit except that it encouraged it. And it has not left any path that leads to the rectification of religious and worldly affairs except that it urged toward it. Likewise, it has not left any corruption, evil, or harm except that it warned against it. ** Majmu Mu'allafaith (6/166), [Al-Baseerah Islamic Center in Plainfield, New Jersey] == Sufyan al-Thawri == * I have not seen anything that takes away from religion, diminishes manliness, wastes pleasure, or preoccupies the heart more than enmity. ** Mukhtasar Tarikh Dimashq (25/104), [Al-Baseerah Islamic Center in Plainfield, New Jersey] == Ibn Taymiyya == === Refuting the Murji'ah === * Whoever says or does that which is kufr, then he becomes a kafir by that itself, even if he did not intend to become a disbeliever, for no one intends kufr except whom Allah wills. ** As-Sarim al-Maslul, 2/339 [[Category:Islamic Studies]] [[Category:Atcovi's Work]] 3512glfoz6w1qe9hyly5596hc0slf2k 2 332700 2844504 2843396 2026-11-04T18:32:35Z ~2026-56788-08 3113246 /* als */ get source from previous userspace page I created 2844504 wikitext text/x-wiki == als == CBC the fifth estate Presents: ALS Country One woman’s fight to understand why her husband died of ALS and the public health investigation into a possible cluster of Lou Gehrig's disease. check out the following: * [[RNA interference/Medical#section Treating Lou Gehrig's disease]] * [[Introduction to Brain-Computer Interfaces/Module 6: Clinical Applications]] * [[World War I -- Life Histories/Section 018/H. Gaston Carney]] [[w:ALS|enwp article about Amyotrophic lateral sclerosis (ALS)]] == womens health == p5oxkg8l58wyugo6fxbktem2hcr2n6h 0 332799 2844456 2844452 2026-11-04T12:35:59Z MathXplore 2888076 Added {{[[Template:BookCat|BookCat]]}} using [[User:1234qwer1234qwer4/BookCat.js|BookCat.js]] 2844456 wikitext text/x-wiki == Introduction == The Nemesis System is a video game mechanic developed by Monolith Productions and was first introduced in the 2014 game ''Middle-earth: Shadow of Mordor''. It allows non-player characters, otherwise known as NPCs, to remember previous encounters the player had with them and develop unique characteristics. Enemies may be promoted to higher positions in their army after defeating the player, and they can develop rivalries or become allies with the player. <ref name=":1">{{Cite web|url=https://www.engadget.com/2014-01-23-research-is-a-vehicle-for-revenge-in-middle-earth-shadow-of-mor.html|title=Research is a vehicle for revenge in Middle-earth: Shadow of Mordor|last=Hinkle|first=David|date=2014-01-23|website=Engadget|language=en-US|access-date=2026-11-04}}</ref> The system uses procedural generation to create new enemies, so each enemy will be different from one another. It programs the enemies to remember the prior encounters with the player through dialogue, increase in strength, or visual differences. The system was later expanded upon in the sequel, ''Middle-earth: Shadow of War'', released in 2017. Warner Bros. Entertainment Inc., the company that published the game, later obtained a patent covering aspects of the system in 2021. == History and Development == The Nemesis System was developed by Monolith Productions for ''Middle-earth: Shadow of Mordor''. The developers wanted to create a game where enemies that weren't a main part of the story, like main characters or bosses, could remember their interactions with the player, rather than having them simply disappear after being defeated. This gave more depth to the open-world gameplay as players progressed through the story. The idea for the Nemesis System was mainly influenced by sports and role-playing games. Sports games inspired the rivalries that develop over time, where winning or losing to an opponent does not end their appearance in the game. On the other hand, role-playing games inspired the developers to give enemies distinctive personalities, abilities, and characteristics that would make them memorable to the player.<ref name=":2">{{Cite web|url=https://www.gamespot.com/articles/shadow-of-mordor-s-nemesis-system-was-inspired-by/1100-6423740/|title=Shadow of Mordor's Nemesis System Was Inspired by Batman and Sports Games|website=GameSpot|language=en-US|access-date=2026-11-04}}</ref> The framework for the Nemesis System was laid out approximately three months into the game’s three-year development. Initially, the developers experimented with more complicated features, such as relationships between enemies, including rivalries and conflicts within their ranks. However, these ideas were eventually simplified to focus more on individual enemies and their relationships with the player.<ref name=":2" /> Monolith was previously working on a Batman game, codenamed ''Apollo''. The project included an earlier version of the Nemesis System, which would have allowed enemies to remember their encounters with Batman. The game's production was cancelled, however, so Monolith eventually developed Shadow of Mordor, where the system became one of the game’s main features.<ref>{{Cite web|url=https://www.gamespot.com/articles/more-footage-for-batman-game-that-became-shadow-of-mordor-surfaces-online/1100-6522392/|title=More(dor) Batman Footage For Game That Became Shadow Of Mordor Surfaces Online|website=GameSpot|language=en-US|access-date=2026-11-04}}</ref> == How the Nemesis System Works == The Nemesis System keeps track of interactions between the player and individual enemies. Regular enemies can become more important later on by defeating the player or surviving an encounter with them.<ref name=":3">{{Cite web|url=https://gameinformer.com/games/middle-earth_shadow_of_mordor/b/playstation4/archive/2014/09/25/middle-earth-shadow-of-mordor-game-informer-review|title=Middle-earth: Shadow of Mordor Review: Emergent Action Evolves|last=Miller|first=Matt|date=2014-09-30|website=Game Informer|language=en|access-date=2026-11-04}}</ref> In ''Middle-earth: Shadow of Mordor'', these enemies, called orcs or ''Uruks'', can be promoted to captains within Sauron's army. If an Uruk kills the player, it can gain additional strength and move up in the army hierarchy. This makes future encounters with that same enemy more challenging. Enemies who survive previous battles may also remember the player and develop physical scars or new characteristics based on what happened. For example, an Uruk who was burned during an encounter may return with burn scars and a desire for revenge.<ref name=":4">{{Cite web|url=https://www.gamesradar.com/shadow-mordor-nemesis-system-amazing-how-works/|title=Shadow of Mordor's Nemesis system is amazing--here's how it works|date=2014-08-29|website=GamesRadar+|language=en|access-date=2026-11-04}}</ref> Another feature of the Nemesis system, specifically in Shadow of Mordor, is the ability to ally with enemies and turn them against Sauron's army. Players can recruit Uruks and create conflicts between enemy groups.<ref name=":1" /> == Procedural Generation and Behavior == Procedural generation plays an important role in the Nemesis System by allowing the game to create enemies from a mix and match of different stats instead of relying entirely on predetermined ones that look the same. Each enemy has a unique name, appearance, personality, strengths, and fears.<ref name=":1" /> These characteristics are combined to create a large variety of possible enemies. Most of the names are combinations of different first names with a title attached to it, for example, "Humgrat the Unkillable".<ref name=":4" /> Programmed changes in behavior of enemies help determine how these they interact with the player and the game's open-world. For example, enemies can recognize the player from a previous encounter, and react differently depending on their past experiences. Some enemies may attempt to avoid the player because of previous defeats, while others may seek revenge. These interactions allow the player's experiences to change over time and create new gameplay situations.<ref name=":3" /> == Warner Bros. Patent == Warner Bros. Entertainment Inc. filed a patent application for the Nemesis System on March 25, 2016. The patent, titled ''Nemesis Characters, Nemesis Forts, Social Vendettas and Followers in Computer Games'', was officially granted on February 23, 2021, under U.S. Patent No. [https://patents.google.com/patent/US10926179B2/en US10926179B2]. The patent provides a description that covers programming NPCs with predetermined behavior, and NPCs that fit the description of a "Nemesis", which is an NPC that remembers previous interactions and evolves throughout the game, per the Nemesis System. It covers aspects such as enemy hierarchies, character relationships, and followers that can influence events within a game. However, the patent does not give Warner Bros. ownership over every possible mechanic involving enemies that remember players. The approval of the patent attracted attention because the Nemesis System was considered a unique gameplay mechanic that other game developers might want to experiment with. Warner Bros. faced multiple attempts and lots of editing before they could obtain approval for the patent.<ref>{{Cite web|url=https://www.ign.com/articles/wb-games-nemesis-system-patent-was-approved-this-week-after-multiple-attempts|title=WB Games' Nemesis System Patent Was Approved This Week After Multiple Attempts|last=Ryan|first=Jon|date=2021-02-05|website=IGN|language=en|access-date=2026-11-04}}</ref> == Effects on the Video Game Industry == The Warner Bros.' patent on the Nemesis System has raised a discussion on how intellectual property rights can affect innovation in the video game industry specifically. The Nemesis System has demonstrated how enemy interactions and procedural generation in video games creates an immersive gameplay experience. Developers and players believe that similar systems could improve other games by creating more interesting enemies and unique side quests. However, this patent and many like it may discourage developers from experimenting with mechanics that resemble the patented inventions because of potential legal complications.<ref>{{Cite news|url=https://www.eurogamer.net/warner-bros-is-trying-to-patent-middle-earth-shadow-of-mordors-nemesis-system|title=Warner Bros is trying to patent Middle-Earth: Shadow of Mordor's Nemesis system|date=2021-01-30|work=Eurogamer.net|access-date=2026-11-04|language=en}}</ref> This is especially challenging for small game studios, who may not have the resources to deal with many legal disputes. Although developers can still create their own systems that build from the Nemesis System, they must be careful not to infringe on existing patent claims. ==References== {{reflist|liststyle=decimal}} {{BookCat}} dbqgaqa1p1wi1lounvmhirn1trj64xf 3 332800 2844457 2026-11-04T12:46:28Z MathXplore 2888076 advert1 ([[m:User:ZbVl/VD|Vandoom]]) 2844457 wikitext text/x-wiki == 2026-11-04 == <div class="mw-content-ltr" dir="ltr" style="text-align: left" lang="en">[[File:Information.svg|25px|alt=Information icon]] Hello. Apologies for writing this in English, but I wanted to let you know that one or more of [[Special:Contributions/Sreenidhi U|your recent contributions]] have been undone because they appeared to be promotional. [[:m:en:WP:SOAPBOX|Advertising or using <span style="white-space:nowrap">Wikiversity</span> as a "soapbox"]] are not permitted. Take a look at the welcome pages to learn more about <span style="white-space:nowrap">Wikiversity</span>. Thanks. </div><!-- Glow-advert1 @ 1793795913244.7s --><nowiki></nowiki> [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 12:46, 4 November 2026 (UTC) l1t08f7zfg1cgqmyn2cajqa27yecmx1 3 332801 2844459 2026-11-04T12:49:49Z MathXplore 2888076 vandalism1 ([[m:User:ZbVl/VD|Vandoom]]) 2844459 wikitext text/x-wiki == 2026-11-04 == [[File:Information.svg|25px|alt=Information icon]] Hello, I’m letting you know that one or more of your recent contributions have been reverted because they did not appear constructive. If you would like to experiment, please use the [[Wikiversity:Sandbox|sandbox]] or ask for assistance at the [[Wikiversity:Colloquium|Colloquium]]. Thank you.<!-- Glow-vandalism1 @ 1793796587276.3s --><nowiki></nowiki> [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 12:49, 4 November 2026 (UTC) mt0oaxo95b9yokq3lwy3gfdsaa74sy0 6 332802 2844465 2026-11-04T13:31:50Z U3292769 3106878 {{Information |Description= |Source= |Date= |Author= |Permission= }} 2844465 wikitext text/x-wiki == Summary == {{Information |Description= |Source= |Date= |Author= |Permission= }} == Licensing == {{subst:nld}} 9q75p5jdj3ikvbycoisf34ouwgd6m21 2844474 2844465 2026-11-04T13:40:36Z U3292769 3106878 File summary addition 2844474 wikitext text/x-wiki == Summary == {{Information |Description= |Prompt= |Date= |Source= |Author= |Permission= }} == Licensing == {{subst:nld}} kzjvmfiyw4ox35hpogwhcbzr5nu1qgy 2844475 2844474 2026-11-04T13:43:02Z U3292769 3106878 /* Summary */ Image edit summary 2844475 wikitext text/x-wiki == Summary == {{Information |Description=Physiological and psychological changes of excitement based on two-factor theory |Prompt="please make me a cartoon style image with text based on two-theory on the physiological and psychological changes of excitement' |Date=16 October 2026 |Source=Own work |Author=ChatGPT |Permission= }} == Licensing == {{subst:nld}} 1zqq3ttra4qseq8133dmcbmfjok1z5h 6 332803 2844467 2026-11-04T13:33:06Z Young1lim 21186 {{Information |Description=Carry Lookahead Adders 2B traditional (20261103 - 20261102) |Source={{own|Young1lim}} |Date=2026-11-04 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2844467 wikitext text/x-wiki == Summary == {{Information |Description=Carry Lookahead Adders 2B traditional (20261103 - 20261102) |Source={{own|Young1lim}} |Date=2026-11-04 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} isoj0ocuccleyqp2zmycysmagnfvx27 6 332804 2844469 2026-11-04T13:33:58Z Young1lim 21186 {{Information |Description=Carry Lookahead Adders 2B traditional (20261104 - 20261103) |Source={{own|Young1lim}} |Date=2026-11-04 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2844469 wikitext text/x-wiki == Summary == {{Information |Description=Carry Lookahead Adders 2B traditional (20261104 - 20261103) |Source={{own|Young1lim}} |Date=2026-11-04 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} o82e3raippxjh1x3ve6rpw4mt7fvcqm 6 332805 2844470 2026-11-04T13:34:44Z Young1lim 21186 {{Information ||Description=Block CLA 2C Multi Level (20261103 - 20261102) |Source={{own|Young1lim}} |Date=2026-11-04 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2844470 wikitext text/x-wiki == Summary == {{Information ||Description=Block CLA 2C Multi Level (20261103 - 20261102) |Source={{own|Young1lim}} |Date=2026-11-04 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} f551lgtc01ypfgh4yy5gtea5ewrdzgi 6 332806 2844472 2026-11-04T13:35:31Z Young1lim 21186 {{Information ||Description=Block CLA 2C Multi Level (20261104 - 20261103) |Source={{own|Young1lim}} |Date=2026-11-04 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2844472 wikitext text/x-wiki == Summary == {{Information ||Description=Block CLA 2C Multi Level (20261104 - 20261103) |Source={{own|Young1lim}} |Date=2026-11-04 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} s8fxv0mqivnqg5k35mndioqdiaw55q1 6 332807 2844477 2026-11-04T13:54:43Z Young1lim 21186 {{Information |Description=C04.SA0: Address and Dereference Operators (20261031 - 20261030) |Source={{own|Young1lim}} |Date=2026-11-04 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2844477 wikitext text/x-wiki == Summary == {{Information |Description=C04.SA0: Address and Dereference Operators (20261031 - 20261030) |Source={{own|Young1lim}} |Date=2026-11-04 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} k4ibifytfvanf48o31bdzycy1mi556g 6 332808 2844479 2026-11-04T13:55:30Z Young1lim 21186 {{Information |Description=C04.SA0: Address and Dereference Operators (20261102 - 20261031) |Source={{own|Young1lim}} |Date=2026-11-04 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2844479 wikitext text/x-wiki == Summary == {{Information |Description=C04.SA0: Address and Dereference Operators (20261102 - 20261031) |Source={{own|Young1lim}} |Date=2026-11-04 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} r81pg3wk7y49kxikqzsxid2rlpyirth 0 332809 2844482 2026-11-04T16:17:26Z Calixto Rosa 3111607 Created page with "A practical guide for people who work on research software as part of a team: students and interns who are joining a project, and researchers and developers who already work on one. It covers what to look at when you arrive, how to work so that others can follow and continue your work, and how to leave the project ready for the next person. This guide is about '''contributing to research software that already exists''', in a team whose members change over time, rather t..." 2844482 wikitext text/x-wiki A practical guide for people who work on research software as part of a team: students and interns who are joining a project, and researchers and developers who already work on one. It covers what to look at when you arrive, how to work so that others can follow and continue your work, and how to leave the project ready for the next person. This guide is about '''contributing to research software that already exists''', in a team whose members change over time, rather than about designing new software. It does not teach programming. The software engineering practices it describes (version control, reproducible environments, tests, documentation, traceability and licensing) are among the most basic ones. None of them is advanced, but together they are what makes it possible for someone else to understand, run and continue your work. The examples use Python, the most common language in the projects this guide was written for. The practices apply to any language, and each section mentions equivalent tools for other languages where it matters. == What is research software == The term ''research software'' refers to the source code files, scripts, algorithms, computational workflows and executables created during the research process or for a research purpose (Barker et al., 2022). It differs from conventional software in two ways. First, it is closely tied to the algorithms and models of a specific scientific domain. Second, it is often developed and maintained by researchers of that domain, who frequently have no training in software engineering (Storer, 2017). Because of this, a few concerns deserve special attention: * '''Reproducibility.''' A researcher may need to run an analysis again years later, or a reviewer may ask how a result was produced. This is only possible if the software and its environment can be rebuilt. * '''Traceability.''' For any result, it should be possible to know which version of the code, which parameters, which data and which environment produced it. * '''Continuity.''' Students and interns stay for months; the software often stays for years. Knowledge that is only in someone's head or on their computer is lost when they leave. * '''Licensing.''' Software can only be legally reused, studied and shared if it has a license. When research software is shared, a free software license makes these uses possible. == Principles == # '''Good enough first.''' Each section ends with a checklist split into ''essential'' and ''recommended'' items. Start with the essential ones, and improve later. # '''Improve what you touch.''' You do not need to fix everything at once. Leave the parts you work on a bit clearer than you found them. # '''Write things down.''' Decisions, commands, versions and known problems belong in the repository, in an issue or in the wiki, not only in your memory. == Contents == New to a project? Start with section 1. Otherwise, go to the section you need. {| class="wikitable" |- ! # ! Section ! Summary |- | 1 | [[Contributing_to_Research_Software/Joining_an_existing_project|Joining an existing project]] | Understand and run what exists, then decide whether to continue, refactor or rewrite. |- | 2 | [[Contributing_to_Research_Software/A_minimally_organized_repository|A minimally organized repository]] | Directory layout and the files every project needs: README, LICENSE, .gitignore, CONTRIBUTING, CITATION. |- | 3 | [[Contributing_to_Research_Software/Git,_issues_and_collaboration|Git, issues and collaboration]] | Commits, branches, merge requests, issues and code review. |- | 4 | [[Contributing_to_Research_Software/Reproducible_environments|Reproducible environments]] | Pinned dependencies, virtual environments, Docker, and other tools such as Guix. |- | 5 | [[Contributing_to_Research_Software/Automating_with_a_Makefile|Automating with a Makefile]] | One entry point for building, running and testing. |- | 6 | [[Contributing_to_Research_Software/Tests|Tests]] | Why research code needs tests, and how to protect existing behaviour before changing it. |- | 7 | [[Contributing_to_Research_Software/Code_quality_and_continuous_integration|Code quality and continuous integration]] | Linters and automatic checks on every change. |- | 8 | [[Contributing_to_Research_Software/Documentation|Documentation]] | README, docstrings, input and output formats: what goes where. |- | 9 | [[Contributing_to_Research_Software/Traceability_and_provenance|Traceability and provenance]] | Linking each result to the code, parameters and environment that produced it. |- | 10 | [[Contributing_to_Research_Software/Licensing,_citation_and_open_science|Licensing, citation and open science]] | Choosing a license and making software citable. |- | 11 | [[Contributing_to_Research_Software/Leaving_the_project|Leaving the project]] | A hand-over checklist for the next person. |} pwghf5v67xc0yga57a132anxz7ibiya 2844483 2844482 2026-11-04T16:36:46Z Calixto Rosa 3111607 2844483 wikitext text/x-wiki A practical guide for people who work on research software as part of a team: students and interns who are joining a project, and researchers and developers who already work on one. It covers what to look at when you arrive, how to work so that others can follow and continue your work, and how to leave the project ready for the next person. This guide is about '''contributing to research software that already exists''', in a team whose members change over time, rather than about designing new software. It does not teach programming. The software engineering practices it describes (version control, reproducible environments, tests, documentation, traceability and licensing) are among the most basic ones. None of them is advanced, but together they are what makes it possible for someone else to understand, run and continue your work. The examples use Python, the most common language in the projects this guide was written for. The practices apply to any language, and each section mentions equivalent tools for other languages where it matters. == What is research software == The term ''research software'' refers to the source code files, scripts, algorithms, computational workflows and executables created during the research process or for a research purpose (Barker et al., 2022). It differs from conventional software in two ways. First, it is closely tied to the algorithms and models of a specific scientific domain. Second, it is often developed and maintained by researchers of that domain, who frequently have no training in software engineering (Storer, 2017). Because of this, a few concerns deserve special attention: * '''Reproducibility.''' A researcher may need to run an analysis again years later, or a reviewer may ask how a result was produced. This is only possible if the software and its environment can be rebuilt. * '''Traceability.''' For any result, it should be possible to know which version of the code, which parameters, which data and which environment produced it. * '''Continuity.''' Students and interns stay for months; the software often stays for years. Knowledge that is only in someone's head or on their computer is lost when they leave. * '''Licensing.''' Software can only be legally reused, studied and shared if it has a license. When research software is shared, a free software license makes these uses possible. == Principles == # '''Good enough first.''' Each section ends with a checklist split into ''essential'' and ''recommended'' items. Start with the essential ones, and improve later. # '''Improve what you touch.''' You do not need to fix everything at once. Leave the parts you work on a bit clearer than you found them. # '''Write things down.''' Decisions, commands, versions and known problems belong in the repository, in an issue or in the wiki, not only in your memory. == Contents == New to a project? Start with section 1. Otherwise, go to the section you need. {| class="wikitable" |- ! # ! Section ! Summary |- | 1 | [[Contributing_to_Research_Software/Joining_an_existing_project|Joining an existing project]] | Understand and run what exists, then decide whether to continue, refactor or rewrite. |- | 2 | [[Contributing_to_Research_Software/A_minimally_organized_repository|A minimally organized repository]] | Directory layout and the files every project needs: README, LICENSE, .gitignore, CONTRIBUTING, CITATION. |- | 3 | [[Contributing_to_Research_Software/Git,_issues_and_collaboration|Git, issues and collaboration]] | Commits, branches, merge requests, issues and code review. |- | 4 | [[Contributing_to_Research_Software/Reproducible_environments|Reproducible environments]] | Pinned dependencies, virtual environments, Docker, and other tools such as Guix. |- | 5 | [[Contributing_to_Research_Software/Automating_with_a_Makefile|Automating with a Makefile]] | One entry point for building, running and testing. |- | 6 | [[Contributing_to_Research_Software/Tests|Tests]] | Why research code needs tests, and how to protect existing behaviour before changing it. |- | 7 | [[Contributing_to_Research_Software/Code_quality_and_continuous_integration|Code quality and continuous integration]] | Linters and automatic checks on every change. |- | 8 | [[Contributing_to_Research_Software/Documentation|Documentation]] | README, docstrings, input and output formats: what goes where. |- | 9 | [[Contributing_to_Research_Software/Traceability_and_provenance|Traceability and provenance]] | Linking each result to the code, parameters and environment that produced it. |- | 10 | [[Contributing_to_Research_Software/Licensing,_citation_and_open_science|Licensing, citation and open science]] | Choosing a license and making software citable. |- | 11 | [[Contributing_to_Research_Software/Leaving_the_project|Leaving the project]] | A hand-over checklist for the next person. |} == References == * Barker M. et al. (2022). [https://doi.org/10.1038/s41597-022-01710-x Introducing the FAIR Principles for research software]. ''Scientific Data'' 9, 622. * Storer T. (2017). [https://doi.org/10.1145/3084225 Bridging the chasm: a survey of software engineering practice in scientific programming]. ''ACM Computing Surveys'' 50(4). f7us2ysd969ez9ufyrxe1pm1t9gfnpf 2844525 2844483 2026-11-04T22:33:51Z MathXplore 2888076 added [[Category:Software]] using [[Help:Gadget-HotCat|HotCat]] 2844525 wikitext text/x-wiki A practical guide for people who work on research software as part of a team: students and interns who are joining a project, and researchers and developers who already work on one. It covers what to look at when you arrive, how to work so that others can follow and continue your work, and how to leave the project ready for the next person. This guide is about '''contributing to research software that already exists''', in a team whose members change over time, rather than about designing new software. It does not teach programming. The software engineering practices it describes (version control, reproducible environments, tests, documentation, traceability and licensing) are among the most basic ones. None of them is advanced, but together they are what makes it possible for someone else to understand, run and continue your work. The examples use Python, the most common language in the projects this guide was written for. The practices apply to any language, and each section mentions equivalent tools for other languages where it matters. == What is research software == The term ''research software'' refers to the source code files, scripts, algorithms, computational workflows and executables created during the research process or for a research purpose (Barker et al., 2022). It differs from conventional software in two ways. First, it is closely tied to the algorithms and models of a specific scientific domain. Second, it is often developed and maintained by researchers of that domain, who frequently have no training in software engineering (Storer, 2017). Because of this, a few concerns deserve special attention: * '''Reproducibility.''' A researcher may need to run an analysis again years later, or a reviewer may ask how a result was produced. This is only possible if the software and its environment can be rebuilt. * '''Traceability.''' For any result, it should be possible to know which version of the code, which parameters, which data and which environment produced it. * '''Continuity.''' Students and interns stay for months; the software often stays for years. Knowledge that is only in someone's head or on their computer is lost when they leave. * '''Licensing.''' Software can only be legally reused, studied and shared if it has a license. When research software is shared, a free software license makes these uses possible. == Principles == # '''Good enough first.''' Each section ends with a checklist split into ''essential'' and ''recommended'' items. Start with the essential ones, and improve later. # '''Improve what you touch.''' You do not need to fix everything at once. Leave the parts you work on a bit clearer than you found them. # '''Write things down.''' Decisions, commands, versions and known problems belong in the repository, in an issue or in the wiki, not only in your memory. == Contents == New to a project? Start with section 1. Otherwise, go to the section you need. {| class="wikitable" |- ! # ! Section ! Summary |- | 1 | [[Contributing_to_Research_Software/Joining_an_existing_project|Joining an existing project]] | Understand and run what exists, then decide whether to continue, refactor or rewrite. |- | 2 | [[Contributing_to_Research_Software/A_minimally_organized_repository|A minimally organized repository]] | Directory layout and the files every project needs: README, LICENSE, .gitignore, CONTRIBUTING, CITATION. |- | 3 | [[Contributing_to_Research_Software/Git,_issues_and_collaboration|Git, issues and collaboration]] | Commits, branches, merge requests, issues and code review. |- | 4 | [[Contributing_to_Research_Software/Reproducible_environments|Reproducible environments]] | Pinned dependencies, virtual environments, Docker, and other tools such as Guix. |- | 5 | [[Contributing_to_Research_Software/Automating_with_a_Makefile|Automating with a Makefile]] | One entry point for building, running and testing. |- | 6 | [[Contributing_to_Research_Software/Tests|Tests]] | Why research code needs tests, and how to protect existing behaviour before changing it. |- | 7 | [[Contributing_to_Research_Software/Code_quality_and_continuous_integration|Code quality and continuous integration]] | Linters and automatic checks on every change. |- | 8 | [[Contributing_to_Research_Software/Documentation|Documentation]] | README, docstrings, input and output formats: what goes where. |- | 9 | [[Contributing_to_Research_Software/Traceability_and_provenance|Traceability and provenance]] | Linking each result to the code, parameters and environment that produced it. |- | 10 | [[Contributing_to_Research_Software/Licensing,_citation_and_open_science|Licensing, citation and open science]] | Choosing a license and making software citable. |- | 11 | [[Contributing_to_Research_Software/Leaving_the_project|Leaving the project]] | A hand-over checklist for the next person. |} == References == * Barker M. et al. (2022). [https://doi.org/10.1038/s41597-022-01710-x Introducing the FAIR Principles for research software]. ''Scientific Data'' 9, 622. * Storer T. (2017). [https://doi.org/10.1145/3084225 Bridging the chasm: a survey of software engineering practice in scientific programming]. ''ACM Computing Surveys'' 50(4). [[Category:Software]] 7hwx1ijc4lst6xmtvkhzz5ki0q5bur2 2844526 2844525 2026-11-04T22:34:00Z MathXplore 2888076 added [[Category:Research]] using [[Help:Gadget-HotCat|HotCat]] 2844526 wikitext text/x-wiki A practical guide for people who work on research software as part of a team: students and interns who are joining a project, and researchers and developers who already work on one. It covers what to look at when you arrive, how to work so that others can follow and continue your work, and how to leave the project ready for the next person. This guide is about '''contributing to research software that already exists''', in a team whose members change over time, rather than about designing new software. It does not teach programming. The software engineering practices it describes (version control, reproducible environments, tests, documentation, traceability and licensing) are among the most basic ones. None of them is advanced, but together they are what makes it possible for someone else to understand, run and continue your work. The examples use Python, the most common language in the projects this guide was written for. The practices apply to any language, and each section mentions equivalent tools for other languages where it matters. == What is research software == The term ''research software'' refers to the source code files, scripts, algorithms, computational workflows and executables created during the research process or for a research purpose (Barker et al., 2022). It differs from conventional software in two ways. First, it is closely tied to the algorithms and models of a specific scientific domain. Second, it is often developed and maintained by researchers of that domain, who frequently have no training in software engineering (Storer, 2017). Because of this, a few concerns deserve special attention: * '''Reproducibility.''' A researcher may need to run an analysis again years later, or a reviewer may ask how a result was produced. This is only possible if the software and its environment can be rebuilt. * '''Traceability.''' For any result, it should be possible to know which version of the code, which parameters, which data and which environment produced it. * '''Continuity.''' Students and interns stay for months; the software often stays for years. Knowledge that is only in someone's head or on their computer is lost when they leave. * '''Licensing.''' Software can only be legally reused, studied and shared if it has a license. When research software is shared, a free software license makes these uses possible. == Principles == # '''Good enough first.''' Each section ends with a checklist split into ''essential'' and ''recommended'' items. Start with the essential ones, and improve later. # '''Improve what you touch.''' You do not need to fix everything at once. Leave the parts you work on a bit clearer than you found them. # '''Write things down.''' Decisions, commands, versions and known problems belong in the repository, in an issue or in the wiki, not only in your memory. == Contents == New to a project? Start with section 1. Otherwise, go to the section you need. {| class="wikitable" |- ! # ! Section ! Summary |- | 1 | [[Contributing_to_Research_Software/Joining_an_existing_project|Joining an existing project]] | Understand and run what exists, then decide whether to continue, refactor or rewrite. |- | 2 | [[Contributing_to_Research_Software/A_minimally_organized_repository|A minimally organized repository]] | Directory layout and the files every project needs: README, LICENSE, .gitignore, CONTRIBUTING, CITATION. |- | 3 | [[Contributing_to_Research_Software/Git,_issues_and_collaboration|Git, issues and collaboration]] | Commits, branches, merge requests, issues and code review. |- | 4 | [[Contributing_to_Research_Software/Reproducible_environments|Reproducible environments]] | Pinned dependencies, virtual environments, Docker, and other tools such as Guix. |- | 5 | [[Contributing_to_Research_Software/Automating_with_a_Makefile|Automating with a Makefile]] | One entry point for building, running and testing. |- | 6 | [[Contributing_to_Research_Software/Tests|Tests]] | Why research code needs tests, and how to protect existing behaviour before changing it. |- | 7 | [[Contributing_to_Research_Software/Code_quality_and_continuous_integration|Code quality and continuous integration]] | Linters and automatic checks on every change. |- | 8 | [[Contributing_to_Research_Software/Documentation|Documentation]] | README, docstrings, input and output formats: what goes where. |- | 9 | [[Contributing_to_Research_Software/Traceability_and_provenance|Traceability and provenance]] | Linking each result to the code, parameters and environment that produced it. |- | 10 | [[Contributing_to_Research_Software/Licensing,_citation_and_open_science|Licensing, citation and open science]] | Choosing a license and making software citable. |- | 11 | [[Contributing_to_Research_Software/Leaving_the_project|Leaving the project]] | A hand-over checklist for the next person. |} == References == * Barker M. et al. (2022). [https://doi.org/10.1038/s41597-022-01710-x Introducing the FAIR Principles for research software]. ''Scientific Data'' 9, 622. * Storer T. (2017). [https://doi.org/10.1145/3084225 Bridging the chasm: a survey of software engineering practice in scientific programming]. ''ACM Computing Surveys'' 50(4). [[Category:Software]] [[Category:Research]] fvcdkzg9758t8f19w4o460w2i7hkso2 0 332810 2844484 2026-11-04T16:38:01Z Calixto Rosa 3111607 Created page with "''Section 1 of [[Contributing_to_Research_Software|Contributing to Research Software: A Practical Guide]].'' == Why it matters == When you join a project, the software already has a history: choices made for reasons that may not be written anywhere, behaviours that published results depend on, and people who use it. Taking time to understand it first helps you choose the best way forward, whether that is continuing the existing code, improving its structure or starting..." 2844484 wikitext text/x-wiki ''Section 1 of [[Contributing_to_Research_Software|Contributing to Research Software: A Practical Guide]].'' == Why it matters == When you join a project, the software already has a history: choices made for reasons that may not be written anywhere, behaviours that published results depend on, and people who use it. Taking time to understand it first helps you choose the best way forward, whether that is continuing the existing code, improving its structure or starting a new version. == Steps == === 1. Read what exists === Start with the README, then the documentation (wiki or <code>docs/</code>), the open issues and merge requests, the dependency files (<code>pyproject.toml</code>, <code>Dockerfile</code>...) and the tests. Section 2 describes what each of these files usually contains. === 2. Run it as it is === Try to run the software without changing it, on a small example input. Write down every command you used, every error you met and how you solved it. If you needed steps that are not in the README, they can become a new section of it. If you cannot run it (missing dependencies, an old language version), let the team know early. Setting up a reproducible environment may be the first task (see section 4). === 3. Write down what you found === Write a short note, in an issue or in the wiki. You can use this template: <pre class="markdown">## State of &lt;project&gt; (&lt;date&gt;) **What it does:** &lt;in your own words&gt; **How to run it:** &lt;the commands that worked&gt; **What is missing:** &lt;e.g. tests, license, README, pinned dependencies&gt; **Behaviours to confirm:** &lt;outputs you did not expect, and why&gt; **Open questions:** &lt;for the team&gt;</pre> If an output looks unexpected, record it and ask before changing it: published results may depend on that behaviour. === 4. Talk to the team === * Who uses the software, and for what? Which outputs matter most? * Who wrote it, and can you reach them? * Have results (papers, reports) been produced with it that must stay reproducible? * Where should your work go: which repository, which branch, and who reviews it? === 5. Decide how to continue === With what you learned, choose an approach together with your supervisor: {| class="wikitable" |- ! Approach ! Good fit when ! Points of attention |- | '''Continue''' (add to the existing code) | The software runs, its structure suits your task, and the change is small. | Keep the parts you touch clear and documented. |- | '''Refactor''' (improve the structure step by step, keeping the same behaviour) | The software works, but its structure makes changes difficult. | Write tests that capture the current outputs before changing anything (section 6). |- | '''Rewrite''' (build a new version) | The current code can no longer run, its technology is no longer supported, or the new requirements differ a lot from the original ones. | Use the current outputs as a reference, and compare the new version against them. |} Whatever the approach, record the decision and the reasons for it (section 9). The next person will want to know. == Checklist == === Essential === * ☐ I read the README, the documentation and the open issues. * ☐ I ran the software on an example input. * ☐ I wrote down how to run it and what was missing. * ☐ I recorded unexpected behaviours and asked the team about them. * ☐ I chose an approach with my supervisor and recorded why. === Recommended === * ☐ I know who uses the outputs and who reviews my work. * ☐ My notes are shared in an issue or in the wiki. 3zgricb4lsjx4ya3gzj87yczph1ccxf 0 332811 2844486 2026-11-04T16:51:51Z Calixto Rosa 3111607 Created page with "A practical guide for people who work on research software as part of a team: students and interns who are joining a project, and researchers and developers who already work on one. It covers what to look at when you arrive, how to work so that others can follow and continue your work, and how to leave the project ready for the next person. This guide is about '''contributing to research software that already exists''', in a team whose members change over time, rather t..." 2844486 wikitext text/x-wiki A practical guide for people who work on research software as part of a team: students and interns who are joining a project, and researchers and developers who already work on one. It covers what to look at when you arrive, how to work so that others can follow and continue your work, and how to leave the project ready for the next person. This guide is about '''contributing to research software that already exists''', in a team whose members change over time, rather than about designing new software. It does not teach programming. The software engineering practices it describes (version control, reproducible environments, tests, documentation, traceability and licensing) are among the most basic ones. None of them is advanced, but together they are what makes it possible for someone else to understand, run and continue your work. The examples use Python, the most common language in the projects this guide was written for. The practices apply to any language, and each section mentions equivalent tools for other languages where it matters. == What is research software == The term ''research software'' refers to the source code files, scripts, algorithms, computational workflows and executables created during the research process or for a research purpose (Barker et al., 2022). It differs from conventional software in two ways. First, it is closely tied to the algorithms and models of a specific scientific domain. Second, it is often developed and maintained by researchers of that domain, who frequently have no training in software engineering (Storer, 2017). Because of this, a few concerns deserve special attention: * '''Reproducibility.''' A researcher may need to run an analysis again years later, or a reviewer may ask how a result was produced. This is only possible if the software and its environment can be rebuilt. * '''Traceability.''' For any result, it should be possible to know which version of the code, which parameters, which data and which environment produced it. * '''Continuity.''' Students and interns stay for months; the software often stays for years. Knowledge that is only in someone's head or on their computer is lost when they leave. * '''Licensing.''' Software can only be legally reused, studied and shared if it has a license. When research software is shared, a free software license makes these uses possible. == Principles == # '''Good enough first.''' Each section ends with a checklist split into ''essential'' and ''recommended'' items. Start with the essential ones, and improve later. # '''Improve what you touch.''' You do not need to fix everything at once. Leave the parts you work on a bit clearer than you found them. # '''Write things down.''' Decisions, commands, versions and known problems belong in the repository, in an issue or in the wiki, not only in your memory. == Contents == New to a project? Start with section 1. Otherwise, go to the section you need. {| class="wikitable" |- ! # ! Section ! Summary |- | 1 | [[/Joining_an_existing_project|Joining an existing project]] | Understand and run what exists, then decide whether to continue, refactor or rewrite. |- | 2 | [[/A_minimally_organized_repository|A minimally organized repository]] | Directory layout and the files every project needs: README, LICENSE, .gitignore, CONTRIBUTING, CITATION. |- | 3 | [[/Git,_issues_and_collaboration|Git, issues and collaboration]] | Commits, branches, merge requests, issues and code review. |- | 4 | [[/Reproducible_environments|Reproducible environments]] | Pinned dependencies, virtual environments, Docker, and other tools such as Guix. |- | 5 | [[/Automating_with_a_Makefile|Automating with a Makefile]] | One entry point for building, running and testing. |- | 6 | [[/Tests|Tests]] | Why research code needs tests, and how to protect existing behaviour before changing it. |- | 7 | [[/Code_quality_and_continuous_integration|Code quality and continuous integration]] | Linters and automatic checks on every change. |- | 8 | [[/Documentation|Documentation]] | README, docstrings, input and output formats: what goes where. |- | 9 | [[/Traceability_and_provenance|Traceability and provenance]] | Linking each result to the code, parameters and environment that produced it. |- | 10 | [[/Licensing,_citation_and_open_science|Licensing, citation and open science]] | Choosing a license and making software citable. |- | 11 | [[/Leaving_the_project|Leaving the project]] | A hand-over checklist for the next person. |} == References == * Barker M. et al. (2022). [https://doi.org/10.1038/s41597-022-01710-x Introducing the FAIR Principles for research software]. ''Scientific Data'' 9, 622. * Storer T. (2017). [https://doi.org/10.1145/3084225 Bridging the chasm: a survey of software engineering practice in scientific programming]. ''ACM Computing Surveys'' 50(4). 2qib2jzyqlgmy0ocsjufyu6ho9t3h8k 0 332812 2844487 2026-11-04T16:52:44Z Calixto Rosa 3111607 Created page with "''Section 1 of [[../|Contributing to Research Software: A Practical Guide]].'' == Why it matters == When you join a project, the software already has a history: choices made for reasons that may not be written anywhere, behaviours that published results depend on, and people who use it. Taking time to understand it first helps you choose the best way forward, whether that is continuing the existing code, improving its structure or starting a new version. == Steps ==..." 2844487 wikitext text/x-wiki ''Section 1 of [[../|Contributing to Research Software: A Practical Guide]].'' == Why it matters == When you join a project, the software already has a history: choices made for reasons that may not be written anywhere, behaviours that published results depend on, and people who use it. Taking time to understand it first helps you choose the best way forward, whether that is continuing the existing code, improving its structure or starting a new version. == Steps == === 1. Read what exists === Start with the README, then the documentation (wiki or <code>docs/</code>), the open issues and merge requests, the dependency files (<code>pyproject.toml</code>, <code>Dockerfile</code>...) and the tests. Section 2 describes what each of these files usually contains. === 2. Run it as it is === Try to run the software without changing it, on a small example input. Write down every command you used, every error you met and how you solved it. If you needed steps that are not in the README, they can become a new section of it. If you cannot run it (missing dependencies, an old language version), let the team know early. Setting up a reproducible environment may be the first task (see section 4). === 3. Write down what you found === Write a short note, in an issue or in the wiki. You can use this template: <pre class="markdown">## State of &lt;project&gt; (&lt;date&gt;) **What it does:** &lt;in your own words&gt; **How to run it:** &lt;the commands that worked&gt; **What is missing:** &lt;e.g. tests, license, README, pinned dependencies&gt; **Behaviours to confirm:** &lt;outputs you did not expect, and why&gt; **Open questions:** &lt;for the team&gt;</pre> If an output looks unexpected, record it and ask before changing it: published results may depend on that behaviour. === 4. Talk to the team === * Who uses the software, and for what? Which outputs matter most? * Who wrote it, and can you reach them? * Have results (papers, reports) been produced with it that must stay reproducible? * Where should your work go: which repository, which branch, and who reviews it? === 5. Decide how to continue === With what you learned, choose an approach together with your supervisor: {| class="wikitable" |- ! Approach ! Good fit when ! Points of attention |- | '''Continue''' (add to the existing code) | The software runs, its structure suits your task, and the change is small. | Keep the parts you touch clear and documented. |- | '''Refactor''' (improve the structure step by step, keeping the same behaviour) | The software works, but its structure makes changes difficult. | Write tests that capture the current outputs before changing anything (section 6). |- | '''Rewrite''' (build a new version) | The current code can no longer run, its technology is no longer supported, or the new requirements differ a lot from the original ones. | Use the current outputs as a reference, and compare the new version against them. |} Whatever the approach, record the decision and the reasons for it (section 9). The next person will want to know. == Checklist == === Essential === * ☐ I read the README, the documentation and the open issues. * ☐ I ran the software on an example input. * ☐ I wrote down how to run it and what was missing. * ☐ I recorded unexpected behaviours and asked the team about them. * ☐ I chose an approach with my supervisor and recorded why. === Recommended === * ☐ I know who uses the outputs and who reviews my work. * ☐ My notes are shared in an issue or in the wiki. 2x16odt5kum6yetnhfoqau27ked8qmu 0 332813 2844488 2026-11-04T16:53:49Z Calixto Rosa 3111607 Created page with "''Section 2 of [[../|Contributing to Research Software: A Practical Guide]].'' == Why it matters == A predictable layout helps people find things without asking. When code, tests and documentation are always in the same places, a newcomer understands a project faster, and tools (test runners, packaging, continuous integration) work with little configuration. A few files are also expected in every project: they answer the first questions anyone has. What is this? Can I..." 2844488 wikitext text/x-wiki ''Section 2 of [[../|Contributing to Research Software: A Practical Guide]].'' == Why it matters == A predictable layout helps people find things without asking. When code, tests and documentation are always in the same places, a newcomer understands a project faster, and tools (test runners, packaging, continuous integration) work with little configuration. A few files are also expected in every project: they answer the first questions anyone has. What is this? Can I use it? How do I contribute? How do I cite it? == How to do it == === Directory layout === A common layout for a Python project: <syntaxhighlight lang="text">my-project/ ├── README.md # what it is, how to install, run and cite it ├── LICENSE # the license text ├── CONTRIBUTING.md # how to contribute ├── CITATION.cff # how to cite the software ├── .gitignore # files Git must ignore ├── pyproject.toml # package metadata and dependencies ├── Makefile # common commands (section 5) ├── my_project/ # source code │ ├── __init__.py │ └── ... ├── tests/ # automated tests (section 6) ├── docs/ # documentation beyond the README └── docker/ └── Dockerfile # reproducible environment (section 4)</syntaxhighlight> Other languages have their own conventions; follow the one of your language or of the existing project. What matters is that the layout is consistent and explained in the README. Some general rules for names: * Use lower case, no spaces and no accents in file and directory names. * Choose names that say what something is (<code>network_builder.py</code>, not <code>script2_final.py</code>). * Keep generated files (results, logs, build outputs) out of the source directories. === Files every project should have === {| class="wikitable" |- ! File ! Purpose ! Minimum content |- | <code>README.md</code> | Entry point for any visitor. | What the project does, how to install it, how to run it, how to cite it, license. |- | <code>LICENSE</code> | Says what others may do with the code. | The full text of the chosen license (section 10). |- | <code>.gitignore</code> | Keeps generated or private files out of Git. | Virtual environments, caches, outputs, local configuration. |- | <code>CONTRIBUTING.md</code> | Tells contributors how to work on the project. | How to set up the environment, run the tests, name branches and open a merge request. |- | <code>CITATION.cff</code> | Tells people how to cite the software. | Title, authors, version, license, repository URL (section 10). |- | <code>CHANGELOG.md</code> | Lists notable changes between versions. | One entry per version (section 9). |} === A README template === <pre class="markdown"># &lt;Project name&gt; &lt;One or two sentences: what it does and for whom.&gt; ## Installation &lt;Commands to install it, or to build the Docker image.&gt; ## Usage &lt;The command to run it on an example input, and what output to expect.&gt; ## Tests &lt;The command to run the tests.&gt; ## Citation &lt;How to cite it, or &quot;see CITATION.cff&quot;.&gt; ## License &lt;License name, with a link to the LICENSE file.&gt;</pre> === A .gitignore example (Python) === <syntaxhighlight lang="text"># virtual environments .venv/ venv/ # Python caches __pycache__/ *.pyc # test and coverage outputs .pytest_cache/ .coverage htmlcov/ # generated results output/ # local configuration (may contain passwords or tokens) .env</syntaxhighlight> Ready-made templates for many languages are available in the [https://github.com/github/gitignore github/gitignore] repository. == Checklist == === Essential === * ☐ The repository has a README with installation and usage instructions. * ☐ The repository has a LICENSE file. * ☐ The repository has a <code>.gitignore</code>, and no generated or private files are committed. * ☐ Source code, tests and documentation are in separate directories. === Recommended === * ☐ The repository has a <code>CONTRIBUTING.md</code> and a <code>CITATION.cff</code>. * ☐ File and directory names are descriptive, in lower case, without spaces. * ☐ The README explains the layout if it is not obvious. == Further reading == * [https://www.makeareadme.com/ Make a README] * [https://github.com/github/gitignore github/gitignore]: <code>.gitignore</code> templates. * [https://packaging.python.org/en/latest/tutorials/packaging-projects/ Python Packaging User Guide: Packaging Python Projects] * Wilson G. et al. (2017). [https://doi.org/10.1371/journal.pcbi.1005510 Good enough practices in scientific computing], section &quot;Project organization&quot;. 9mcfst8hn9w5vlm05ulj4smfc9ykv0b 0 332814 2844489 2026-11-04T16:54:32Z Calixto Rosa 3111607 Created page with "''Section 3 of [[../|Contributing to Research Software: A Practical Guide]].'' == Why it matters == Version control keeps the full history of a project: what changed, when, by whom and why. It lets several people work at the same time, makes it possible to go back to any earlier version, and links each result to the exact code that produced it (section 9). Issues and merge requests add the conversation around the code: what needs to be done, what was decided, and what..." 2844489 wikitext text/x-wiki ''Section 3 of [[../|Contributing to Research Software: A Practical Guide]].'' == Why it matters == Version control keeps the full history of a project: what changed, when, by whom and why. It lets several people work at the same time, makes it possible to go back to any earlier version, and links each result to the exact code that produced it (section 9). Issues and merge requests add the conversation around the code: what needs to be done, what was decided, and what was reviewed. Together, they let someone who joins later understand how the project got to where it is. == How to do it == === 1. The basic workflow === <syntaxhighlight lang="bash">git clone https://gitlab.com/<group>/<project>.git # get a copy of the repository cd <project> git switch -c fix/encoding-author-names # create a branch for your task # ... edit files ... git status # see what changed git diff # see the changes line by line git add path/to/file.py # choose what goes in the commit git commit -m "fix: read author names as UTF-8" # record the change git push -u origin fix/encoding-author-names # send the branch to the server</syntaxhighlight> Then open a merge request on GitLab (called a ''pull request'' on GitHub). To bring the latest changes from <code>main</code> into your branch: <syntaxhighlight lang="bash">git switch main git pull git switch fix/encoding-author-names git merge main</syntaxhighlight> === 2. Write good commits === * '''One logical change per commit.''' A commit that fixes a bug, renames a file and reformats a module is hard to review and hard to undo. * '''Commit often.''' Small commits are easier to understand and to revert. * '''Explain the change.''' The first line says what changed, in a few words. If needed, the body explains why. Many projects follow the [https://www.conventionalcommits.org/ Conventional Commits] format, <code>type(scope): description</code>: <syntaxhighlight lang="text">fix(parser): read author names as UTF-8 Names with accents were split into two tokens when the corpus was read with the system default encoding. Fixes #15</syntaxhighlight> Common types: <code>feat</code> (new feature), <code>fix</code> (bug fix), <code>docs</code>, <code>test</code>, <code>refactor</code> (change of structure without change of behaviour), <code>chore</code> (maintenance). === 3. Use branches === * Create one branch per task, named after it: <code>&lt;type&gt;/&lt;short-description&gt;</code>, for example <code>feat/export-csv</code> or <code>docs/installation-guide</code>. * Keep <code>main</code> working at all times: changes reach it only through reviewed merge requests. On GitLab and GitHub, <code>main</code> can be ''protected'' so that nobody pushes to it directly. * Delete branches once they are merged. === 4. Use issues === An issue records something to be done or discussed: a bug, a task, an idea, a question. It gives the work a number that commits and merge requests can refer to. A useful bug report says: <pre class="markdown">**What happened:** &lt;the error or the unexpected output&gt; **What I expected:** &lt;the correct behaviour&gt; **How to reproduce:** &lt;the command, the input file, the parameters&gt; **Version:** &lt;commit, tag or release&gt;</pre> A useful task description says what should be done, why, and how to know it is finished. Tips: * Search existing issues before opening a new one. * Keep one topic per issue. * Use labels (<code>bug</code>, <code>documentation</code>, <code>good first issue</code>...) to organize them. * Close issues from commits or merge requests with <code>Fixes #15</code> or <code>Closes #15</code>. === 5. Open merge requests === A merge request asks for your branch to be integrated into <code>main</code>. It is where your work is reviewed. * Keep it small: one task, a few hundred lines at most when possible. * Describe what changes, why, and how you tested it. Link the issue. * Open it as a ''draft'' early if you want feedback before it is finished. * Make sure the tests pass (sections 6 and 7). === 6. Review and be reviewed === Code review is one of the best ways to learn a project and to share knowledge about it. As an author: * Read your own changes once before asking for a review. * Answer every comment, even with &quot;done&quot; or &quot;I prefer to keep it because...&quot;. As a reviewer: * Check that the change does what it says, that it is tested and documented, and that you understand it. * Be specific and kind. Comment on the code, not on the person. Questions (&quot;What happens if the file is empty?&quot;) often work better than orders. == Checklist == === Essential === * ☐ I work on a branch named after my task, never directly on <code>main</code>. * ☐ My commits are small and their messages explain the change. * ☐ My task has an issue, and my commits or merge request refer to it. * ☐ Someone else reviewed my merge request before it was merged. === Recommended === * ☐ My merge request has a description, a link to the issue and passing tests. * ☐ <code>main</code> is protected against direct pushes. * ☐ Merged branches are deleted. == Further reading == * [https://swcarpentry.github.io/git-novice/ Software Carpentry: Version Control with Git] * Chacon S., Straub B. [https://git-scm.com/book/en/v2 Pro Git] (free online book). * [https://www.conventionalcommits.org/ Conventional Commits] * [https://docs.gitlab.com/user/project/merge_requests/ GitLab documentation: Merge requests] nppwatffiehcxwcwju1e9ckkaqjmjr5 0 332815 2844490 2026-11-04T16:55:15Z Calixto Rosa 3111607 Created page with "''Section 4 of [[../|Contributing to Research Software: A Practical Guide]].'' == Why it matters == The same code can give different results, or not run at all, on another computer or a few years later. The language version, the libraries and their versions, and the operating system all form the software's ''environment''. If the environment is not described, nobody can be sure to rebuild it, and &quot;it works on my machine&quot; becomes a real obstacle to reproducing..." 2844490 wikitext text/x-wiki ''Section 4 of [[../|Contributing to Research Software: A Practical Guide]].'' == Why it matters == The same code can give different results, or not run at all, on another computer or a few years later. The language version, the libraries and their versions, and the operating system all form the software's ''environment''. If the environment is not described, nobody can be sure to rebuild it, and &quot;it works on my machine&quot; becomes a real obstacle to reproducing results. == How to do it == === 1. Declare the dependencies === In Python, dependencies are declared in <code>pyproject.toml</code>: <pre class="toml">[project] name = &quot;my-project&quot; version = &quot;1.2.0&quot; requires-python = &quot;&gt;=3.11&quot; dependencies = [ &quot;pandas&gt;=2.2,&lt;3&quot;, &quot;networkx&gt;=3.3,&lt;4&quot;, ] [project.optional-dependencies] test = [&quot;pytest&gt;=8&quot;]</pre> Version ranges (<code>&gt;=2.2,&lt;3</code>) say which versions are expected to work. To record the '''exact''' versions used to produce a result, generate a ''lock file'' with a tool such as [https://pip-tools.readthedocs.io/ pip-tools] (<code>pip-compile</code>) or [https://docs.astral.sh/uv/ uv] (<code>uv lock</code>), and commit it. At minimum, save the output of <code>pip freeze</code> next to important results. === 2. Use a virtual environment === A virtual environment keeps the project's libraries separate from the system and from other projects: <syntaxhighlight lang="bash">python3 -m venv .venv # create it (once) source .venv/bin/activate # activate it (in each new terminal) pip install -e ".[test]" # install the project and its test dependencies</syntaxhighlight> Add <code>.venv/</code> to <code>.gitignore</code>: the environment is rebuilt from <code>pyproject.toml</code>, it is not committed. === 3. Use a container === A virtual environment isolates Python libraries, but still depends on the operating system and its system libraries. A container packages the whole environment: operating system, system libraries, language and dependencies. With [https://docs.docker.com/get-started/ Docker]: * a '''Dockerfile''' is the recipe that describes the environment; * an '''image''' is the result of building the recipe; * a '''container''' is a running instance of an image. A minimal Dockerfile for a Python project: <syntaxhighlight lang="dockerfile"># A fixed base image version, not "latest" FROM python:3.12-slim-bookworm WORKDIR /app # Copy the dependency description first, so this layer is cached COPY pyproject.toml README.md ./ COPY my_project ./my_project RUN pip install --no-cache-dir ".[test]" ENTRYPOINT ["python", "-m", "my_project"]</syntaxhighlight> Build and run it: <syntaxhighlight lang="bash">docker build -t my-project -f docker/Dockerfile . docker run --rm -v "$PWD/data:/data" my-project /data/input.csv</syntaxhighlight> Some good practices: * Use a specific base image tag (<code>python:3.12-slim-bookworm</code>), never <code>latest</code>. * Keep the Dockerfile in the repository, and build it in continuous integration (section 7). * Do not put passwords or tokens in the Dockerfile or the image. A Dockerfile describes ''how'' to build an environment, not exactly ''what'' it contains: installing a package today and next year may give different versions. Lock files and fixed base images reduce this problem. === 4. Other tools === Depending on the project and on how strict reproducibility must be, other tools may be a better fit. They are only listed here; see their documentation for details. {| class="wikitable" |- ! Tool ! What it offers |- | [https://docs.conda.io/ conda] / [https://mamba.readthedocs.io/ mamba] | Environments with packages from several languages (Python, R, C libraries), common in scientific computing. |- | [https://guix.gnu.org/ GNU Guix] | Declarative environments, defined by a list of packages and the exact version of the package collection; the same environment can be rebuilt bit for bit years later (<code>guix time-machine</code>). It can also produce Docker images. |- | [https://nixos.org/ Nix] | A similar approach to Guix: declarative and reproducible environments. |- | [https://apptainer.org/ Apptainer] (formerly Singularity) | Containers designed for shared computing clusters (HPC), where Docker is often not allowed. |- | [https://mybinder.org/ Binder] | Runs the notebooks of a public repository in the browser, in an environment built from the repository's files. |} === 5. Document the environment === In the README, say: * which language version and which tools are required; * the commands to set up the environment (or to build the image); * how to check that it works (for example, by running the tests). == Checklist == === Essential === * ☐ Dependencies are declared in the repository, with versions. * ☐ The project runs in a virtual environment or a container. * ☐ No passwords or tokens are in the environment files or images. * ☐ The README explains how to set up the environment. === Recommended === * ☐ Exact versions are recorded (lock file, or at least <code>pip freeze</code>) for results that matter. * ☐ Container base images use a fixed version, not <code>latest</code>. == Further reading == * [https://book.the-turing-way.org/reproducible-research/renv The Turing Way: Reproducible Environments] * [https://packaging.python.org/ Python Packaging User Guide] * [https://docs.docker.com/get-started/ Docker: Get started] * [https://guix.gnu.org/manual/en/html_node/Replicating-Guix.html GNU Guix manual: Replicating Guix] ewyav36ngpfx9klmb9qg6g3fh5mjvbc 0 332816 2844491 2026-11-04T16:55:45Z Calixto Rosa 3111607 Created page with "''Section 5 of [[../|Contributing to Research Software: A Practical Guide]].'' == Why it matters == Building an image, running the software on an example, running the tests: each task needs a command, often a long one with several options. When these commands live only in someone's memory or terminal history, newcomers have to rediscover them. A <code>Makefile</code> gives the project a single entry point. <code>make test</code> runs the tests, <code>make run</code> r..." 2844491 wikitext text/x-wiki ''Section 5 of [[../|Contributing to Research Software: A Practical Guide]].'' == Why it matters == Building an image, running the software on an example, running the tests: each task needs a command, often a long one with several options. When these commands live only in someone's memory or terminal history, newcomers have to rediscover them. A <code>Makefile</code> gives the project a single entry point. <code>make test</code> runs the tests, <code>make run</code> runs the software, whatever the commands behind them. It also works as documentation that can be executed: if the command changes, it is updated in one place. == How to do it == === 1. Anatomy of a rule === <syntaxhighlight lang="makefile">target: prerequisites command another command</syntaxhighlight> * <code>target</code> is the name you type after <code>make</code> (<code>make test</code>). * <code>prerequisites</code> are other targets that must run first (optional). * Each command line '''must start with a tab character''', not spaces. This is the most common error with Makefiles. Targets that are not file names (almost all of them in this use) are declared as <code>.PHONY</code>, so <code>make</code> always runs them. === 2. An example === <syntaxhighlight lang="makefile"># Common commands for this project. Run `make help` to list them. IMAGE ?= my-project JOB ?= tests/data/example/job.yaml .PHONY: help help: ## List the available targets @grep -E '^[a-zA-Z_-]+:.*## ' $(MAKEFILE_LIST) | \ awk 'BEGIN {FS = ":.*## "}; {printf " %-14s %s\n", $$1, $$2}' .PHONY: install install: ## Install the project and its test dependencies in the active environment pip install -e ".[test]" .PHONY: docker-build docker-build: ## Build the Docker image docker build -t $(IMAGE) -f docker/Dockerfile . .PHONY: run run: ## Run the software on an example (JOB=path/to/job.yaml) python -m my_project $(JOB) .PHONY: test test: ## Run the tests pytest tests/ .PHONY: lint lint: ## Check the code style ruff check . .PHONY: clean clean: ## Remove generated files rm -rf output/ .pytest_cache/ htmlcov/</syntaxhighlight> Usage: <syntaxhighlight lang="bash">make help # list the targets make test # run the tests make run JOB=tests/data/other.yaml # override a variable</syntaxhighlight> * <code>?=</code> sets a default value that can be changed from the command line. * <code>$(VARIABLE)</code> uses a variable. In commands, a literal <code>$</code> is written <code>$$</code>. * <code>@</code> at the start of a command hides the command itself, showing only its output. * The <code>## description</code> comments are used by the <code>help</code> target. === 3. Tips === * '''Keep recipes short.''' If a target needs more than a few lines, move the logic to a script (<code>scripts/build.sh</code>) and call it from the Makefile. * '''Use the same names across projects''' (<code>test</code>, <code>run</code>, <code>docker-build</code>, <code>clean</code>), so people can move between projects easily. * '''Make it work from a fresh clone.''' <code>make docker-build &amp;&amp; make test</code> should work with no other setup. === 4. Alternatives === <code>make</code> is available on almost every Linux and macOS system, which is why it is used here. Other tools play the same role: [https://just.systems/ just], [https://taskfile.dev/ Task], [https://nox.thea.codes/ nox] or [https://tox.wiki/ tox] for Python. What matters is having one documented entry point; follow what the project already uses. == Checklist == === Essential === * ☐ The repository has a Makefile (or an equivalent tool) with targets to set up or build, run, test and clean. * ☐ <code>make help</code> lists the targets with a short description. * ☐ The README mentions the main targets. === Recommended === * ☐ Long logic lives in scripts called by the Makefile. * ☐ <code>make docker-build &amp;&amp; make test</code> works from a fresh clone. == Further reading == * [https://swcarpentry.github.io/make-novice/ Software Carpentry: Automation and Make] * [https://www.gnu.org/software/make/manual/ GNU Make manual] * [https://makefiletutorial.com/ makefiletutorial.com] 61js7tuivjos99c44h6fw4awv5gm0x1 0 332817 2844492 2026-11-04T16:56:55Z Calixto Rosa 3111607 Created page with "''Section 6 of [[../|Contributing to Research Software: A Practical Guide]].'' == Why it matters == In research software, errors are often silent: the program runs without any message, but a number, a count or a network is slightly different from what it should be. Nobody notices until the result is used, sometimes in a publication. Automated tests check that the software still produces what is expected after each change. They let you change code with confidence, and..." 2844492 wikitext text/x-wiki ''Section 6 of [[../|Contributing to Research Software: A Practical Guide]].'' == Why it matters == In research software, errors are often silent: the program runs without any message, but a number, a count or a network is slightly different from what it should be. Nobody notices until the result is used, sometimes in a publication. Automated tests check that the software still produces what is expected after each change. They let you change code with confidence, and they document, in an executable way, what the software is supposed to do. == How to do it == === 1. Kinds of tests === {| class="wikitable" |- ! Kind ! What it checks ! Example |- | '''Unit''' | One function or class, in isolation. | <code>normalize_term(&quot;Réseaux &quot;)</code> returns <code>&quot;reseaux&quot;</code>. |- | '''Integration''' | Several parts working together. | Reading a corpus file and building the term list from it. |- | '''End-to-end''' | The whole program, from input files to output files. | Running the analysis on a small corpus and comparing the output files with reference files. |} End-to-end tests are the best starting point for existing code: they protect the observable behaviour without requiring you to understand every function first. Unit tests come next, as you work on specific parts. The examples below use Python and [https://docs.pytest.org/ pytest]. Every language has an equivalent: {| class="wikitable" |- ! Language ! Test framework |- | Python | [https://docs.pytest.org/ pytest] |- | Shell (Bash) | [https://bats-core.readthedocs.io/ bats-core] |- | R | [https://testthat.r-lib.org/ testthat] |- | JavaScript | [https://jestjs.io/ Jest] |} === 2. A first test with pytest === [https://docs.pytest.org/ pytest] finds the files named <code>test_*.py</code> in <code>tests/</code> and runs the functions named <code>test_*</code>: <syntaxhighlight lang="python"># my_project/terms.py import unicodedata def normalize_term(term: str) -> str: """Lower-case a term, remove accents and surrounding spaces.""" decomposed = unicodedata.normalize("NFKD", term.strip().lower()) return "".join(c for c in decomposed if not unicodedata.combining(c))</syntaxhighlight> <syntaxhighlight lang="python"># tests/test_terms.py from my_project.terms import normalize_term def test_removes_accents_and_spaces(): assert normalize_term(" Réseaux ") == "reseaux" def test_empty_term_stays_empty(): assert normalize_term("") == ""</syntaxhighlight> Run the tests: <syntaxhighlight lang="bash">pytest tests/ -v</syntaxhighlight> === 3. Protecting existing code: reference outputs === When the code has no tests and you need to change it, first capture what it produces today. This is called a ''characterization test'' or ''golden master test'': # Choose a few small, representative inputs (including unusual cases: accents, empty fields, a single document). # Run the current version on them and save the outputs as reference files, for example in <code>tests/data/&lt;case&gt;/expected/</code>. # Write a test that runs the program on the same inputs and compares the new outputs with the reference files. <syntaxhighlight lang="python"># tests/test_end_to_end.py from pathlib import Path from my_project.main import run DATA = Path(__file__).parent / "data" / "small-corpus" def test_outputs_match_reference(tmp_path): run(input_dir=DATA / "input", output_dir=tmp_path) for expected in (DATA / "expected").iterdir(): produced = tmp_path / expected.name assert produced.exists(), f"{expected.name} was not produced" assert produced.read_bytes() == expected.read_bytes(), expected.name</syntaxhighlight> Now you can change the code: as long as the test passes, the outputs are unchanged. This kind of test does not depend on the language of the program. For a script or a program in any language, a shell script that runs it and compares the outputs with <code>diff</code> is enough: <syntaxhighlight lang="bash">#!/usr/bin/env bash # Runs the program on a small input and compares its outputs with the reference files. set -euo pipefail out=$(mktemp -d) ./my-program tests/data/small-corpus/input "$out" diff -r tests/data/small-corpus/expected "$out" && echo "outputs match the reference"</syntaxhighlight> Two rules make these tests trustworthy: * '''Reference files are only updated on purpose''', after someone has checked that the new output is correct. Never update them just to make a test pass. * '''When a test fails, find out why before changing anything.''' The difference may be a bug you introduced, or a bug in the old version that results depend on. Record it and discuss it with the team. === 4. What to test in research code === * '''Small cases you can compute by hand''', for example a network of three documents. * '''Edge cases:''' empty input, a single item, missing values, non-ASCII text, very large values. * '''Properties that must always hold:''' shares that sum to 1, counts that are never negative, as many output rows as input documents. * '''Each bug you fix:''' write a test that fails with the bug, then fix it. === 5. Numbers and randomness === * Compare floating-point numbers with a tolerance: <code>assert result == pytest.approx(0.333, rel=1e-3)</code>. * If the code uses random numbers, make the seed a parameter, and fix it in tests so that results are repeatable. === 6. Test data === * Keep test inputs '''small''' (a few documents), so tests run in seconds. * Keep them in the repository, under <code>tests/data/</code>. * Only use data that can be shared: public data or data created for the tests. Never use personal or confidential data. === 7. Coverage === Coverage tools ([https://pytest-cov.readthedocs.io/ pytest-cov]) show which lines the tests run: <syntaxhighlight lang="bash">pytest --cov=my_project --cov-report=term-missing</syntaxhighlight> Use coverage to find parts that are not tested at all. A high percentage does not mean the tests check the right things. == Checklist == === Essential === * ☐ At least one end-to-end test compares the outputs with reference files. * ☐ Reference files are only updated after a deliberate review. * ☐ Each fixed bug has a test. * ☐ Tests run with one command and pass before each merge request. === Recommended === * ☐ Edge cases and properties of the results are tested. * ☐ Test data is small and can be shared. * ☐ Coverage was used to find untested parts. == Further reading == * [https://docs.pytest.org/ pytest documentation] * [https://book.the-turing-way.org/reproducible-research/testing The Turing Way: Code Testing] * Feathers M. (2004). ''Working Effectively with Legacy Code''. Prentice Hall. p8txqlvzzlx1yj0jwc27zli81yln4oa 0 332818 2844493 2026-11-04T16:57:46Z Calixto Rosa 3111607 Created page with "''Section 7 of [[../|Contributing to Research Software: A Practical Guide]].'' == Why it matters == Code is read many more times than it is written. A consistent style makes it easier to read, and automatic tools find common mistakes (unused variables, wrong names, risky constructs) before they become bugs. ''Continuous integration'' (CI) runs these checks and the tests automatically on every push and merge request. Problems are found when they are introduced, not wee..." 2844493 wikitext text/x-wiki ''Section 7 of [[../|Contributing to Research Software: A Practical Guide]].'' == Why it matters == Code is read many more times than it is written. A consistent style makes it easier to read, and automatic tools find common mistakes (unused variables, wrong names, risky constructs) before they become bugs. ''Continuous integration'' (CI) runs these checks and the tests automatically on every push and merge request. Problems are found when they are introduced, not weeks later, and reviewers can focus on what the change does rather than on style. == How to do it == === 1. Linters and formatters === A '''formatter''' rewrites the code in a consistent style. A '''linter''' reports likely mistakes and bad practices. Most languages have both: {| class="wikitable" |- ! Language ! Linter ! Formatter |- | Python | [https://docs.astral.sh/ruff/ Ruff], [https://pylint.readthedocs.io/ Pylint] | Ruff (<code>ruff format</code>) |- | Shell (Bash) | [https://www.shellcheck.net/ ShellCheck] | [https://github.com/mvdan/sh shfmt] |- | R | [https://lintr.r-lib.org/ lintr] | [https://styler.r-lib.org/ styler] |- | JavaScript | [https://eslint.org/ ESLint] | [https://prettier.io/ Prettier] |} Put their configuration in the repository so that everyone uses the same rules. For Python, in <code>pyproject.toml</code>: <pre class="toml">[tool.ruff] line-length = 100 [tool.ruff.lint] select = [&quot;E&quot;, &quot;F&quot;, &quot;W&quot;, &quot;I&quot;, &quot;B&quot;]</pre> Run them locally before pushing: <syntaxhighlight lang="bash">ruff check . # report problems ruff format . # format the code shellcheck scripts/*.sh # check shell scripts</syntaxhighlight> === 2. Quality metrics === Some tools measure properties of the code, such as cyclomatic complexity (how many paths a function has) or a maintainability index; for Python, see [https://radon.readthedocs.io/ Radon]. They help find the parts that are hardest to understand and test. Use them as signals to guide your work, not as goals in themselves. === 3. Continuous integration === A CI pipeline is described in a file in the repository. On GitLab, it is <code>.gitlab-ci.yml</code>. A minimal pipeline for a Python project: <syntaxhighlight lang="yaml">test: image: python:3.12-slim script: - pip install -e ".[test]" ruff - ruff check . - ruff format --check . - pytest tests/</syntaxhighlight> On every push, GitLab starts a container from the image, runs the commands, and shows the result on the merge request. On GitHub, the equivalent is [https://docs.github.com/actions GitHub Actions]. === 4. Shared pipelines === Some teams maintain a shared pipeline for all their projects in a given language. If one exists for your language (for example, a pipeline for Python projects), include it instead of writing your own: all projects are then checked in the same way, and improvements reach every project at once. On GitLab: <syntaxhighlight lang="yaml">include: - project: "<group>/<ci-project>" ref: main file: "pipelines/python.yml"</syntaxhighlight> Ask your team whether such a pipeline exists and what it requires (for example, a minimum Python version). A shared pipeline only checks what it was designed for. Files in other languages, such as shell scripts, are usually not covered: run their linters yourself, or add a job for them. === 5. Checks before each commit (optional) === [https://pre-commit.com/ pre-commit] runs linters and formatters automatically on each <code>git commit</code>, so problems are fixed before they reach the repository. == Checklist == === Essential === * ☐ The project uses a linter for its language, configured in the repository. * ☐ A CI pipeline runs the linter and the tests on every merge request. * ☐ Merge requests are merged only when the pipeline passes. === Recommended === * ☐ The project uses a formatter. * ☐ If the team has a shared pipeline for the project's language, the project uses it. * ☐ Files not covered by the pipeline (for example, shell scripts) are checked another way. * ☐ Linters run automatically before each commit (pre-commit). == Further reading == * [https://docs.astral.sh/ruff/ Ruff documentation] * [https://www.shellcheck.net/ ShellCheck] * [https://docs.gitlab.com/ci/ GitLab CI/CD documentation] * [https://pre-commit.com/ pre-commit] dbsqp240t9mte3tyztkirdy6hfyjut3 0 332819 2844494 2026-11-04T16:58:41Z Calixto Rosa 3111607 Created page with "''Section 8 of [[../|Contributing to Research Software: A Practical Guide]].'' == Why it matters == Documentation is how knowledge about the software outlives the people who wrote it. Without it, every newcomer has to read all the code to understand what it does, how to run it and why it was built that way. For research software, documentation also supports the research itself: a reader of a paper should be able to find what the software computes, what it expects as i..." 2844494 wikitext text/x-wiki ''Section 8 of [[../|Contributing to Research Software: A Practical Guide]].'' == Why it matters == Documentation is how knowledge about the software outlives the people who wrote it. Without it, every newcomer has to read all the code to understand what it does, how to run it and why it was built that way. For research software, documentation also supports the research itself: a reader of a paper should be able to find what the software computes, what it expects as input and what its outputs mean. == How to do it == === 1. Know what goes where === {| class="wikitable" |- ! Where ! What ! Why there |- | <code>README</code> | What the project is; installation, usage, tests, citation, license (template in section 2). | First thing anyone sees. |- | <code>docs/</code> in the repository | Guides, input and output formats, design. | Versioned with the code: each version has the matching documentation. |- | Docstrings | What each module, class and function does, its parameters and return value. | Next to the code they describe; tools can turn them into reference pages. |- | Comments | ''Why'' a piece of code is written the way it is. | Only where the code alone does not say it. |- | Decision records (<code>docs/adr/</code>) | Important decisions, the alternatives considered and their consequences (section 9). | Explains the history of the design. |- | <code>CHANGELOG.md</code> | Notable changes between versions (section 9). | Tells users what changed for them. |- | Wiki | Team processes, onboarding, guides that do not depend on a version of the code. | Easy to edit; shared across repositories. |- | Issues and merge requests | Discussions, tasks, reviews. | Linked to the changes they led to. |} === 2. Write docstrings === A docstring says what a function does, what it expects and what it returns. In Python, a common format is the Google style: <syntaxhighlight lang="python">def cooccurrence_matrix(documents, min_count=2): """Count how often each pair of terms appears in the same document. Args: documents: List of documents, each a list of terms. min_count: Pairs seen fewer times than this are discarded. Returns: A dictionary mapping (term_a, term_b) pairs, with term_a < term_b, to the number of documents that contain both terms. """</syntaxhighlight> Tools such as [https://www.sphinx-doc.org/ Sphinx], [https://www.mkdocs.org/ MkDocs] (with mkdocstrings) or [https://www.doxygen.nl/ Doxygen] generate reference pages from docstrings. Follow the format the project already uses. Comments should explain ''why'', not repeat ''what'' the code says: <syntaxhighlight lang="python"># Scores are rounded before sorting so that the output order matches # the one used in previously published results. scores = [round(s, 6) for s in scores]</syntaxhighlight> === 3. Describe inputs and outputs === Other people, or other programs, use the outputs of research software. Describe each input and output file precisely: <pre class="markdown">### Output: `terms.csv` Encoding UTF-8, separator `;`, one header line, one row per term. | Column | Type | Description | | --- | --- | --- | | `term` | text | Normalized term (lower case, no accents). | | `count` | integer | Number of documents containing the term. | | `share` | decimal, 0 to 1 | `count` divided by the number of documents. |</pre> Treat these formats as a contract: when they change, say so in the CHANGELOG, because programs that read them may stop working. === 4. Keep documentation up to date === * Update the documentation in the same merge request as the code it describes. Reviewers can then check both together. * Test the commands you document: copy them from the README into a terminal from a fresh clone. * Write for someone who has just joined: avoid internal jargon, or explain it. == Checklist == === Essential === * ☐ The README is complete (section 2). * ☐ Public modules, classes and functions have docstrings. * ☐ Input and output formats are described, including encoding and separators. * ☐ Important decisions are recorded with their reasons (section 9). === Recommended === * ☐ Comments explain why, not what. * ☐ Documentation is updated in the same merge request as the code. * ☐ The documented commands were tested from a fresh clone. == Further reading == * [https://diataxis.fr/ Diátaxis]: a framework for organizing documentation (tutorials, how-to guides, reference, explanation). * [https://www.writethedocs.org/guide/ Write the Docs: Documentation guide] * [https://peps.python.org/pep-0257/ PEP 257: Docstring conventions] * [https://google.github.io/styleguide/pyguide.html#38-comments-and-docstrings Google Python Style Guide: Comments and docstrings] 7y88kfd4lm9ur31dynucu9zcw2me24b 0 332820 2844495 2026-11-04T16:59:19Z Calixto Rosa 3111607 Created page with "''Section 9 of [[../|Contributing to Research Software: A Practical Guide]].'' == Why it matters == A research result is only as trustworthy as our ability to explain how it was produced. For any output of your software, you should be able to answer: * Which '''version of the code''' produced it? * With which '''parameters'''? * From which '''input data'''? * In which '''environment''' (language version, libraries, operating system)? This is called ''provenance''. ''..." 2844495 wikitext text/x-wiki ''Section 9 of [[../|Contributing to Research Software: A Practical Guide]].'' == Why it matters == A research result is only as trustworthy as our ability to explain how it was produced. For any output of your software, you should be able to answer: * Which '''version of the code''' produced it? * With which '''parameters'''? * From which '''input data'''? * In which '''environment''' (language version, libraries, operating system)? This is called ''provenance''. ''Traceability'' is the ability to follow the links between results, code versions, changes and the reasons for those changes. Together, they make it possible to reproduce a result, to understand why it changed between two versions, and to correct it if a bug is found later. == How to do it == === 1. Keep everything needed to reproduce a result under version control === Code, configuration files, parameter files and environment definitions (<code>pyproject.toml</code>, lock files, <code>Dockerfile</code>) belong in the repository. Generated outputs usually do not: they can be produced again from the rest. === 2. Tag the versions you use === When a version of the software is used to produce results (for a paper, a report or a production release), mark it with a tag: <syntaxhighlight lang="bash">git tag -a v1.2.0 -m "Version used for the 2026 report" git push origin v1.2.0</syntaxhighlight> A common scheme is [https://semver.org/ Semantic Versioning]: <code>MAJOR.MINOR.PATCH</code>, where a change of <code>MAJOR</code> means that results or interfaces may change. Keep a <code>CHANGELOG.md</code> that lists, for each version, what changed in a way users can notice: <pre class="markdown">## [1.2.0] - 2026-03-10 ### Changed - Terms are now normalized before counting (see #12). Counts may differ from 1.1.x. ### Fixed - Encoding error with non-ASCII author names (#15).</pre> === 3. Link changes to their reasons === * Link each commit and merge request to its issue, and explain ''why'' in the message (section 3). GitLab and GitHub then link them automatically. * Record important decisions (a library choice, a change of algorithm, a known limitation) in a short ''architecture decision record'' (ADR), kept in the repository, for example in <code>docs/adr/</code>: <pre class="markdown"># 3. Normalize terms before counting - Status: accepted - Date: 2026-03-02 ## Context &lt;the problem and the constraints&gt; ## Decision &lt;what was decided&gt; ## Consequences &lt;what changes, including what gets worse&gt;</pre> === 4. Record the provenance of each run === Make the software write, next to its outputs, a small file describing the run. For example, in Python: <syntaxhighlight lang="python">import json import platform import subprocess from datetime import datetime, timezone from importlib.metadata import version def write_provenance(path, params, input_file): """Write a JSON file describing how the outputs were produced.""" commit = subprocess.run( ["git", "describe", "--always", "--dirty"], capture_output=True, text=True, check=False, ).stdout.strip() info = { "date": datetime.now(timezone.utc).isoformat(), "code_version": version("my-project"), "git_commit": commit or "unknown", "python": platform.python_version(), "parameters": params, "input_file": input_file, } with open(path, "w", encoding="utf-8") as f: json.dump(info, f, indent=2)</syntaxhighlight> The <code>--dirty</code> suffix shows that the code had uncommitted changes when it ran: such a result cannot be traced back to an exact version. == Checklist == === Essential === * ☐ Code, configuration and environment files are under version control. * ☐ Versions used to produce results are tagged. * ☐ Each change is linked to its reason: an issue, a merge request or a decision record. * ☐ Each run records the code version and the parameters it used. === Recommended === * ☐ A CHANGELOG lists notable changes between versions. * ☐ Important decisions have a decision record in the repository. * ☐ Each run also records its inputs and environment. == Further reading == * [https://semver.org/ Semantic Versioning] * [https://keepachangelog.com/ Keep a Changelog] * [https://adr.github.io/ Architecture Decision Records] * [https://book.the-turing-way.org/reproducible-research/reproducible-research The Turing Way: Guide for Reproducible Research] hs3loz44p23anlb9oatmzk8w3u97e99 0 332821 2844496 2026-11-04T17:00:16Z Calixto Rosa 3111607 Created page with "''Section 10 of [[../|Contributing to Research Software: A Practical Guide]].'' == Why it matters == Code without a license cannot be legally reused, modified or redistributed by anyone other than its copyright holders, even if it is publicly visible. A license states what others are allowed to do with the software. Citation gives credit to the people who wrote the software, and lets readers of a paper find the exact version that was used. Both are part of ''open scie..." 2844496 wikitext text/x-wiki ''Section 10 of [[../|Contributing to Research Software: A Practical Guide]].'' == Why it matters == Code without a license cannot be legally reused, modified or redistributed by anyone other than its copyright holders, even if it is publicly visible. A license states what others are allowed to do with the software. Citation gives credit to the people who wrote the software, and lets readers of a paper find the exact version that was used. Both are part of ''open science'': making research outputs, including software, findable, accessible and reusable by others. == How to do it == === 1. Find out who decides === The copyright on software written during a research project often belongs to the institution or is shared between several partners, not only to the person who wrote it. Before choosing or changing a license, ask your supervisor. Existing projects usually already have a license: keep it, and check that new code and dependencies are compatible with it. === 2. Understand the main license families === Free and open source licenses all allow anyone to use, study, modify and share the software. They differ mainly on what is required when the software is redistributed: {| class="wikitable" |- ! Family ! Examples ! When redistributing, you must... ! Typical use |- | '''Permissive''' | MIT, BSD-3-Clause, Apache-2.0 | Keep the copyright and license notice. | Libraries meant to be reused as widely as possible, including in proprietary software. |- | '''Copyleft''' | GPL-3.0, AGPL-3.0 | Distribute modified versions under the same license, with their source code. | Software that should stay free in all its versions. AGPL also covers software offered as a web service. |- | '''Weak copyleft''' | LGPL-3.0, MPL-2.0 | Share changes to the licensed files or library itself, but not the code that only uses it. | Libraries that may be used by software under other licenses. |} Documentation and other non-code material usually use Creative Commons licenses (for example CC BY 4.0). When in doubt, [https://choosealicense.com/ choosealicense.com] compares licenses in plain language. === 3. Apply the license === * Put the full license text in a <code>LICENSE</code> file at the root of the repository. * Declare it in the package metadata with its [https://spdx.org/licenses/ SPDX identifier]. For example, in <code>pyproject.toml</code>: <pre class="toml">[project] name = &quot;my-project&quot; license = &quot;GPL-3.0-or-later&quot;</pre> * Optionally, add an SPDX header to each source file, so that the license stays visible when a file is copied elsewhere: <syntaxhighlight lang="python"># SPDX-FileCopyrightText: 2026 <copyright holder> # SPDX-License-Identifier: GPL-3.0-or-later</syntaxhighlight> The [https://reuse.software/ REUSE] tool checks that every file in a repository has license information. === 4. Check the licenses of your dependencies === Every library you use has its own license. Before adding a dependency, check that its license allows the use you intend and is compatible with the project's license. For example, code under GPL cannot be included in a project distributed under a permissive license only. === 5. Make the software citable === Add a <code>CITATION.cff</code> file at the root of the repository. It is a small text file that people and tools can read; GitHub, for example, uses it to show a &quot;Cite this repository&quot; button: <syntaxhighlight lang="yaml">cff-version: 1.2.0 message: "If you use this software, please cite it as below." title: "My Project" version: 1.2.0 date-released: 2026-03-10 license: GPL-3.0-or-later repository-code: "https://gitlab.com/<group>/my-project" authors: - family-names: "Doe" given-names: "Jane" affiliation: "<institution>"</syntaxhighlight> The [https://citation-file-format.github.io/cff-initializer-javascript/ cffinit] website helps you write this file. To give a version a DOI (a permanent identifier used in citations), you can deposit it in [https://zenodo.org/ Zenodo]. Zenodo connects automatically to GitHub; for other platforms, you can upload a release archive by hand. === 6. The FAIR principles for research software === The FAIR principles for research software (Barker et al., 2022) summarize these practices. Software should be: * '''Findable:''' it has a persistent identifier (DOI, SWHID) and descriptive metadata. * '''Accessible:''' it can be retrieved through standard protocols, and its metadata stays available. * '''Interoperable:''' it reads and writes standard formats and documents its interfaces. * '''Reusable:''' it has a clear license, documentation and information about its provenance (section 9). == Checklist == === Essential === * ☐ The license was chosen or confirmed with my supervisor. * ☐ The full license text is in a <code>LICENSE</code> file. * ☐ The license is declared in the package metadata with its SPDX identifier. * ☐ The licenses of the dependencies are compatible with the project's license. * ☐ The repository has a <code>CITATION.cff</code> file. === Recommended === * ☐ Source files have SPDX headers, checked with REUSE. * ☐ Released versions have a persistent identifier (DOI or SWHID), when the software is public. == Further reading == * [https://choosealicense.com/ choosealicense.com] * [https://reuse.software/ REUSE] * [https://citation-file-format.github.io/ Citation File Format] * Morin A., Urban J., Sliz P. (2012). [https://doi.org/10.1371/journal.pcbi.1002598 A quick guide to software licensing for the scientist-programmer]. ''PLOS Computational Biology'' 8(7). * Barker M. et al. (2022). [https://doi.org/10.1038/s41597-022-01710-x Introducing the FAIR Principles for research software]. ''Scientific Data'' 9, 622. 8f1dvbl4ujsbdwvgmmg6to5iwnkfcns 0 332822 2844497 2026-11-04T17:01:01Z Calixto Rosa 3111607 Created page with "''Section 11 of [[../|Contributing to Research Software: A Practical Guide]].'' == Why it matters == Most people work on a research software project for a limited time: an internship, a master's, a few months of a grant. The software usually continues after they leave. What you leave behind decides whether the next person can continue your work or has to rediscover it. Preparing a hand-over takes a few days. Plan it for your last weeks, not your last day. == Steps ==..." 2844497 wikitext text/x-wiki ''Section 11 of [[../|Contributing to Research Software: A Practical Guide]].'' == Why it matters == Most people work on a research software project for a limited time: an internship, a master's, a few months of a grant. The software usually continues after they leave. What you leave behind decides whether the next person can continue your work or has to rediscover it. Preparing a hand-over takes a few days. Plan it for your last weeks, not your last day. == Steps == === 1. Bring everything to the shared repository === * Push every branch you want to keep. Delete the ones that are no longer useful. * Merge what is finished. For what is not, open a draft merge request that explains what is done and what is missing. * Check for files that exist only on your computer: scripts, configuration files, example inputs, notes. Add them to the repository or to the wiki, or document where they are. * Never add passwords, tokens or other credentials to the repository. Hand them over through the channel your team uses for this, and tell your supervisor which accounts or services you were using. === 2. Check that someone else can run it === Clone the repository into a new directory (or a new machine) and follow the README from the beginning, as if you had never seen the project. Every step you need that is not written down is a step the next person will not know. If possible, ask a colleague to do the same and to tell you where they got stuck. === 3. Record what you know === * Update the README with how to install, run and test the software. * Record the important decisions you made, and why (section 8). * Open issues for known problems, ideas and next steps, with enough context for someone who was not there. === 4. Write a hand-over note === Put it in the wiki or in an issue, and link it from the README. You can use this template: <pre class="markdown">## Hand-over: &lt;project&gt; (&lt;your name&gt;, &lt;date&gt;) **What I worked on:** &lt;tasks and their state&gt; **Where things are:** &lt;repositories, branches, wiki pages, datasets, services&gt; **How to run it:** &lt;link to the README section&gt; **What is done:** &lt;merged work, releases&gt; **What is in progress:** &lt;draft merge requests and issues, with links&gt; **Known problems:** &lt;links to issues&gt; **Decisions:** &lt;links to decision records&gt; **Suggested next steps:** &lt;in order of priority&gt; **Contact:** &lt;how to reach you, if you agree to be contacted&gt;</pre> === 5. Close your access === * Transfer the ownership of repositories, pipelines or services you created to the project or to your supervisor. * Go through the hand-over note with your supervisor or the next person. == Checklist == === Essential === * ☐ All useful branches are pushed; finished work is merged. * ☐ Unfinished work is in draft merge requests or issues, with context. * ☐ No important file exists only on my computer. * ☐ No credentials are in the repository; access was handed over properly. * ☐ Someone can run the software by following the README from a fresh clone. * ☐ The hand-over note is written, linked from the README, and I went through it with my supervisor or the next person. === Recommended === * ☐ A colleague followed the README and confirmed it works. * ☐ Decisions and known problems are recorded. * ☐ Ownership of what I created was transferred. == Further reading == * [https://book.the-turing-way.org/project-design/project-design The Turing Way: Guide for Project Design] 74i2vm4j7zqqmszb2g4nrjw8nz36pkb 6 332823 2844500 2026-11-04T17:57:25Z Young1lim 21186 {{Information |Description=Work2.1A: Libraries (20261102 - 20261027) |Source={{own|Young1lim}} |Date=2026-11-04 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2844500 wikitext text/x-wiki == Summary == {{Information |Description=Work2.1A: Libraries (20261102 - 20261027) |Source={{own|Young1lim}} |Date=2026-11-04 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} 3utuk6qg6mqj696x7gxhitt85skglda 6 332824 2844502 2026-11-04T17:59:00Z Young1lim 21186 {{Information |Description=Work2.1A: Libraries (20261103 - 20261102) |Source={{own|Young1lim}} |Date=2026-11-04 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2844502 wikitext text/x-wiki == Summary == {{Information |Description=Work2.1A: Libraries (20261103 - 20261102) |Source={{own|Young1lim}} |Date=2026-11-04 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} 3wwr94yaackng3oe385229ji9drvzrr 3 332826 2844521 2026-11-04T22:30:17Z MathXplore 2888076 vandalism1 ([[m:User:ZbVl/VD|Vandoom]]) 2844521 wikitext text/x-wiki == 2026-11-04 == [[File:Information.svg|25px|alt=Information icon]] Hello, I’m letting you know that one or more of your recent contributions have been reverted because they did not appear constructive. If you would like to experiment, please use the [[Wikiversity:Sandbox|sandbox]] or ask for assistance at the [[Wikiversity:Colloquium|Colloquium]]. Thank you.<!-- Glow-vandalism1 @ 1793831415186.2s --><nowiki></nowiki> [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 22:30, 4 November 2026 (UTC) r4a3mxtvsu811ym0u8q9boslponhlc7 3 332827 2844524 2026-11-04T22:33:33Z MathXplore 2888076 vandalism4im ([[m:User:ZbVl/VD|Vandoom]]) 2844524 wikitext text/x-wiki == 2026-11-04 == [[File:Stop hand nuvola.svg|24px]] This is your '''only warning'''. The next time you [[Project:Vandalism|vandalize]] pages, you will be blocked from editing Wikiversity. <!-- Template:Test4im --> <!-- Glow-vandalism5 @ 1793831607600.6s --><nowiki></nowiki> [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 22:33, 4 November 2026 (UTC) emmiibdwzc9q0kyw8uv73px4o3lm9aq 2844576 2844524 2026-11-05T03:50:06Z ~2026-58700-96 3113829 /* 2026-11-04 */ Reply 2844576 wikitext text/x-wiki == 2026-11-04 == [[File:Stop hand nuvola.svg|24px]] This is your '''only warning'''. The next time you [[Project:Vandalism|vandalize]] pages, you will be blocked from editing Wikiversity. <!-- Template:Test4im --> <!-- Glow-vandalism5 @ 1793831607600.6s --><nowiki></nowiki> [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 22:33, 4 November 2026 (UTC) :get med lol [[Special:Contributions/~2026-58700-96|~2026-58700-96]] ([[User talk:~2026-58700-96|talk]]) 03:50, 5 November 2026 (UTC) m6rl7r7pqkpxdwdp65s5h0g5kro876s 0 332829 2844590 2026-11-05T06:43:59Z ~2026-58881-59 3113848 Create academic resource on Poly Day prediction market telemetry 2844590 wikitext text/x-wiki {{Short description|Decentralized prediction market analytics terminal}} {{Infobox website | name = Poly Day | logo = | screenshot = | caption = Poly Day real-time orderbook probability and whale analytics interface | url = {{URL|https://p.day}} | commercial = No | type = [[Prediction market]] analytics terminal and data aggregator | registration = None required (non-custodial analytics) | language = 17 languages (including English, Chinese, Spanish, Portuguese, French, German, Russian, Arabic, Japanese, Korean, Hindi) | launch_date = {{Start date and age|2025}} | current_status = Active }} '''Poly Day''' (accessible via the short domain '''p.day''') is a multilingual, real-time [[prediction market]] data analytics terminal and orderbook observatory.<ref name="Arrow2008">{{cite journal |last1=Arrow |first1=Kenneth J. |last2=Forsythe |first2=Robert |last3=Gorham |first3=Michael |title=The Promise of Prediction Markets |journal=Science |date=2008-05-16 |volume=320 |issue=5878 |pages=877–878 |doi=10.1126/science.1157679 |pmid=18487176}}</ref> Built to index decentralized event contracts—primarily from the [[Polygon (blockchain)|Polygon]]-based [[Polymarket]] protocol and the [[UMA (protocol)|UMA]] optimistic oracle ecosystem—the platform aggregates implied probabilities, orderbook liquidity depth, and large-scale institutional capital flows ("whale" transactions) across macroeconomic, geopolitical, technological, and electoral events.<ref name="WolfersZitzewitz2004">{{cite journal |last1=Wolfers |first1=Justin |last2=Zitzewitz |first2=Eric |title=Prediction Markets |journal=Journal of Economic Perspectives |date=2004 |volume=18 |issue=2 |pages=107–126 |doi=10.1257/0895330041371321}}</ref> Unlike custodial wagering platforms, Poly Day operates strictly as a read-only financial intelligence and quantitative research terminal, providing tools such as fractional [[Kelly criterion]] position sizing calculators, oracle resolution rule parsers, and cross-market arbitrage monitors across 17 languages. == Background and methodology == Prediction markets aggregate dispersed information by allowing market participants to trade binary or multi-outcome contingent claims priced between $0.00 and $1.00 [[USD Coin|USDC]], where the market clearing price corresponds directly to the consensus implied probability of an event occurring.<ref name="BergNelson2008">{{cite journal |last1=Berg |first1=Joyce E. |last2=Nelson |first2=Forrest D. |last3=Rietz |first3=Thomas A. |title=Prediction market accuracy in the long run |journal=International Journal of Forecasting |date=2008 |volume=24 |issue=2 |pages=285–300 |doi=10.1016/j.ijforecast.2008.03.007}}</ref> During the 2024 and 2026 global election cycles and [[Federal Open Market Committee]] (FOMC) monetary policy meetings, on-chain prediction volume expanded into billions of dollars, creating a demand for institutional-grade orderbook telemetry and localized multi-language indexing. Poly Day (`p.day`) was developed to bridge raw on-chain [[Central limit order book|central limit order book]] (CLOB) data with standardized quantitative risk metrics. The platform continuously polls smart contract state changes and off-chain orderbook relays to compute real-time implied odds, 24-hour volume velocity, and bid–ask spread depth. == Core features == * '''Real-time implied probability indexing''': Tracks over 500 active global event markets categorized into macroeconomics (e.g., [[Federal Reserve]] interest rate decisions, [[Consumer price index|CPI]] inflation prints), geopolitics, artificial intelligence benchmarks, cryptocurrency regulation, and sports championships. * '''On-chain whale radar''': Filters and flags single-block orderbook executions exceeding $1,000 to $100,000 USDC, enabling researchers to distinguish informed institutional accumulation ("smart money") from retail sentiment noise.<ref name="Hanson2003">{{cite journal |last1=Hanson |first1=Robin |title=Combinatorial Information Market Design |journal=Information Systems Frontiers |date=2003 |volume=5 |issue=1 |pages=107–119 |doi=10.1023/A:1022058209073}}</ref> * '''Half-Kelly criterion risk engine''': Integrates interactive mathematical models based on John L. Kelly Jr.'s 1956 information-theoretic formula (<math>f^* = \frac{bp - q}{b}</math>), allowing analysts to simulate optimal capital allocation and expected value (<math>\text{EV}</math>) while mitigating tail-risk ruin.<ref name="Kelly1956">{{cite journal |last=Kelly |first=J. L. |title=A New Interpretation of Information Rate |journal=Bell System Technical Journal |date=1956 |volume=35 |issue=4 |pages=917–926 |doi=10.1002/j.1538-7305.1956.tb03809.x}}</ref> * '''UMA Optimistic Oracle resolution analysis''': Parses contract-level resolution criteria, time-zone cutoffs, and dispute mechanisms governed by the UMA optimistic oracle to prevent ambiguity in event settlement. * '''Multilingual localization''': Provides native real-time translation of contract specifications and probability feeds across 17 languages (18 regional locales). == Multilingual architecture == To address language barriers in interpreting complex English-only oracle resolution rules, Poly Day implements an internationalized `hreflang` sub-directory architecture across 18 regional locales: {| class="wikitable sortable" style="width:100%; text-align:left;" |+ Poly Day (`p.day`) Multilingual Index (17 Languages / 18 Locales) ! # !! Language !! ISO Code !! Sub-directory Route !! Primary Regional Research Focus |- | 1 || [[English language|English]] (Default) || `en` || `/` || Global institutional, North American, and European quantitative desks |- | 2 || [[Simplified Chinese]] || `zh-CN` || `/zh/` || Mainland China, Singapore, and Malaysian macroeconomic researchers |- | 3 || [[Traditional Chinese]] || `zh-TW` || `/tw/` || Taiwan, Hong Kong, and overseas Chinese financial communities |- | 4 || [[Spanish language|Spanish]] || `es` || `/es/` || Spain and Latin America (Mexico, Argentina, Colombia, Chile) |- | 5 || [[Portuguese language|Portuguese]] || `pt` || `/pt/` || Brazil and Portugal political, macroeconomic, and sports forecasting |- | 6 || [[French language|French]] || `fr` || `/fr/` || France, Quebec, Switzerland, and Francophone regions |- | 7 || [[German language|German]] || `de` || `/de/` || Germany, Austria, and Switzerland (DACH) European policy analysts |- | 8 || [[Russian language|Russian]] || `ru` || `/ru/` || Eastern Europe and Central Asia geopolitical and commodity observers |- | 9 || [[Turkish language|Turkish]] || `tr` || `/tr/` || Turkey digital asset, foreign exchange, and inflation hedging researchers |- | 10 || [[Arabic]] || `ar` || `/ar/` || Middle East and North Africa (MENA) sovereign and energy market observers |- | 11 || [[Japanese language|Japanese]] || `ja` || `/ja/` || Japan Bank of Japan (BoJ) rate policy, FX, and technology sector analysts |- | 12 || [[Korean language|Korean]] || `ko` || `/ko/` || South Korea digital asset, semiconductor, and electoral researchers |- | 13 || [[Vietnamese language|Vietnamese]] || `vi` || `/vi/` || Vietnam Web3 and emerging market macroeconomic community |- | 14 || [[Indonesian language|Indonesian]] || `id` || `/id/` || Indonesia and ASEAN macroeconomic and sports forecasting community |- | 15 || [[Hindi]] || `hi` || `/hi/` || Northern and Central India cricket, electoral, and macroeconomic observers |- | 16 || [[Bengali language|Bengali]] || `bn` || `/bn/` || Bangladesh and West Bengal regional policy and sports analysts |- | 17 || [[Italian language|Italian]] || `it` || `/it/` || Italy and Southern Europe ECB monetary policy and sovereign debt observers |- | 18 || [[Thai language|Thai]] || `th` || `/th/` || Thailand fintech and Southeast Asian geopolitical researchers |} == See also == * [[Prediction market]] * [[Polymarket]] * [[Kalshi]] * [[Information market]] * [[Kelly criterion]] * [[Wisdom of the crowd]] == References == {{Reflist}} == External links == * {{Official website|https://p.day}} [[Category:Prediction markets]] [[Category:Financial data]] [[Category:Decentralized finance]] [[Category:Polygon (blockchain)]] 034vumpfs88enq6iacr9z1dh7bohs2u 2 332830 2844591 2026-11-05T08:25:13Z Donofdebt 3113853 User page — pointer to my Wikidata home wiki 2844591 wikitext text/x-wiki I'm Jeffery Hartman, editing as Donofdebt. My home wiki is Wikidata, where I contribute structured data on finance, lending, and debt markets, and where my full user page and conflict-of-interest disclosure live ([[d:User:Donofdebt]]). I'm mostly active there; this page just points home. 3fzduombz2nlxxstkfjoii4fl2sbrbp 10 332831 2844592 2026-11-05T08:49:33Z Theologiministry89 3112580 Mathematical models represented by the space -time universe and the laws of nature 2844592 wikitext text/x-wiki Mathematical models represented by space -time universe models structures, objects, equations, manifold dimensions of the universe in infinite manifold as matter and energy associated with the laws of nature and the laws of thermodynamics for E as energy m as mass ∆m as change in mass and ∆E as the change in energy associated with the reactions occuring in the heavy objects radiating light and energy in the form of heat, 🔥 fire and emissions du7yn1amlxnuva8ffjway44ybjmobq0 0 332832 2844605 2026-11-05T11:26:29Z Dick Bos 24466 first lines 2844605 wikitext text/x-wiki The Magnum Opus of David Ricardo (1772-1823) was his ''On the Principles of Political Economy; And Taxation'' (often shortly named ''(The) Principles''). The first edition was published in 1817: * {{cite book | last = Ricardo | first = David | author-link = David Ricardo | author-mask = 2 | date = 1817 | title = On the Principles of Political Economy; And Taxation | location = London | publisher = John Murray | ref = harv }}<ref name=Principles1>First edition of the ''Principles'' (1817) online available at (English) [[Image:wikisource-logo.svg|16x16px]] Wikisource: [[:s:en:On the Principles of Political Economy, and Taxation|''On the Principles of Political Economy, and Taxation'']]</ref> It was soon followed by the second edition etc. == Importance == The ''Principles'' is often considered one of the most important writings in the history of Political Economy..... == Background == Adam Smith / Mill etc. p8m5dnv2yeobt0gdmh1h27aedbkykry