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 2844613 2844604 2026-11-05T12:40:49Z MathXplore 2888076 Reverted edits by [[Special:Contributions/Theologiministry89|Theologiministry89]] ([[User_talk:Theologiministry89|talk]]) to last version by [[User:~2026-56788-08|~2026-56788-08]] using [[Wikiversity:Rollback|rollback]] 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 102 1808 2844646 2844584 2026-11-05T17:25:26Z ~2026-59142-52 3113866 2844646 wikitext text/x-wiki 2844647 2844646 2026-11-05T17:27:03Z ~2026-59142-52 3113866 2844647 wikitext text/x-wiki 2844648 2844647 2026-11-05T17:28:05Z ~2026-59142-52 3113866 LT1MS 2844648 wikitext text/x-wiki = edits courtesy of LT1MS = {{proposed deletion|since a large portion of Wikiversity are learning materials, it is not clear what a portal is too achieve here}} = I eat dirt -mathxplore<!-- This portal was created using subst:box portal skeleton --> = <div style="float:right; width:100%"> = poop poopty scoop poop poop mm yummy poop -MathXplore = {{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}}]] bjchjkrrrws9pdmxv9xt292ac9qptr2 2844649 2844648 2026-11-05T17:28:54Z ~2026-59142-52 3113866 2844649 wikitext text/x-wiki 2844650 2844649 2026-11-05T17:31:29Z ~2026-59142-52 3113866 lollolol 2844650 wikitext text/x-wiki = edits courtesy of LT1MS MATHXPLORE SUX!!!!!! MATERIALSCIENTIST SUX!!!!! seriously this is what you get for letting anyone edit = {{proposed deletion|since a large portion of Wikiversity are learning materials, it is not clear what a portal is too achieve here}} = I eat dirt -mathxplore<!-- This portal was created using subst:box portal skeleton --> = <div style="float:right; width:100%"> = poop poopty scoop poop poop mm yummy poop -MathXplore = {{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}}]] mjbwsuvkqnoa785l295ugvowpdj9kza 2844651 2844650 2026-11-05T17:31:53Z Quinlan83 2913823 Reverted edits by [[Special:Contributions/~2026-59142-52|~2026-59142-52]] ([[User_talk:~2026-59142-52|talk]]) to last version by [[User:Morkoz|Morkoz]] 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 2844652 2844651 2026-11-05T17:35:43Z ~2026-59142-52 3113866 LT1MS 2844652 wikitext text/x-wiki = MORKOZ SUX!!!!! LOL YOU CAN'T GET ME!!! I'LL JUST CHANGE TO A DIFFERENT IP! LOL IN YOUR FACE MORKOZ, MATHXPLORE, AND MATERIALSCIENTIST! - LT1MS! = {{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}}]] 475nxcwbqfn5b9237xgwacus1pwja6u 2844653 2844652 2026-11-05T17:35:49Z Quinlan83 2913823 Reverted edit by [[Special:Contributions/~2026-59142-52|~2026-59142-52]] ([[User_talk:~2026-59142-52|talk]]) to last version by [[User:Quinlan83|Quinlan83]] 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 2844655 2844653 2026-11-05T17:36:24Z ~2026-59142-52 3113866 MMM 2844655 wikitext text/x-wiki 2844656 2844655 2026-11-05T17:36:53Z Quinlan83 2913823 Restored revision 2844584 by [[Special:Contributions/Morkoz|Morkoz]] ([[User talk:Morkoz|talk]]): Restore (TwinkleGlobal) 2844656 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 2844664 2844656 2026-11-05T19:55:38Z Codename Noreste 2969951 Protected "[[Portal:Learning Materials]]": Excessive vandalism ([Edit=Allow only autoconfirmed users] (expires 19:55, 5 February 2027 (UTC)) [Move=Allow only autoconfirmed users] (expires 19:55, 5 February 2027 (UTC))) 2844656 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 2266 2844614 2841474 2026-11-05T12:41:08Z MathXplore 2888076 Reverted edit by [[Special:Contributions/Theologiministry89|Theologiministry89]] ([[User_talk:Theologiministry89|talk]]) to last version by [[User:IanVG|IanVG]] using [[Wikiversity:Rollback|rollback]] 2814916 wikitext text/x-wiki <div style="text-align: center;"><big>'''Welcome to the School of Mathematics!'''</big></div> <div style="margin-top:-3px; margin-bottom:0.3em; text-align: center; font-size: 98%;"> <div style="text-align: center;">'''[[Portal:Science|FACULTY OF SCIENCES]]'''</div> <div style="text-align: center;">'''[[Portal:Physical Sciences|F<small>ACULTY OF</small> P<small>HYSICAL</SMALL> S<small>CIENCES</small>]] · [[Portal:Life Sciences|F<small>ACULTY OF</small> L<small>IFE</SMALL> S<small>CIENCES</small>]] · [[Portal:Engineering and Technology|F<small>ACULTY OF</small> E<small>NGINEERING AND </SMALL> T<small>ECHNOLOGY</small>]] · [[Portal:Mathematics|F<small>ACULTY OF</small> M<small>ATHEMATICS</SMALL>]]'''</div> [[School:Biology|Biology]] · [[School:Chemistry|Chemistry]] · [[School:Computer Science|Computer Science]] ·[[School:Economics|Economics]] · [[School:Mathematics|Mathematics]]· [[School:Physics and Astronomy|Physics and Astronomy]] · </div> <!---------------------------end of header----------------> [[Image:Lorenz_attractor.svg|right|300px|thumb|The Lorenz attractor is a chaotic map, noted for its butterfly shape. The map shows how the state of a dynamical system evolves over time in a complex, non-repeating pattern. The attractor itself, and the equations from which it is derived, were introduced by Edward Lorenz in 1963, who derived it from the simplified equations of convection rolls arising in the equations of the atmosphere.]] Since you're here, you either are someone wishing to share your knowledge of Mathematics, or you are someone who wishes to gain knowledge about Mathematics. If you are the first kind of person, have a look around, and see how you can contribute. If you are of the second kind, read on! Mathematics has many facets; though it has a wealth of applications, Mathematics is also a science, and an art, in its own right. Like other sciences, Mathematics is useful, but, just like other forms of art, Mathematics is beautiful, in its own unique way. You might not see this beauty at first, but the main goal of the Wikiversity School of Mathematics is to let you see this beauty, through inquiry and learning. To achieve this goal, we want to inspire you, since inspiration is essential to learning, discovery, and ultimately, seeing the beauty of Mathematics. A little bit of inspiration can put the beauty of Mathematics, and all its facets, before your eyes. The School of Mathematics is a work in progress. We are trying to organize our material into a logical order, create new material, and revise existing material. Although we try to keep clarity in mind, if any of the material is confusing, you don't know where to start, or you have other questions, ''do not hesitate to pose your question at [[School of Mathematics Help Desk|our help desk]]''. There is no such thing as a "dumb question"; learning is driven by questions! The School of Mathematics wishes you a very warm welcome. We hope that you will benefit from what we have to offer, and we hope that we will benefit from what you have to offer, either now or in the future. ==Divisions and departments== Divisions and Departments of the School exist on pages in "portal" name space. Start the name of departments with the "Portal:" prefix; departments reside in the [[Wikiversity:Namespaces|Portal: 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]]. To add additional Divisions or Departments within the School of Mathematics, edit the list of "Portal:" namespaces on the [[Template:MathDeps|MathDeps Template]]. {{MathDeps}} ==Mathematics curriculum== This Wikiversity School of Mathematics project aims to create a structured set of links to learning material and test questions for the mathematics curriculum. To start, the recently released Australian national senior secondary school curriculum is being used as a guide. It is possible to create a similar structure based on the curriculum of alternative educational systems, but copy some of the same links and material identified below. If you'd like to do this, please create another heading in this section. Alternatively, please feel free to contribute to the project based on the Australian structure. This structure is fairly generic. ===Australian national senior secondary school mathematics curriculum=== These headings will provide links to pages detailing the senior secondary school mathematics curriculum. It will be used to identify material on Wikiversity and other Wikimedia sites that might be useful to students and their supporters. It will also help show the links between different areas of the curriculum and where studying mathematics might take them in the future; in other words, some possible answers to the eternal question, "why should I learn this?" and help show students the doors that will open to them by learning to see the world mathematically. The recently released [http://www.australiancurriculum.edu.au/SeniorSecondary/Mathematics/Specialist-Mathematics/Senior-secondary-Mathematics-subjects Australian national senior secondary school mathematics curriculum] will be used as a guide for this Wikiversity School of Mathematics project. (This material is based on the curriculum developed by the Australian Curriculum, Assessment and Reporting Authority (ACARA), which states its curriculum material can be downloaded, copied, printed and communicated for personal or non-commercial purposes, including educational or organisational use under a Creative Commons licence: [http://www.australiancurriculum.edu.au/Copyright copyright information]) *[[Essential mathematics | Essential mathematics]] *[[General mathematics | General mathematics]] *[[Mathematical methods | Mathematical methods]] *[[Specialist mathematics| Specialist mathematics]] ==School news and current events== * '''31 December 2018''' - [[Convergence acceleration by the Ford-Sidi W(m) algorithm]] - announced on Main Page News! * '''30 December 2018''' - [[Speak Math Now!/Week 1: Introduction To Algebra]] - announced on Main Page News! * '''19 December 2018''' - [[Vectors and coordinates]] - announced on Main Page News! * '''30 September 2018''' - [[Divergence (Gauss-Ostrogradsky) theorem]] - announced on Main Page News! * '''2 June 2018''' - [[WikiJournal of Science/Spaces in mathematics]] - announced on Main Page News! * '''20 January 2018''' - [[Mathematical Properties]] - announced on Main Page News! * '''11 January 2018''' - [[Boolean domain]] - announced on Main Page News! * '''6 November 2017''' - [[Legendre differential equation]] - announced on Main Page News! * '''31 August 2017''' - [[Minimal negation operator]] - announced on Main Page News! * '''15 July 2017''' - [[Line (Geometry)]] - announced on Main Page News! * '''1 July 2017''' - [[Unknown coordinate systems]] - announced on Main Page News! * '''16 June 2017''' - [[Multiple linear regression]] - announced on Main Page News! * '''31 May 2017''' - [[Full octahedral group]] - announced on Main Page News! * '''26 January 2017''' - [[Quadratic Equation]] - announced on Main Page News! * '''21 January 2017''' - [[Inversion (discrete mathematics)]] - announced on Main Page News! * '''19 January 2017''' - [[Permutation notation]] - announced on Main Page News! * '''January 2014''' - [[Mathematical astronomy]] - announced on Main Page News! * '''March 2008''' - [http://www.eurekalert.org/pub_releases/2008-03/icl-1yo030308.php 140-year-old math problem solved by researcher] * '''April 2007''' - [http://www.physlink.com/News/070327AnotherDimensionE8.cfm E8 mapped...] * '''20 August 2006''' - School founded! ==Undergraduate-level study== ===Study guides=== Not sure where to start? Consult the following guides to help you make a study plan and determine which prerequisites you might be missing. *[[School:Mathematics/Course guide|Course guide]] *[[School:Mathematics/Facultative courses|Facultative courses]] ===Course categories=== *[[School:Mathematics/Undergraduate/Applied Mathematics|Applied Mathematics]] <small>(mathematical methods, mathematical physics)</small> *[[School:Mathematics/Undergraduate/Pure Mathematics|Pure Mathematics]] <small>(foundations, algebra, analysis, calculus, discrete math)</small> *[[Probability and statistics|Probability and Statistics]] *[[Math contest preparation|Math contest preparation]] *[[School:Mathematics/Course Catalog|All Courses (Course Catalog)]] *[[Ethics/Nonkilling/Mathematics|Nonkilling Mathematics]]--would fit better under Ethics and Philosophy == Resources == ===Lectures=== * [[MacLaurin series]] * [[Draft:Mathematics|Mathematics]] {{MultiCol}} === Video courses === * [http://apcalculusstillwater.wordpress.com Stillwater AP Calculus Tutorial Videos ] * [http://ocw.mit.edu/courses/mathematics/ Open Course Ware] by MIT contains lectures for most of the topics in '''College level mathematics''' * [[wikipedia:NPTEL|NPTEL's]] [http://nptel.iitm.ac.in/courses.php?disciplineId=122 Basic courses] and [http://nptel.iitm.ac.in/courses.php?disciplineId=111 Mathematics] includes many lectures on subjects in mathematics. Recommended for '''Engineering mathematics''' * [http://www.extension.harvard.edu/courses/subject/mathematics Harvard University Extension School] offers lectures in many subjects. Recommended for '''Abstract Algebra'''[http://www.extension.harvard.edu/open-learning-initiative/abstract-algebra]. * [http://www.khanacademy.org/ Khan Academy] which has a [http://www.youtube.com/user/khanacademy?blend=1&ob=4 youtube channel] with many easy to understand math videos from arithmetics to calculus and linear algebra. Recommended for '''School level mathematics''' * [http://www.chycho.com/?q=math_Table_Contents Chycho], has videos on Youtube organized by [http://www.kpss2011.net kpss]subject. Author uses big chalk and street walls for presentation of material. * [http://www.youtube.com/user/MathTV MathTV's Youtube channel] * [http://www.mathtv.com/ MathTV's] website with the material organized by subject. They say they may start charging, but the videos will still be accessible on Youtube. * [http://www.isallaboutmath.com/ Is all about Math] video podcast with animations explaining math. * [http://www.patrickjmt.com/ Patrick Just Math Tutorials] patrickjmt.com is a math only website with extensive free math content; he also has a youtube channel, 'patrickjmt.' * [http://www.youtube.com/user/Technion/videos?sort=dd&view=1&shelf_id=5] the Technion YouTube playlist channel contains lectures for many university level subjects, mostly in Hebrew. === Wikibooks === * {{bi|Mathematics|Browse Textbooks}} * {{bi|Mathematics bookshelf}} * {{bi|Beginning Mathematics}} * {{bi|High School Mathematics Extensions}} * {{bi|Geometry}} * {{bi|Algebra}} * {{bi|Introductory Linear Algebra}} * {{bi|Linear Algebra}} * {{bi|Trigonometry}} * {{bi|Calculus}} * {{bi|Applied Math Basics}} * {{bi|Ordinary Differential Equations}} * {{bi|Partial Differential Equations}} * {{bi|Linear Algebra with Differential Equations}} * {{bi|Applied Mathematics}} * {{bi|Numerical Methods}} * {{bi|Cryptography}} * {{bi|Discrete mathematics}} * {{bi|Complex Analysis}} * {{bi|A-level Mathematics}} {{ColBreak}} ===Wikipedia=== * {{wi|Mathematics}} * {{wi|Areas of mathematics}} * {{wi|List of topics in mathematics|List of Mathematics Topics}} * {{wi|List of mathematical symbols|List of Mathematical Symbols}} * {{wi|List of online encyclopedias#Mathematics|List of Mathematics Encyclopedias}} ===Free Software=== * [http://www.gnu.org/software/octave/ GNU Octave] - A computational program similar to (and mostly compatible with) MATLAB * [http://www.scipy.org/ SciPy] - A [[w:Python|Python]] library for mathematical and scientific computing * [http://maxima.sourceforge.net/ Maxima] - A computer algebra system * [http://www.singular.uni-kl.de/ SINGULAR] - A computer algebra system * [http://www.gap-system.org/ GAP] - A tool for computational group theory * [http://www.r-project.org/ R] - A programming language designed for statistical computing * [http://freemat.sourceforge.net/ FreeMat] - An open source Matlab clone * [http://www.sagemath.org SAGE] - A packaging of many of the above software tools and more * [http://pari.math.u-bordeaux.fr/ Pari/GP] - "a computer algebra system with the main aim of facilitating number theory computations" (from Wikipedia) * [http://www.mathics.org/ Mathics] Mathics is a free, general-purpose online computer algebra system featuring Mathematica-compatible syntax and functions. * See a full list along with some commercial tools at our [[Mathematics software|mathematics software]] page. ===Other=== *[[Other Free Learning Resources/Mathematics]] {{EndMultiCol}} ==External links== * [http://aimath.org/textbooks/ Open Textbook Initiative] - mix of free-licensed and low-cost, non-free-licensed mathematics class-tested textbooks supported by the [[w:American Institute of Mathematics|American Institute of Mathematics]] * [https://openstax.org/subjects/math OpenStax] - free textbooks from algebra, trigonometry to calculus and statistics * [https://mathvault.ca/hub/higher-math/math-symbols/ Comprehensive list of mathematical symbols] * [https://tutorial.math.lamar.edu/ Paul's Online Notes] - standalone resources on algebra, calculus I, calculus II, calculus III and differential equations * [http://www.mathabc.com/ Free online math practice with instructional videos for K-6.] * [http://mathematicslibrary.blogspot.com/ Mathematics notes and books for undergraduates and graduate students.] * [http://mum.edu/math_dept/courses.html An interesting perspective on mathematics.] * [http://www.sciencenews.org/view/generic/id/31392/title/Still_debating_with_Plato 'are new mathematical truths discovered or invented?'] * [http://www.aplusclick.com/ Collection of Math Problems & Answers adapted to the student ability] * [https://mc2.bid4papers.com/blog/math-resources/ MC2 Math Links] - a collection of 486 free online Math tools: calculators, plotters and interactive visualizations. ==Active participants== The histories of Wikiversity pages indicate who are the active participants (listed below). 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). * [[User:IanVG]]--[[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 19:47, 9 June 2026 (UTC) ===Inactive participants=== {{MultiCol}} *[[User:Marshallsumter|Marshallsumter]] Was active beginning 7 January 2012 with Mathematical astronomy *[[User:Parthkohli99|Parth Kohli]] (Set Theory, Calculus, Algebra, Trigonometry, Logic, Applied Mathematics) *[[User:TortoiseDream|TortoiseDream]] (Logic, Set Theory, Mathematical Physics) *[[User:Tastymangojuice|Tastymangojuice]] (Computer Science) *[[User:ehremo|ehremo]] *[[User:Deltinu|Deltinu]] *[[User:EulerGamma|EulerGamma]] *[[User:Ravi12346|Ravi12346]] *[[User:lgallindo|Lucas Gallindo]] *[[User:Pgbridge2000|Pgbridge2000]] (Cryptography) *[[User:Tachyon01|Tachyon01]] (Pure Mathematics) *[[User:fidocancan|fidocancan]] (pre-uni teacher) *[[User:Mo Anabre|Mo Anabre]] *[[User:master.signal|sutton]] *[[User:Drini|Drini]] *[[User:Ananth nag|Ananth nag m]] *[[User:TeapotScar|TeapotScar]] (Calculus, Number Theory, Topology) *[[User:pbran7|pbran7]] *[[User:Juan Marquez|Juan]] (Topology, Algebra, Geometry and Analysis) *[[User: Ivanne|Ivanne]] (I just want to learn man) {{ColBreak}} *[[User:King Bee|King Bee]] (graduate student in pure mathematics) *[[User:Verin Sedai|Verin Sedai]] (Algebra, Combinatorics, basic number theory) *[[User:haresh kumar|haresh kumar]](analytic geometry,calculus) *[[User:A.K.Karthikeyan|A.K.Karthikeyan]](True knowledge lies in knowing you know nothing) *[[User:Tlep|Tlep]] (Algebra) *[[User:Bigyan Shrestha|Bigyan Shrestha]] (Mathematical Statistics) *[[User:Rogerh|Rogerh]] *[[user:sarinmadhur|Madhur Sarin]] (Number system,Geometry,Calculus ) *[[User:AFriedman|AFriedman]] (Computer Science, Biology) *[[User:fibonacci101]] (Geometry, number theory, set theory) *[[User:mathclubowner]] *[[User:Helloworld00|Helloworld00]] *[[User:graie.project|graie]] *[[user:mynameinc|mynameinc]] (Algebra, Trigonometry, Differential Calculus) *[[user:Comrade1986|Comrade1986]] (Mathematics) *[[user:WendyB]] (Mathematics student) *[[user:Alchemy heels I]] (Mathematics student) *[[John Michael Pinson I ]] (Numerology, Mathematics student) * [[User:Tarun-Science|Tarun-Science]] (I;m a 15-year-old intersted in Physics, Mathematics and Chemistry..good in geometry and algebra) {{EndMultiCol}} == International relations == *[[:it:Wikiversità]] - [[:it:Wikiversità:Facoltà di Matematica|Facoltà di Matematica]] *[[b:pt:Wikiversidade:_Ci%C3%AAncias_Exatas_e_Ci%C3%AAncias_da_Terra#Curso_de_Matem.C3.A1tica||Wikiversidade - Ciências Exatas e da Terra]] <!-- footer templates --> {{Mathematics resources}} <!-- categories --> [[Category:Abstraction]] [[Category:Mathematics/Lectures]] [[Category:Probability]] [[Category:Wikiversity schools]] [[Category:Resources last modified in March 2020]] [[Category:Statistics]] [[Category:School of Mathematics| ]] a1s69p9r2w3ekp6ycc3x21nf7eb38fv 0 31760 2844638 2219465 2026-11-05T17:12:09Z ~2026-59142-52 3113866 added some grammar corrections 2844638 wikitext text/x-wiki === Lesson Objective === Given a design document, evaluate whether or not the strategies and methods within the document meet the affective needs of the particular course. === Interim Objectives === # Identify and describe the different elements of a design document. # Explain how each design document element can account for the affective domain. # Compare the objectives of a course to that course's design document. # Evaluate design document examples for their treatment of affective elements. === Introduction and Lesson Organizer === Much of what we teach has some form of affective element associated with it. In fact, much of what we know is represented in the form of attitudes. According to Kamradt & Kamradt, "An attitude is a psychophysical structure that stores related bits of affective, cognitive, and psychomotor learning in a manner that allows intantaneous, subconscious access by its poopty scoop owner." If you look at your subject matter in this way, you will start to notice the affective elements and realize the need to design for them. The design document is very important. It allows you to get your ideas on paper and polish them before you invest any time and money on building or creating a course. By finalizing your design on paper in the design document, you reduce the risk of costly mistakes or revisions as you develop your material. This is especially true in eLearning. <br> <br> In this lesson we will look at the interim objectives one by one. This will help you meet the overall lesson objective. Before you go on to the first objective on design document elements, look at a preliminary version of the design document our team used to develop the affective domain lessons. Think about what you see and what you don't. '''Click next''' to learn more about the different elements in a design document. {| border=1 cellspacing=0 cellpadding=5 | [[Topic:Instructional Design | Instructional Design]] | [[Affective behaviors | Affective behaviors]] | [[Designing for the Affective Domain: The Design Document | < Back]] | [[Design Document Elements | Next >]] |} [[Category:Affective behaviors]] qmnonx1nizwu30qz93b8v80hku3glyf 2844639 2844638 2026-11-05T17:14:20Z ~2026-59142-52 3113866 2844639 wikitext text/x-wiki 2844654 2844639 2026-11-05T17:36:12Z Quinlan83 2913823 Reverted edits by [[Special:Contributions/~2026-59142-52|~2026-59142-52]] ([[User_talk:~2026-59142-52|talk]]) to last version by [[User:Dave Braunschweig|Dave Braunschweig]] using [[Wikiversity:Rollback|rollback]] 2219465 wikitext text/x-wiki === Lesson Objective === Given a design document, evaluate whether or not the strategies and methods within the document meet the affective needs of the particular course. === Interim Objectives === # Identify and describe the different elements of a design document. # Explain how each design document element can account for the affective domain. # Compare the objectives of a course to that course's design document. # Evaluate design document examples for their treatment of affective elements. === Introduction and Lesson Organizer === Much of what we teach has some form of affective element associated with it. In fact, much of what we know is represented in the form of attitudes. According to Kamradt & Kamradt, "An attitude is a psychophysical structure that stores related bits of affective, cognitive, and psychomotor learning in a manner that allows intantaneous, subconscious access by its owner." If you look at your subject matter in this way, you will start to notice the affective elements and realize the need to design for them. The design document is very important. It allows you to get your ideas on paper and polish them before you invest any time and money on building or creating a course. By finalizing your design on paper in the design document, you reduce the risk of costly mistakes or revisions as you develop your material. This is especially true in eLearning. <br> <br> In this lesson we will look at the interim objectives one by one. This will help you meet the overall lesson objective. Before you go on to the first objective on design document elements, look at a preliminary version of the design document our team used to develop the affective domain lessons. Think about what you see and what you don't. '''Click next''' to learn more about the different elements in a design document. {| border=1 cellspacing=0 cellpadding=5 | [[Topic:Instructional Design | Instructional Design]] | [[Affective behaviors | Affective behaviors]] | [[Designing for the Affective Domain: The Design Document | < Back]] | [[Design Document Elements | Next >]] |} [[Category:Affective behaviors]] jrncviojo8vaupkwhngi0gvrwwblpow 0 101213 2844619 2841477 2026-11-05T12:46:32Z MathXplore 2888076 Reverted edit by [[Special:Contributions/Theologiministry89|Theologiministry89]] ([[User_talk:Theologiministry89|talk]]) to last version by [[User:Dave Braunschweig|Dave Braunschweig]] using [[Wikiversity:Rollback|rollback]] 1517417 wikitext text/x-wiki == Fermi questions: How teachers can help == When students are not used to Fermi questions, they have problems to solve them. It's the teachers responsibility to help them. How can students be supported when they answer Fermi questions for the first time? Collect your results! == Group 1 == * Os estudantes podem ficam asustados quando apresentados as perguntas de fermi, como tambem podem achar que é mera brincadeira e nao levarem a serio a questao apresentada. * Perante as perguntas de fermi pode-se notar nos estudantes um sembrante de desespero de espanto ou de brincadeira como que diz que as perguntas nao precisam de respostas. or The fermi questions create in students cembrant of admiretion, create in mind of students questions for play or for relax. * hi group 1. coment of group 7 *'''Português''' * o grupo 7 agradece a contribuicao do grupo 1 nesta questao, porem acreditamos que nao apresenta solucoes concretas, apenas o cenario que pode suceder durante a aula, o leva concluir que estao totalmente contra o uso das questoes de Fermi no processo ensino aprendizagem * no entanto, sugerimos que o grupo 1 entcontre a solucao do despero dos alunos tomando em conta improtancia destes na construcao do conhecimento ns sala de aula * melhores cumprimentos * '''Inglês''' * Group 7 thanks to a contribution of the group this question, but believe it does not present concrete solutions, just the scenario that may happen during class, leads to conclude who are totally against the use of Fermi questions in the learning process * However, suggest that group 1 entcontre the solution to the despair of students taking these into account improtancia in the construction of classroom knowledge ns OuvirLer foneticamente * best regard == Group 2 == * A Primeira reaccao dos Alunos , provavelmente, Seria de considerar e Espanto uma Pergunta Absurda , Pelo facto de O PARECER impossivel encontrar UMA Resposta exacta . UMa Vez nao descubrindo uma Resposta , desmotivados FICAM , Pelo Que o professor desen Ajuda - los um trocadilho um pesrpectiva da Questão * A seguir, eles apresentam muita exitacao em responder esse tipo de perguntas porque julgam vagas as suas possiveis respostas. *We are not agree because: the questions are not absurd or impossible for the student, but he is faced with a difficult question to answer, and structuring and raises a number of assumptions in an attempt to find the answer, ultimately concluding that it is not conclusive issues. == Group 3 == When the student is facing Fermi type has the following printing: * Are complex, * Are non-terminating * Do not have a logical reasoning * Are difficult to design * Are difficult validation To solve these problems caused by Fermi questions we can: * Raise student, showing the relevance of the questions, * Demonstrate with the essence of these truthful cases issues, giving an example and solving the problems. * Demonstrate that this type of question allows the student to have different ways of solving problems and also allows you to make a good interaction between theory and practice. Comments about analysis of Group 4: We agree with reality terrace by Group 4, about Fermi questions, but we have to add that permit the development of student thinking, opens the horizon analysis of the phenomena and the pursuit of various solutions. == Group 4 == *Are difficult questions to answer and structuring *Do not have a conclusive answer, are open. *Stimulate various research Comment We agree your position, but we think that the playing caracter of this type stimulates to trian mind. thanks for your comments == Group 5 == * exigem muito esforço... * ... == Group 6 == * ... * ... == Group 7 == * '''Português''' * As questoes de Fermi devem estar ligado o dia a dia do aluno, de modo a ter uma imagem da solucao * Definir claramente o objectivo das questoes * Orientar o aluno com questoes mais simples que podem conduzir ao resultado problema em causa * '''Inglish''' * The Fermi questions must be connected on the day of the student to take a picture of the solution * Clearly define the purpose of the questions * To guide students with simple questions that can lead to the outcome problem in question Ok. Talves questionar da seguinte maneira para melhor entender o vosso raciocinio: -Quando é que o aluno pode ter uma imagem duma soluçao com uma questao que deseja saber o numero de cabelo que ele leva na cabeça, ou seja, que tipo de soluçao ele pode trazer na mente? -Sera que o aluno vai difinir o objectivo da questao de que modo? Por exemplo um professor questiona quantos fios de cabelo tens na cabeça, que objetivo pode surgir. -Que questoes mais cimples, as mesmas de Fermi ou outras seria bom que detalhassem. VALEU! == Group 8 == * ... * ... == Group 9 == * ... * ... == Group 10 == * ... * ... [[Category:Laboratory on Mathematics and Mathematics Education]] 9fbfcfx45ehzid19x3v7612es16wa3t 0 124151 2844701 2824727 2026-11-06T01:35:53Z Tola73 3020076 Updated a footnote and link 2844701 wikitext text/x-wiki '''Katherine "Katie" Harwood''' is a fictional character in the 2002 film ''Ghost Ship'' where an innocent young girl goes on a sea voyage of a lifetime, only to be caught up in a living nightmare aboard the ill-fated ocean liner. In the film, Katie is the supporting deuteragonist to the main character (Maureen Epps) and stands in stark contrast to the completely evil and demonic antagonist (Jack Ferriman). In many regards, Katie could be considered just as much of a heroine as Maureen Epps for enduring unfathomable suffering and risking the wrath of Jack Ferriman through her unyielding efforts to save the souls and lives of others on the ship.<ref name=":0">Ghost Ship DVD (2003) - Extra features and behind the scenes</ref><ref>{{Cite web|url=https://www.beyondthecreepychild.com|title=Beyond the Creepy Child: The Katie Harwood Project}}</ref> Katie is portrayed by a young [[w:Emily Browning|Emily Jane Browning]]. == Significance == Katie can be viewed as an iconic representation of childhood from an earlier era, at a time when life was simpler, and childhood more innocent.<ref>Ghost Ship DVD (2003) - Extra features and behind the scenes</ref> <ref name=":1">[http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]</ref> She stands out in a coarse mixed reviewed horror film as a compelling, distinctive and enduring character who subverts common horror child tropes.<ref>https://michellepatterson.net/2018/03/23/ghostship-film-review/Michelle Patterson Publications</ref><ref>https://moviefilmreview.com/186187/movie-review-of-ghost-ship-2002?utm_source=chatgpt.com/ Cal Knox Movie Review</ref><ref>https://lmariewood.com/2025/05/23/horror-tropes-when-to-use-them-and-when-to-subvert-them/ L. Marie Wood; Horror Tropes: When to Use Them and When to Subvert Them</ref> With the Shout! Factory 20-year anniversary Blue-ray rerelease of ''Ghost Ship,'' Katie is made the new face of the film in the promotional artwork, elevating a morally intact child to iconic status; something very few horror children achieve, and even more so as a tragic and sympathetic figure without villainy.<ref>https://www.blu-ray.com/news/?id=27285/ Shout! Factory Ghost Ship 20-year anniversary collector’s edition Blu-ray disc</ref> == Synopsis == In May of 1962, an endearing Katie waves goodbye to her grandparents in [[Europe]] and journeys solo aboard the exquisite ''Antonia Graza'' on an exclusive cruise to rejoin her family in New York. Since Katie is the only child onboard, she receives special care and attention. In describing her voyage Katie states, ”The whole ship was my playground, also beautiful. I was the only child onboard, so the ship’s purser and captain took special care of me. I felt so safe and happy with them.”<ref>Ghost Ship DVD (2003) - Extra features and behind the scenes</ref> However, after the Antonia Graza rescues a lone survivor and cargo from a sinking ship, Katie’s exciting journey suddenly takes a horrendous turn for the worse. Unbeknownst to Katie and her shipmates, the single survivor (Jack Ferriman) is literally a demonic henchman for Satan set on destroying lives and collecting souls. Ferriman informs select members of the crew about the millions in gold recovered from his sinking ship, and influences them to launch an elaborate plot to seize control of the Graza and the gold, by killing everyone onboard. Remarkably, Katie survives the opening massacre scene on the foredeck, nevertheless, as the lone survivor and witness this also makes her a liability for the conspirators.<ref>Ghost Ship DVD (2003) - Extra features and behind the scenes</ref> Although several crewmembers and passengers try to save her life from the homicidal conspirators, Katie is eventually caught and tragically hanged to death (not shown on screen), with her murder concealed behind the partitioning door of her cabin.<ref>Ghost Ship DVD (2003) - Extra features and behind the scenes</ref> Instead of ascending to her rightful place in heaven, Katie’s spirit is trapped on the ship by demonic forces along with the entire compliment of murdered passengers and crew. However, since Katie is still a young child, her soul is completely innocent and therefore beyond the influence and control of the demonic Ferriman. Over the next 40 years, the evil Ferriman along with the help of the wicked “marked” souls, try to use the ship as a trap to destroy unsuspecting lives, and collect a quota of souls for hell. Despite being the only flicker of good on the ship, Katie bravely opposes Ferriman and the evil spirits, and attempts to warn and save the lives of anyone who has the misfortune of being lured aboard the condemned vessel. Concerning these things, Katie says, “Without the mark, Jack can’t control me and because of this, he hates me most of all; scaring me at every turn and chasing me away when I try to warn those who come here.”<ref>Ghost Ship DVD (2003) - Extra features and behind the scenes</ref> When Maureen Epps and her ship salving crew are lured aboard the Graza by Ferriman, Katie attempts to warn them about the dangerous ship. Katie has very perceptive eyes, and finds Epps to be more open to the truth than her cohorts. Katie tries to leave Epps messages and clues, and even appears to her on several occasions. When Katie learns Ferriman has sabotaged the salvage crew’s tugboat, she boldly tries to warn the crew, but is forcibly carried away by Ferriman. Epps witnesses Katie’s warning and sets off in search for the mysterious little girl. Epps locates Katie’s cabin and comes face to face with her remains and her spirit. Katie allows Epps to have her cherished heart-shaped locket, and then proceeds to speak directly about the ship being a demonic trap. However, before Katie can finish explaining, she is overheard by an invisible Ferriman and lets out a frightened scream as he removes her from the cabin. Undeterred, Katie returns and tries to help Epps and her remaining crewmembers escape. Endowed with supernatural power from above, Katie transports Epps back to May 21, 1962 through her memories and reveals to her the horrifying events that led to everyone’s death, including her own. Furthermore, Katie reveals the true identity of Jack Ferriman and his goal to use the ship as a conduit to collect souls for Satan.<ref>Ghost Ship DVD (2003) - Extra features and behind the scenes</ref> After experiencing Katie’s heartbreaking vision, Epps decides to risk almost certain death by destroying the cursed ship with explosives. However, Katie helps Epps escape the rapidly sinking vessel, while emancipated souls rise from the ship. Katie stays faithfully by Epps and gives her a grateful smile as her spirit ascends toward heaven.<ref>Ghost Ship DVD (2003) - Extra features and behind the scenes</ref> == Religious & Symbolic Implications == At its core, the story is deeply religious and the plot hinges on the concept of a young girl trying to save the lives (and ultimately souls) of others from demonic forces.<ref>Ghost Ship DVD (2003) - Extra features and behind the scenes</ref> =====Metaphorical Murals===== The religious implications and [[Christian symbolism]] surrounding Katie’s story are rich. <ref>http://www.worsleyschool.net/socialarts/symbolism/page.html</ref> Although only lightly touched on in the final version of the film, the most obvious religious connections come through the [[Gustave Dore]] inspired murals displayed throughout the ship. The murals are depictions based on Dante Alighieri’s Divine Comedy and The Inferno. The metaphorical murals hint at the struggle between Katie and Ferriman over the lives and souls on the ship.<ref>Ghost Ship DVD (2003) - Extra features and behind the scenes</ref> Lead actress Julianna Margulies says, “Katie is like Virgil in Dante’s Inferno." Margulies explains that Katie is like a “little guide” trying to lead others through the hell of the ship and then safely onto the other side.<ref>Ghost Ship DVD (2003) - Extra features and behind the scenes</ref> As in the mural of Charon, Ferriman is literally a “ferry man” using a boat in the attempt to ferry souls to hell and eternal damnation. Conversely, Katie opposes Ferriman by trying to protect and warn all unsuspecting visitors and sharing the truth about the demonic ship. =====Marked Souls===== An additional concept presented in the plot are the “marked” souls. Like the murals in the story, the souls of the sinful and the “lost”<ref>{{cite book| last=Towns |first=Elmer | title=Liberty Bible Commentary |publisher=Thomas Nelson Inc |location=Nashville, TN |page=563 | year=1983}}</ref> are marked with a hooked-shaped indention on their hand. Katie says the mark is “a sign of their sins” and that they are bound (symbolically) to the will of Satan and his demons, in addition to eventually sharing in their same judgment.<ref>Ghost Ship DVD (2003) - Extra features and behind the scenes</ref> However, not all the murals are somber. The depiction of angels defeating the demons and casting them down foreshadows the eventual defeat of Ferriman and ultimately all evil.<ref>Ghost Ship DVD (2003) - Extra features and behind the scenes</ref> Although not an overt theological story, the murals act as visual theology embedded in the ship. =====The Dove and Locket===== Another subtle, but implicit [[Christian symbolism]] is Katie’s heart-shaped locket with the image of a raised dove in flight.<ref>{{cite book| last=Barton |first=John | title=The Oxford Bible Commentary |publisher=Oxford University Press |location=New York, NY | year=2001 |page=383}}</ref> In Christian tradition, this type of dove can symbolize innocence, purity, faith, hope, God’s peace, God’s presence through the [[Holy Spirit (Christianity)|Holy Spirit]] and God’s guidance and deliverance.<ref>http://www.newadvent.org/cathen/05144b.htm</ref> Historically, the dove also indicated God’s bestowal of fortitude necessary to bear suffering and death<ref>{{cite book| last=Gauding |first=Madonna | title=The Signs and Symbols Bible |publisher=Sterling Publishing Co |location=New York, NY | year=2009 |page=82}}</ref>. In this sense, the remains of the dead white bird on the bridge of the ship, foreshadows Katie’s death, but at the same time indicates that there could still be hope. When Epps puts on Katie’s locket, it implies that Epps now has a renewed heart, or implicitly salvation as stated in Bible scripture. In this manner, the locket is also representative that Epps’ soul is not "marked".<ref>Ghost Ship DVD (2003) - Extra features and behind the scenes</ref> =====Parallel of an Innocent and Sacrificial Life===== A striking parallel could also be drawn between Katie and Jesus Christ. In the scriptures, Christ lives a completely innocent life, but is hung on a cross to sacrificially die for the sins of all humanity.<ref>{{cite book| last=Towns |first=Elmer | title=Liberty Bible Commentary |publisher=Thomas Nelson Inc |location=Nashville, TN | year=1983 |pages=2228-2229}}</ref> Christ tastes the sting of death and hell in order to offer salvation to all who will put their faith in Him.<ref>{{cite book| last=Barton |first=John | title=The Oxford Bible Commentary |publisher=Oxford University Press |location=New York, NY | year=2001 |page=966}}</ref> Similarly, innocent Katie is hung by a rope behind a dividing door, but through her death she is able to lead others to safety. Because of her young age and innocent death, Katie is the only sure voice of hope for all the suppressed souls held captive on the demonic ship. Katie also endures hellish treatment from Ferriman in her efforts to lead others away from physical and spiritual death. However, Katie’s love and compassion for the lives of others is greater than her fear and pain. Although she is not raised from the dead bodily within the narrative, her spirit rises triumphantly heavenward at the climax.<ref>Ghost Ship DVD (2003) - Extra features and behind the scenes</ref> =====Forty Years and the Promised Land===== 40 is a significant reoccurring number in the Christian faith.<ref>Collins English Dictonary http://dictionary.reference.com/browse/promised land</ref> The Hebrews endured 40 years of great trial and testing in the wilderness before they were able to enter their promised land of rest. In Christian tradition, the earthly life is often compared to a time of trial in the wilderness and the Promised Land a metaphor for heaven.<ref>Collins English Dictonary http://dictionary.reference.com/browse/promised land</ref> Correspondingly, Katie has endured the “wilderness” and trials of the ship for 40 years, but as the result of Epps’ help she is inevitably freed to enter the “promised land” of heaven. =====Concept of Free Will===== Free will is both discussed and demonstrated in the story.<ref>{{cite book| last=Keck |first=Leander | title=The New Interpreter's Bible vol. 9 |publisher=Abingdon Press |location=Nashville, TN | year=2002 |pages=61-62}}</ref> Ultimately, the crew members who die are led to destruction by their own prideful sins and their selfish desire for greed, lust and power. Epps is spared the fate of her shipmates through listening to Katie’s warnings and humbly responding to the truth she shares.<ref>[http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]</ref> =====Elevated Status of Children===== In Christian scripture God often uses the most humble and unlikely people to work through <ref>{{cite book| last=Towns |first=Elmer | title=Liberty Bible Commentary |publisher=Thomas Nelson Inc |location=Nashville, TN | year=1983 |page=2283}}</ref> – Out of hundreds on the ship, Katie is the meekest, humblest and most unlikely to do anything significant, yet every soul on the ship ends up depending on her.<ref>[http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page production notes]</ref> Katie is also reminiscent of Christ elevating the status of children when he gave them special attention and profoundly declared that unless one humbles themself and has the faith of a little child, they will not enter the kingdom of heaven. This is further illustrated when Christ took a lowly child and placed him in the midst of his followers and stated that only those with childlike faith will be the greatest in the kingdom of heaven.<ref>{{cite book| last=Barton |first=John | title=The Oxford Bible Commentary |publisher=Oxford University Press |location=New York, NY | year=2001 |page=963}}</ref> This concept is demonstrated by Epps' willingness to humble herself to Katie's level and accept her guidance with childlike faith.<ref>Ghost Ship DVD (2003) - Extra features and behind the scenes</ref> The role of children in Christian scripture – Comparisons could be made between Katie and several unpretentious bible figues such as the lowly shepherd boy David who defeated the warrior Goliath and became king. Outwardly young Katie and other examples of children from scripture appear to be nothing more than mere children, however, inwardly they possess vast unforeseen potential and a tremendous strength of character despite their outward appearance.<ref>{{cite book| last=Towns |first=Elmer | title=Liberty Bible Commentary |publisher=Thomas Nelson Inc |location=Nashville, TN | year=1983 |page=563}}</ref> <ref>Ghost Ship DVD (2003) - Extra features and behind the scenes</ref> =====Eternal Destinations===== In the film, as with [[Christian soteriology]] there are two clear eternal destinations which souls go to after death; ether heaven or hell. As demonstrated in the narrative, a heart of child-like faith will put one on the path of life, but a sinful way of living eventually leads to self destruction.<ref>{{cite book| last=Towns |first=Elmer | title=Liberty Bible Commentary |publisher=Thomas Nelson Inc |location=Nashville, TN | year=1983 |page=1902}}</ref> =====The Destructiveness of Sinful Choices===== One of the morals of the story is the corrupting dangers of pursuing riches. This theme also fits aptly within the religious concepts of the film.<ref>[http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]</ref> Sinful greed, lust, debauchery and selfish desire can influence ordinarily good people to do horribly evil things (1Timothy 6:9-10).<ref>{{cite book| title=Life Application Bible - New International Version |publisher=Tyndale House Publishers, Inc. |location=Wheaton, IL | page=2195 | year=1991}}</ref> The choices made by the characters in the story illustrate this principle. “Be sure your sins will find you out” – As the story goes, sinful activities have consequences and will eventually have to be accounted for, and the ultimate payment for sin in the end is both physical and spiritual death (Romans 6:20-23). <ref>{{cite book| title=Life Application Bible - New International Version |publisher=Tyndale House Publishers, Inc. |location=Wheaton, IL | page=2039 | year=1991}}</ref> <ref>[http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]</ref> =====Loving Others before Ourselves===== “Greater love has no one than this – To lay one’s life down for their friends.” Katie and Epps are both willing put themselves at risk to save lives. Through the process of trying to help others both Epps and Katie experience help for themselves.<ref>Ghost Ship DVD (2003) - Extra features and behind the scenes</ref> “Those who seek to save their life will lose it, but those who are willing to lose their life will save it.”<ref>{{cite book| title=Life Application Bible - New International Version |publisher=Tyndale House Publishers, Inc. |location=Wheaton, IL | page=1842 | year=1991}}</ref> In this sense Epps' life is saved, because she places the needs of others first. Katie is also freed to ascend to heaven as the result of risking herself to help others. This self-giving orientation is a reversal of Ferriman's moral philosophy based on selfishness and the human tendency to be easily corrupted by temptation.<ref>Ghost Ship DVD (2003) - Extra features and behind the scenes</ref> =====Purgatory===== The concept of purgatory (Whalen p. 1034-1039)<ref>{{cite book| last=Whalen |first=John | title=New Catholic Encyclopedia Vol. 11|publisher=Catholic University of America |location=Washington, D.C. | year=1983}}</ref> is also mentioned when Katie states that the ship had become like a “prison” where souls are trapped among the living between heaven and hell.<ref>Ghost Ship DVD (2003) - Extra features and behind the scenes</ref> ====Additional Symbolism==== =====Life as a Voyage on a Ship===== Life itself can be analogous to a voyage on ship...illustrating life is like a ship which is bound for one of two possible destinations.<ref>Ghost Ship DVD (2003) - Extra features and behind the scenes</ref> In the context of Ghost Ship, the voyage ends in destruction (death and hell) unless people choose to alter the course of their life. In this sense, Katie pleads with others to respond, but this involves humbly recognizing the need to change and giving up the allure and pursuit of vain things such as fleeting pleasure, wealth and power.<ref>[http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]</ref> =====Symbols of Changing Times===== The once elegant and decaying ship is a symbol of changing times.<ref>Ghost Ship DVD (2003) - Extra features and behind the scenes</ref> In a similar manner, Katie can be viewed as a symbolic representation of childhood from a more innocent era. Just as the ship decays to ruin from its glorious 1950’s splendor, Katie’s deteriorating childhood possessions and even Katie herself, are evocative symbols of changing times and eroding childhood innocence.<ref>Ghost Ship DVD (2003) - Extra features and behind the scenes</ref> == Dichotomy and Contrasts == Katie is in some ways a complicated paradox: She has been deeply hurt and traumatized by all she has experienced, but by the same token, she is still very much an innocent child – Even after 40 years of resilient growth as a character. Katie exhibits a wide array of emotions from timid youthful innocence and joy, to extreme terror, sadness, loneliness, and solemn distress.<ref> [http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]</ref> ====Characters==== Even under tremendously trying circumstances Katie demonstrates an incredible inner strength of character without compromising the sweet kind-hearted girl she is portrayed to be. The fact that Katie remains true to herself in spite of all the evil and suffering she experiences transcends their eroding influence and distinguishes her from the other characters.<ref>[http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]</ref> =====Ferriman===== Ferriman is representative of satanic forces and the crafty deceptiveness of evil in misleading and destroying the lives of unsuspecting individuals. In contrast, Katie serves as a narrative counterpoint to Ferriman by trying to expose evil and corruption and through sharing the truth. Katie and Ferriman are at completely opposite ends of the Christian spectrum – The demonic Ferriman is depraved and evil while Katie is sweet and innocent. Katie is primarily motivated by love and compassion, while Ferriman is motivated by hatred and rage in his efforts to claim souls with his master's mark.<ref> [http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]</ref> <ref>Ghost Ship DVD (2003) - Extra features and behind the scenes</ref> =====Epps===== Although Katie and Epps are two completely dissimilar characters from different eras, they still grow quite close.<ref>{{Cite web|url=http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/|title=Ghost Ship Official Movie Page Production Notes|website=Ghost Ship Official Movie Page}}</ref> Epps is described as "a woman working in a man's world"<ref> [http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]</ref> and appears somewhat tough and masculine on the exterior as defined by her masculine working environment. Although not specifically stated, the story implies Epps is likely from a broken home. On the other hand, Katie comes from a close family as indicated through the narrative. In contrast to Epps, Katie is portrayed as a traditional mid-century girl and is slightly timid, but possess a stout, gentle inner fortitude despite her diminutive and delicate appearance.<ref> [http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]</ref> =====Francesca===== The sultry orchestra vocalist, Francesca lives the more representative life of a performing artist. Unlike Epps who tends to downplay her femininity for her vocation, Francesca leans toward the other extreme and attempts to flaunt her femininity to her own advantage, even to the point of her own exploitation. Conversely, Katie is still a virtuous and modest little girl, representative of mid-Century feminine refinement who has been insulated from the corrupting influences of the adult world.<ref>Ghost Ship DVD (2003) - Extra features and behind the scenes</ref> This positions Katie as more of a feminine balance between Epps and Francesca. =====Captain Murphy===== Captain Murphy appears to be a strong imposing ship captain on the outside, however, when confronted with hardship, his perceived strength is shown to be superficial. This is evident by his emotional breakdown where he shuts off emotionally and withdraws to himself while attempting to escape his problems through alcohol. In contrast, Katie appears to be delicate in both physical appearance and personality. However, despite her dainty appearance, Katie proves her true strength and resilience from within through dealing with her pain and problems directly, instead of allowing hardship to define her<ref>Ghost Ship DVD (2003) - Extra features and behind the scenes</ref> ====General Themes==== Light vs. darkness, good vs. evil, childhood innocence vs. wickedness, tragedy vs. triumph, simpler more wholesome times, vs. the complexity and loss of innocence of today.<ref> [http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]</ref> == Conclusion == On the surface, Katie’s story is an emotional tale of heartache and tragedy – Something that no young person should ever have to endure. However, Katie’s story is also essential to the plot. <ref>[http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]</ref> Through helping each other and being willing to put themselves at risk, Katie and Epps together achieve a triumph out of the tragedy and deliver a major blow to Ferriman and the forces of evil. Although the struggle over the souls of people will continue, this victory of the human spirit gives hope even in the face of terrible suffering, evil and even death.<ref>Ghost Ship DVD (2003) - Extra features and behind the scenes</ref> “Katie is really a very sweet girl. She's completely innocent…“ says Emily Jane Browning, who obviously became completely immersed in her character. “…She's been hoping someone would come onto the ship to be her friend, so when Epps arrives she's very excited – they develop a real friendship."<ref>[http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]</ref> ”Emily did an amazing job,” Director Steve Beck enthuses. “She gave Katie a real complexity…she's not just a little girl caught up in a ghost story…” Both Browning and Beck also added that underneath all her heartfelt emotions, Katie longs for vindication and justice.<ref> [http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]</ref> The deeply symbolic and metaphorical aspects surrounding Katie’s story along with the close relationships she develops with others lends support to the notion that the film was originally intended to be more than just another Hollywood “slasher” horror. This is further substantiated by Katie’s tearful and heartfelt story – Emotionally poignant and touching drama not usually associated with a horror picture.<ref>http://www.dailyscript.com/scripts/ghost_ship_info.txt</ref> == References == ===Footnotes=== {{Reflist|colwidth=30em}} ===Cited Texts=== {{refbegin}} *{{cite book| last=Barclay | first=William | title=The Gospel of Mark |publisher=Westminster Press | location=Philadelphia, PA | isbn=0664213022 | year=1975}} *{{cite book| last=Barton |first=John | title=The Oxford Bible Commentary |publisher=Oxford University Press |location=New York, NY | isbn=0-19-875500-7 | year=2001}} *{{cite book| last=Boettner |first=Loraine | title=Roman Catholicism |publisher=Presbyterian & Reformed Publishing Co. |location=Phillipsburg, NJ |isbn=0875521304 | year=1985}} *{{cite book| last=Buttrick |first=George | title=The Interpreter's Bible vol. 12 |publisher=Abingdon Press |location=Nashville, TN |asin=B000HTP248 | year=1957}} *{{cite book |title=Collins English Dictionary - Complete & Unabridged 10th Edition |date=2009 | publisher=Harper Collins |location=New York, NY}} *{{cite book| last=Dummelow |first=J.R. | title=Commentary on the Whole Bible |publisher=Macmillian Publishing Co. |location=New York, NY | year=1936}} *{{cite book| last=Gaebelein |first=Frank | title=The Expositors Bible Commentary vol. 1 |publisher=Zondervan |location=Grand Rapids, MI |isbn=0310364302 | year=1979}} *{{cite book| last=Gauding |first=Madonna | title=The Signs and Symbols Bible |publisher=Sterling Publishing Co. |location=New York, NY |isbn=1402770049| year=2009}} *{{cite book| last=Jamieson |first=Fausset | title=Commentary on the Whole Bible |publisher=Zondervan |location=Grand Rapids, MI| asin=B004BCQP8O | year=1971}} *{{cite book| last=Keck |first=Leander | title=New Interpreter's Bible vol. 9 |publisher=Abingdon Press |location=Nashville, TN |isbn=0687278228 | year=2002}} *{{cite book| title=Life Application Bible - New International Version |publisher=Tyndale House Publishers, Inc. |location=Wheaton, IL | lccn=90-71553| year=1991}} *{{cite book| last=Towns |first=Elmer | title=Liberty Bible Commentary |publisher=Thomas Nelson Inc |location=Nashville, TN |isbn=0840752954 | year=1983}} *{{cite book| last=Wall |first=Robert | title=The New Interpreter's Bible vol. 10 |publisher=Abingdon Press |location=Nashville, TN |isbn=0687278236 | year=2002}} *{{cite book| last=Whalen |first=John | title=New Catholic Encyclopedia Vol. 11 |publisher=Catholic University of America |location=Washington D.C. | lccn=66-22292 | year=1967}} {{refend}} == External links == * [http://www.dailyscript.com/scripts/ghost_ship_info.txt/ Daily Script_Ghost Ship] * [http://www.imdb.com/title/tt0288477/ Internet Movie Database_Ghost Ship 2002] * [http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes] * [http://www.rottentomatoes.com/m/ghost_ship/ Rotten Tomatoes movie review_Ghost Ship] * [https://www.beyondthecreepychild.com Beyond the Creepy Child: The Katie Harwood Project] [[Category:Analysis]] [[Category:Film]] dn0sljpfuw70uvaxkoefw5xhcsi1dc1 0 136794 2844641 2843594 2026-11-05T17:18:02Z Young1lim 21186 /* Copilot Notes */ 2844641 wikitext text/x-wiki This course belongs to the [[Electrical & Computer Engineering Studies]] == Introduction == * Introduction ([[Media:SysP.Intro.20161128.pdf|pdf]]) == File System == * File System ([[Media:SysP.FileSystem.20251023.pdf|pdf]]) * File Pointer ([[Media:SysP..FilePointer.20161103.pdf|pdf]]) * System Calls ([[Media:SysP.File.SysCall.20161128.pdf|pdf]]) * File IO ([[Media:SysP.FileIO.20251023.pdf|pdf]]) === Copilot Notes === * File System ([[Media:glibcFileSystem.20251029-2.pdf|pdf]]) * File Descriptor Table ([[Media:CP.File.Descriptor.20260811.pdf|pdf]]) * File Buffer ([[Media:glibcFileBuffer.20251025-2.pdf|pdf]]) * File IO ([[Media:glibcFileIO.20251025-2.pdf|pdf]]) * File Permission ([[Media:glibcFilePerm.20260122.pdf|pdf]]) * File Control ([[Media:CP.FileCntl.20260428.pdf|pdf]]) :- Overview ([[Media:CP.FileCntl.A.20260824.pdf|pdf]]) :- Duplicating ([[Media:CP.FileCntl.B.20261102.pdf|pdf]]) :- Locking ([[Media:CP.FileCntl.B.20260504.pdf|pdf]]) :- IO ([[Media:CP.FileCntl.C.20260501.pdf|pdf]]) <br> <br> == Process == * Process ([[Media:SysP.Process.20251120.pdf|pdf]]) * Fork ([[Media:SysP.Fork.20251126.pdf|pdf]]) * Copilot: Process Information ([[Media:glibc.Process.1Info.20251101.pdf|pdf]]) * Copilot: Process Control ([[Media:glibc.Process.2Control.20251103.pdf|pdf]]) * Copilot: Process Execution ([[Media:glibc.Proc.3Exec.20251105.pdf|pdf]]) * Copilot: Process Fork ([[Media:glibc.Proc.4Fork.20251106.pdf|pdf]]) * Copilot: Process Context Switching ([[Media:glibc.Proc.5Context.20251107.pdf|pdf]]) * Copilot: Process Exec family of functions ([[Media:glibc.Proc.6ExecCall.20251112.pdf|pdf]]) * Copilot: Process Wait family of functions ([[Media:glibc.Proc.7WaitCall.20251112.pdf|pdf]]) * Copilot: Process Exit ([[Media:glibc.Proc.8Exit.20251113.pdf|pdf]]) </br> == Inter Process Communication== === Signal === * Signal ([[Media:SysP.7.A.Signal.20121206.pdf|pdf]]) * Copilot: Signal 1. Alarm ([[Media:glibc.Signal.Alarm.20251201.pdf|pdf]]) * Copilot: Signal 2. Other Functions ([[Media:glibc.Signal.2Other.20251205.pdf|pdf]]) </br> === Pipe === * Pipe ([[Media:SysP.3.A.IPC.20121115.pdf|pdf]]) * Copilot: Pipe 1. A Special File ([[Media:glibc.Pipe.File.20260307.pdf|pdf]]) </br> === System V IPC === * Message Queue ([[Media:SysP.5.A.MessageQ.20121213.pdf|pdf]]) * Shared Memory ([[Media:SysP.8.A.SharedMem.20121227.pdf|pdf]]) * Semaphore ([[Media:SysP.6.A.Semaphore.20251215.pdf|pdf]]) </br> * Copilot: Message Queue ([[Media:glibc.MessageQ.20251202.pdf|pdf]]) * Copilot: Shared Memory ([[Media:glibc.SharedMem.20251203.pdf|pdf]]) * Copilot: Semaphore ([[Media:glibc.Semaphore.20251215.pdf|pdf]]) </br> === Socket === * Socket ([[Media:SysP.4.A.Socket.20121122.pdf|pdf]]) </br> == Thread == * POSIX thread (pthread) ([[Media:SysP.9.A.Pthread.20130225.pdf|pdf]]) ==External links== * [http://www.tldp.org/LDP/tlk/tlk.html The Linux Kernel] * [http://www.tldp.org/LDP/lpg/lpg.html The Linux Programmer's Guide] * [http://www.cs.cf.ac.uk/Dave/C/ Programming in C - UNIX System Calls and Subroutines using C.] * [http://www.cs.cmu.edu/afs/cs/academic/class/15492-f07/www/pthreads.html POSIX thread (pthread) libraries] * [https://computing.llnl.gov/tutorials/pthreads/#Thread POSIX Threads Programming] [[Category:Linux]] [[Category:Computer programming]] [[Category:C programming language]] ibdf5k866esgona750cstsyko2k20t9 2844643 2844641 2026-11-05T17:19:37Z Young1lim 21186 /* Copilot Notes */ 2844643 wikitext text/x-wiki This course belongs to the [[Electrical & Computer Engineering Studies]] == Introduction == * Introduction ([[Media:SysP.Intro.20161128.pdf|pdf]]) == File System == * File System ([[Media:SysP.FileSystem.20251023.pdf|pdf]]) * File Pointer ([[Media:SysP..FilePointer.20161103.pdf|pdf]]) * System Calls ([[Media:SysP.File.SysCall.20161128.pdf|pdf]]) * File IO ([[Media:SysP.FileIO.20251023.pdf|pdf]]) === Copilot Notes === * File System ([[Media:glibcFileSystem.20251029-2.pdf|pdf]]) * File Descriptor Table ([[Media:CP.File.Descriptor.20260811.pdf|pdf]]) * File Buffer ([[Media:glibcFileBuffer.20251025-2.pdf|pdf]]) * File IO ([[Media:glibcFileIO.20251025-2.pdf|pdf]]) * File Permission ([[Media:glibcFilePerm.20260122.pdf|pdf]]) * File Control ([[Media:CP.FileCntl.20260428.pdf|pdf]]) :- Overview ([[Media:CP.FileCntl.A.20260824.pdf|pdf]]) :- Duplicating ([[Media:CP.FileCntl.B.20261103.pdf|pdf]]) :- Locking ([[Media:CP.FileCntl.B.20260504.pdf|pdf]]) :- IO ([[Media:CP.FileCntl.C.20260501.pdf|pdf]]) <br> <br> == Process == * Process ([[Media:SysP.Process.20251120.pdf|pdf]]) * Fork ([[Media:SysP.Fork.20251126.pdf|pdf]]) * Copilot: Process Information ([[Media:glibc.Process.1Info.20251101.pdf|pdf]]) * Copilot: Process Control ([[Media:glibc.Process.2Control.20251103.pdf|pdf]]) * Copilot: Process Execution ([[Media:glibc.Proc.3Exec.20251105.pdf|pdf]]) * Copilot: Process Fork ([[Media:glibc.Proc.4Fork.20251106.pdf|pdf]]) * Copilot: Process Context Switching ([[Media:glibc.Proc.5Context.20251107.pdf|pdf]]) * Copilot: Process Exec family of functions ([[Media:glibc.Proc.6ExecCall.20251112.pdf|pdf]]) * Copilot: Process Wait family of functions ([[Media:glibc.Proc.7WaitCall.20251112.pdf|pdf]]) * Copilot: Process Exit ([[Media:glibc.Proc.8Exit.20251113.pdf|pdf]]) </br> == Inter Process Communication== === Signal === * Signal ([[Media:SysP.7.A.Signal.20121206.pdf|pdf]]) * Copilot: Signal 1. Alarm ([[Media:glibc.Signal.Alarm.20251201.pdf|pdf]]) * Copilot: Signal 2. Other Functions ([[Media:glibc.Signal.2Other.20251205.pdf|pdf]]) </br> === Pipe === * Pipe ([[Media:SysP.3.A.IPC.20121115.pdf|pdf]]) * Copilot: Pipe 1. A Special File ([[Media:glibc.Pipe.File.20260307.pdf|pdf]]) </br> === System V IPC === * Message Queue ([[Media:SysP.5.A.MessageQ.20121213.pdf|pdf]]) * Shared Memory ([[Media:SysP.8.A.SharedMem.20121227.pdf|pdf]]) * Semaphore ([[Media:SysP.6.A.Semaphore.20251215.pdf|pdf]]) </br> * Copilot: Message Queue ([[Media:glibc.MessageQ.20251202.pdf|pdf]]) * Copilot: Shared Memory ([[Media:glibc.SharedMem.20251203.pdf|pdf]]) * Copilot: Semaphore ([[Media:glibc.Semaphore.20251215.pdf|pdf]]) </br> === Socket === * Socket ([[Media:SysP.4.A.Socket.20121122.pdf|pdf]]) </br> == Thread == * POSIX thread (pthread) ([[Media:SysP.9.A.Pthread.20130225.pdf|pdf]]) ==External links== * [http://www.tldp.org/LDP/tlk/tlk.html The Linux Kernel] * [http://www.tldp.org/LDP/lpg/lpg.html The Linux Programmer's Guide] * [http://www.cs.cf.ac.uk/Dave/C/ Programming in C - UNIX System Calls and Subroutines using C.] * [http://www.cs.cmu.edu/afs/cs/academic/class/15492-f07/www/pthreads.html POSIX thread (pthread) libraries] * [https://computing.llnl.gov/tutorials/pthreads/#Thread POSIX Threads Programming] [[Category:Linux]] [[Category:Computer programming]] [[Category:C programming language]] 1zef8lt5qjac6hk09col21y6aq8wxdq 0 139384 2844623 2844471 2026-11-05T13:53:10Z Young1lim 21186 /* Adder */ 2844623 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.20261105.pdf|B]], [[Media:VLSI.Arith.2C.CLA.20261105.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]] gj10396q4qnxdkjqp3f95xf6vyrnsfo 0 208510 2844715 2463701 2026-11-06T10:25:19Z ~2026-58926-46 3113902 There was only a paragraph of 3 lines on atheism i have made a full article on atheism with out insulting other belief systems and religion. 2844715 wikitext text/x-wiki [[File:Atheismsymbol_endorsed_by_AAI.svg|thumb|120x120px|Symbol designed by Atheist Alliance International in 2007]] = Atheism = '''Atheism''' is broadly defined in the absence of belief in the existence of deities. More narrowly, atheism is specifically the position that there are no deities. Atheism is contrasted with theism, which in its general form is the belief that at least one deity exists. The term ''atheism'' originated from the ancient Greek ἄθεος (''atheos''), meaning "without god(s)". Initially used as a derogatory term applied to those thought to reject the gods worshipped by the larger society, the term came to denote a conscious position regarding the existence of gods following the Enlightenment and the growth of secularism. == Etymology and Definitions == Defining atheism can be complex, as philosophers and scholars draw distinctions between different forms of atheism: * '''Implicit vs. Explicit Atheism:''' Implicit atheism refers to the absence of belief in deities without a conscious rejection of the concept (such as infants or individuals who have never been exposed to religious ideas). Explicit atheism is the conscious rejection or absence of belief after considering the proposition. * '''Positive vs. Negative Atheism:''' Negative (or "weak") atheism generally includes stances where an individual does not believe in the existence of any deity, without necessarily asserting that no deities could possibly exist. Positive (or "strong") atheism explicitly asserts that deities do not exist. * '''Agno-atheism:''' A stance where an individual does not believe in gods (atheist) while also holding that the existence of ultimate reality or deities is unknown or unknowable (agnostic). == History == The history of atheism spans philosophical traditions across various cultures, evolving from early philosophical critique to organized secular movements. === Antiquity === In ancient India, the '''Cārvāka''' (or Lokāyata) school, dating back to at least the 6th century BCE, was a philosophical tradition that rejected supernaturalism, ritualism, and the authority of the Vedas, advocating for strict materialism and empiricism. In ancient Greece, pre-Socratic philosophers attempted to explain the natural world through physical elements rather than divine intervention. Later, philosophers such as '''Epicurus''' and '''Lucretius''' argued that gods, even if they existed, played no role in human affairs and that death was simply the cessation of consciousness. === Enlightenment and Modern Era === The Scientific Revolution and the European Enlightenment marked a significant shift. Thinkers like '''Baruch Spinoza''', '''Denis Diderot''', and later '''David Hume''' and '''Immanuel Kant''' critically examined religious dogmas and arguments for the existence of God. The 19th and 20th centuries saw the formal articulation of comprehensive non-theistic worldviews. '''Karl Marx''' and '''Friedrich Engels''' critiqued religion as a socio-economic tool used by ruling classes to maintain order and dull the alienation of the working class. Concurrently, '''Friedrich Nietzsche''' famously declared the "death of God," highlighting the decline of Christianity's cultural authority in Europe. == Philosophical Arguments == Arguments for atheism generally focus on critiquing traditional proofs of God or highlighting conceptual inconsistencies within theism. * '''The Problem of Evil:''' This argument posits that the existence of unnecessary and immense suffering in the world is logically inconsistent with the simultaneous existence of an omnipotent, omniscient, and wholly benevolent deity. * '''Inconsistent Revelations:''' This argument notes the vast number of mutually contradictory religions and divine revelations throughout human history, suggesting that none have a superior claim to truth. * '''The Burden of Proof:''' Known in philosophy as ''Russell's teapot'' argument, this position asserts that the burden of proof regarding the existence of a deity lies with the claimant, and in the absence of verifiable evidence, the default position is non-belief. * '''Ultimate Boeing 747 gambit:''' Popularized by Richard Dawkins, this argument contends that any entity complex enough to design the universe would itself require an even more complex explanation, rendering divine creation an inadequate explanatory tool. == Demographics == Measuring global atheism presents challenges due to varying definitions, cultural stigma, and government restrictions in certain regions. According to demographic studies by organizations such as the Pew Research Center, populations identifying as "religiously unaffiliated" (which includes atheists, agnostics, and those who identify as "nothing in particular") make up a significant portion of the global population, particularly in parts of Europe and East Asia (such as China, Japan, and South Korea) - Published Kuldeep Singh Bhati == Readings == * [[Wikipedia: Atheism]] == References == {{Reflist}} [[Category:Atheism]] fbsj03klkpr5dpyj6el8cl3bdxw7x2t 10 217174 2844717 2771085 2026-11-06T11:55:46Z CommonsDelinker 9184 Removing [[:c:File:American-flag-2a.jpg|American-flag-2a.jpg]], it has been deleted from Commons by [[:c:User:Ruthven|Ruthven]] because: [[:c:COM:NETCOPYRIGHT|Copyright violation]], found elsewhere on the web and unlikely to be own work ([[:c:COM:CSD#F1|F 2844717 wikitext text/x-wiki <div style="float:left; border:1px solid #ffd700; margin:1px;"> {| cellspacing="0" style="width:238px; background:#fffacd;{{text color default}};" | style="width:45px; height:45px; background:red;{{text color default}}; text-align:center;" | ''' ''' | style="font-size:8pt; padding:4pt; line-height:1.25em; color:black; text-align: center;" | This user is from '''[[Comparative law and justice/United States|the United States]]''' |}</div><noinclude>[[Category:Americans]]</noinclude><includeonly>{{{category|[[Category:Americans|{{PAGENAME}}]]}}}</includeonly> 0l8c5dwdr778wf6kk7llngivigm276v 11 261532 2844632 2440395 2026-11-05T16:05:35Z Dick Bos 24466 /* Updating */ new section 2844632 wikitext text/x-wiki ==Unlink "doi" redlink?== Is there a way to unlink the "[[digital object identifier|doi]]" redlink in this template akin to how it's not linked in {{tlx|cite journal}}? Thanks! —[[User:Bobamnertiopsis|Collin]] (Bobamnertiopsis)^{[[User talk:Bobamnertiopsis|t]] [[Special:Contributions/Bobamnertiopsis|c]]} 23:57, 6 April 2020 (UTC) :{{At|DannyS712}} Is this something you could help with? -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 12:52, 7 April 2020 (UTC) ::I'm in the middle of something on wikidata, but I'll take a look in a few minutes --[[User:DannyS712|DannyS712]] ([[User talk:DannyS712|discuss]] • [[Special:Contributions/DannyS712|contribs]]) 12:53, 7 April 2020 (UTC) ::I'm sorry, I'm not familiar enough with the modules to figure this out. Perhaps we should create the page so it isn't redlinked? --[[User:DannyS712|DannyS712]] ([[User talk:DannyS712|discuss]] • [[Special:Contributions/DannyS712|contribs]]) 13:15, 7 April 2020 (UTC) :::I added a {{tlx|soft redirect}}. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 19:38, 8 April 2020 (UTC) == Generate "(ed.)"/"(eds.)" when citing a chapter in an edited volume? == Right now, using this citation template's <code>editor=</code> and related editor fields (e.g., <code>editor1-last=</code>, etc.) does not result in automatically appending the editor name(s) with "(ed.)" or "(eds.)" which results in a clunky and incomplete-looking citation: * {{cite book|last1=Gonsalves|first1=Sharon|title=Bisexual Politics: Theories, Queries, & Visions|editor1-last=Tucker|editor1-first=Naomi|date=1995|publisher=Harrington Park Press|isbn=1-56023-869-0|chapter=To Give or Not to Give|pages=115–118}} This looks okay but is muddies the editor's role relative to the cited text. It also looks worse if only initials are being used: * {{cite book|last1=Gonsalves|first1=S.|title=Bisexual Politics: Theories, Queries, & Visions|editor1-last=Tucker|editor1-first=N.|date=1995|publisher=Harrington Park Press|isbn=1-56023-869-0|chapter=To Give or Not to Give|pages=115–118}} As this implementation outputs a second period after the last editor's first name regardless of whether a period is already there. The English Wikipedia version of this template outputs something like this: * {{cite book|last1=Gonsalves|first1=Sharon|title=Bisexual Politics: Theories, Queries, & Visions|editor1-last=Tucker|editor1-first=Naomi (ed.)|date=1995|publisher=Harrington Park Press|isbn=1-56023-869-0|chapter=To Give or Not to Give|pages=115–118}} But this "(ed.)" has to be manually included to appear in Wikiversity's version of the template (as it has been in this example: <code>|editor1-first=Naomi (ed.)</code>). The template documentation makes it clear that this is currently the preferred method ("No text is added, so labels such as '[ed.]' have to be supplied by the user.") but this neither seems to be the best nor only solution. Would it be possible to implement English Wikipedia's automatic "(ed.)"/"(eds.)" when editor fields are used? Thanks! —[[User:Bobamnertiopsis|Collin]] (Bobamnertiopsis)^{[[User talk:Bobamnertiopsis|t]] [[Special:Contributions/Bobamnertiopsis|c]]} 16:59, 17 October 2022 (UTC) == Updating == Would there be no possibility to update this template, at the same time as the template on en-wikipedia is updated? Now there are quite a few features that do not work in the same way as they do on wikipedia, or that do not work at all. --[[User:Dick Bos|Dick Bos]] ([[User talk:Dick Bos|discuss]] • [[Special:Contributions/Dick Bos|contribs]]) 16:05, 5 November 2026 (UTC) ghnhcl7sn134hie1ei5a4cujbyjwn8i 0 285380 2844626 2844478 2026-11-05T14:20:49Z Young1lim 21186 /* Applications */ 2844626 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.20261103.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> p2461nm9lbjurhyiyqrpks7tptzjswi 2844628 2844626 2026-11-05T14:22:14Z Young1lim 21186 /* Applications */ 2844628 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.20261104.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> ehdlkax3ksh4ceabbutfgwqh8ph6ua5 2 289273 2844684 2844593 2026-11-05T22:00:47Z Dc.samizdat 2856930 /* General relativity is the geometry of gravity in Euclidean 4-space */ 2844684 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>. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, state its unique conservation law, and model that conservation law geometrically in Euclidean 4-space. That is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in 3-space: it is the Kepler problem, hosted in Euclidean 4-space. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell). The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We conclude that the rotational geometry of the proton is that of the 4-simplex. The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes. The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each element includes that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, there are only 120 distinct atomic elements. The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. It turns out that if you view the 4-simplex as the largest 4-polytope instead of the smallest, you can find smaller instances of all the other 4-polytopes within it. We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. .... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <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}} oikp230sggaoe39r6jal3b9vb2kotvj 2844686 2844684 2026-11-05T22:02:09Z Dc.samizdat 2856930 /* General relativity is the geometry of gravity in Euclidean 4-space */ 2844686 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, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, state its unique conservation law, and model that conservation law geometrically in Euclidean 4-space. That is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in 3-space: it is the Kepler problem, hosted in Euclidean 4-space. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell). The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We conclude that the rotational geometry of the proton is that of the 4-simplex. The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes. The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each element includes that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, there are only 120 distinct atomic elements. The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. It turns out that if you view the 4-simplex as the largest 4-polytope instead of the smallest, you can find smaller instances of all the other 4-polytopes within it. We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. .... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <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}} 0q6jmwae3v0qdwmnt2xjo4al5vwjqw8 2844687 2844686 2026-11-05T22:03:58Z Dc.samizdat 2856930 /* An object's motion in space is the local product of its discrete self-reflections */ 2844687 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. 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, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, state its unique conservation law, and model that conservation law geometrically in Euclidean 4-space. That is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in 3-space: it is the Kepler problem, hosted in Euclidean 4-space. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell). The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We conclude that the rotational geometry of the proton is that of the 4-simplex. The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes. The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each element includes that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, there are only 120 distinct atomic elements. The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. It turns out that if you view the 4-simplex as the largest 4-polytope instead of the smallest, you can find smaller instances of all the other 4-polytopes within it. We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. .... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <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}} 4n83s0vfp3by6jj857gzvq2a3szd86p 2844689 2844687 2026-11-05T22:05:48Z Dc.samizdat 2856930 /* General relativity is the geometry of gravity in Euclidean 4-space */ 2844689 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. 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, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, state its unique conservation law, and model that conservation law geometrically in Euclidean 4-space. That is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in 3-space.<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> It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell). The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We conclude that the rotational geometry of the proton is that of the 4-simplex. The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes. The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each element includes that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, there are only 120 distinct atomic elements. The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. It turns out that if you view the 4-simplex as the largest 4-polytope instead of the smallest, you can find smaller instances of all the other 4-polytopes within it. We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. .... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <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}} dvt96vqdeimjzz7lide9ftftdxv20p7 2844691 2844689 2026-11-05T22:39:09Z Dc.samizdat 2856930 /* General relativity is the geometry of gravity in Euclidean 4-space */ 2844691 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. 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, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space and therefore visualizable, provided we can stretch our dimensional analogy muscles to visualize perspective in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, state its unique conservation law, and model that conservation law geometrically in Euclidean 4-space. That is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in 3-space.<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> It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell). The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We conclude that the rotational geometry of the proton is that of the 4-simplex. The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes. The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each element includes that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, there are only 120 distinct atomic elements. The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. It turns out that if you view the 4-simplex as the largest 4-polytope instead of the smallest, you can find smaller instances of all the other 4-polytopes within it. We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. .... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <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}} 30vt0m7ceh312e9hnhzc9qgvjv01l6b 2844692 2844691 2026-11-05T22:48:00Z Dc.samizdat 2856930 /* General relativity is the geometry of gravity in Euclidean 4-space */ 2844692 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. 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, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space and therefore visualizable, provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, state its unique conservation law, and model that conservation law geometrically in Euclidean 4-space. That is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in 3-space.<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> It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell). The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We conclude that the rotational geometry of the proton is that of the 4-simplex. The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes. The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each element includes that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, there are only 120 distinct atomic elements. The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. It turns out that if you view the 4-simplex as the largest 4-polytope instead of the smallest, you can find smaller instances of all the other 4-polytopes within it. We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. .... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <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}} qobdo7gaao6j5nazfwaig0gpxad1re4 2844693 2844692 2026-11-05T22:50:19Z Dc.samizdat 2856930 /* General relativity is the geometry of gravity in Euclidean 4-space */ 2844693 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. 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, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, state its unique conservation law, and model that conservation law geometrically in Euclidean 4-space. That is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in 3-space.<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> It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell). The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We conclude that the rotational geometry of the proton is that of the 4-simplex. The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes. The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each element includes that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, there are only 120 distinct atomic elements. The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. It turns out that if you view the 4-simplex as the largest 4-polytope instead of the smallest, you can find smaller instances of all the other 4-polytopes within it. We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. .... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <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}} fbcht1v3v8tu88a3ub5jctujp5wpu49 2844694 2844693 2026-11-05T23:07:14Z Dc.samizdat 2856930 /* General relativity is the geometry of gravity in Euclidean 4-space */ 2844694 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. 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, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, state its unique conservation law, and model that conservation law geometrically in Euclidean 4-space. That is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.<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> It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell). The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We conclude that the rotational geometry of the proton is that of the 4-simplex. The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes. The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each element includes that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, there are only 120 distinct atomic elements. The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. It turns out that if you view the 4-simplex as the largest 4-polytope instead of the smallest, you can find smaller instances of all the other 4-polytopes within it. We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. .... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <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}} 6jsq2rkb567yhzb3wiljim62x6c2a2k 2844695 2844694 2026-11-05T23:16:52Z Dc.samizdat 2856930 /* General relativity is the geometry of gravity in Euclidean 4-space */ 2844695 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. 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, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space, the stretch to imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, state its unique conservation law, and model that conservation law geometrically in Euclidean 4-space. That is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.<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> It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell). The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We conclude that the rotational geometry of the proton is that of the 4-simplex. The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes. The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each element includes that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, there are only 120 distinct atomic elements. The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. It turns out that if you view the 4-simplex as the largest 4-polytope instead of the smallest, you can find smaller instances of all the other 4-polytopes within it. We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. .... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <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}} 3hqe8du99swujes4m03bw6esfm575ac 2844696 2844695 2026-11-05T23:18:14Z Dc.samizdat 2856930 /* General relativity is the geometry of gravity in Euclidean 4-space */ 2844696 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. 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, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. It is a challenge to visualize this double rotation. General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space, the stretch to imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, state its unique conservation law, and model that conservation law geometrically in Euclidean 4-space. That is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.<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> It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell). The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We conclude that the rotational geometry of the proton is that of the 4-simplex. The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes. The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each element includes that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, there are only 120 distinct atomic elements. The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. It turns out that if you view the 4-simplex as the largest 4-polytope instead of the smallest, you can find smaller instances of all the other 4-polytopes within it. We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. .... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <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}} fvd5g7p2rd0qzy9wqgdn4xfxz2dfg15 2844710 2844696 2026-11-06T05:26:12Z Dc.samizdat 2856930 /* An object's motion in space is the local product of its discrete self-reflections */ 2844710 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. 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 our universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, these motions occur only at the constant velocity <math>c</math>, and 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, yielding a measurement of the rate of change of the observer's direction of motion through 4-space at velocity <math>c</math>. It is a challenge to visualize this double rotation. General relativity explains gravity geometrically as a non-Euclidean geometry in spacetime that is more complex than special relativity's non-Euclidean Lorentz geometry in spacetime. In Euclidean 4-space the geometry of special relativity becomes simpler Euclidean geometry, dimensionally analogous to familiar perspective geometry in ordinary Euclidean 3-space, and therefore visualizable provided we can stretch our dimensional analogy muscles enough to imagine perspective views in four orthogonal spatial dimensions. A similar stretch is required of us to visualize general relativity in Euclidean 4-space, the stretch to imagine distinct rotations in four spatial dimensions. To understand gravity geometrically in four dimensions, we must identify the distinct rotation the accelerating observer is performing, state its unique conservation law, and model that conservation law geometrically in Euclidean 4-space. That is dimensionally analogous to the problem that Kepler and Newton solved for the rotation of the planets in Euclidean 3-space.<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> It is a [[W:Kepler problem|Kepler problem]], hosted in Euclidean 4-space. ... == Quantum mechanics is the rotational kinematics of Coxeter groups in Euclidean 4-space == Atoms are 4-polytopes, small round 4-dimensional objects, and their rotational symmetry is compounded from that of the [[Polyscheme#Regular convex 4-polytopes|six regular convex 4-polytopes]]. Their geometry is beyond the scope of this paper, but we begin with the proton, which we identify with the rotational symmetry of the 4-simplex polytope. Schläfli discovered that there exists a compound series of six regular convex 4-polytopes, dimensionally analogous to the compound series of five regular convex 3-polytopes, the Platonic solids. Polytopes in each series are naturally ordered by size, as a measure of content for the same radius, which is also a measure of their roundness, the number of their vertices. They form a series of increasingly round polytopes in which each is a compound of multiple instances of its predecessor (with one exception to this progression which will concern us later). The six regular convex 4-polytopes have 5, 8, 16, 24, 120, and 600 vertices respectively. They are the 5-point (5-cell) 4-simplex, the 8-point (16-cell) 4-orthoplex, the 16-point (8-cell) 4-hypercube, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell). The smallest (least round) regular convex 4-polytope is the [[5-cell|4-simplex]]. It has 5 vertices, 10 edges, 10 triangular faces, and 5 tetrahedral cells. The regular 4-simplex is the configuration of 5 mutually equidistant points in Euclidean space, on the surface of a 3-sphere. We conclude that the rotational geometry of the proton is that of the 4-simplex. The largest (roundest) regular convex 4-polytope is called the [[120-cell|120-cell]]. It has 600 vertices and 1200 edges, because it is a regular compound of 120 completely disjoint 4-simplexes, rotated isoclinically with respect to each other on the surface of a 3-sphere. It is also a regular compound of every other regular convex 4-polytope, so that instances of every regular convex polytope which can exist in Euclidean 4-space are found within it. It is the geometrically comprehensive 4-polytope, containing instances of the relationships among all these polytopes. The nucleus of a distinct atomic element contains a characteristic number of protons, its atomic number. The rotational symmetry of each element includes that of a compound of its distinct number of completely disjoint 4-simplex polytopes, that is, the symmetry of a sparsely populated 120-cell. Because only 120 4-simplexes will fit into a 120-cell, there are only 120 distinct atomic elements. The exception to the rule that each 4-polytope in the series compounds to its successor is the 5-point 4-simplex, which does not compound to its successor the 8-point 4-orthoplex, nor to any other 4-polytope in the series except the largest, the 600-point 120-cell. None of the 4-polytopes in the series are divisible into an integral number of 4-simplexes except the 120-cell. Every 4-polytope in the series except the 4-simplex is divisible into an integral number of 4-orthoplexes. Thus there are two alternate integral building blocks of the 120-cell: the 5-point 4-simplex, and the 8-point 4-orthoplex. Only the 4-orthoplex will build the other three 4-polytopes in the series by simple compounding. However, the symmetry group of the 4-simplex also has a foundational relationship to the symmetries of all the 4-polytopes in the series. It turns out that if you view the 4-simplex as the largest 4-polytope instead of the smallest, you can find smaller instances of all the other 4-polytopes within it. We identify the neutron with the rotational symmetry of the 8-point 4-orthoplex. .... == Dimensional relativity == The formulation of Euclidean relativity based on Coxeter's kinetic law of <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}} p1zd1l2ggthhstr98ck8d8qrro9t3hk 0 299624 2844716 2812781 2026-11-06T11:41:00Z Sylvain Ribault 2127778 Exact exponents 2844716 wikitext text/x-wiki In two-dimensional critical loop models, the monochromatic exponents are a series of universal quantities that govern the behaviour of probabilities of connections by a number of disjoint paths. The first non-trivial case is the backbone exponent. The expression for this exponent is more complicated than those for other known critical exponents. In particular, in the special case of critical percolation, the backbone exponent is a transcendent number, while other known exponents are rational numbers.<ref name="nqsz23"/> The determination of the backbone exponent raises two questions: * Determining the other monochromatic exponents. * Finding a simple derivation. In particular, the known derivation couples the loop model to quantum Liouville gravity. Is it possible to avoid this step? == The backbone exponent == === Original formulation === For <math>4<\kappa<8</math> the parameter of a conformal loop ensemble, the backbone exponent <math>\xi</math> obeys<ref name="nqsz23"/> :<math> \sin\left(\tfrac{8\pi}{\kappa}\right) \sqrt{\tfrac12 \kappa \xi +\left(1-\tfrac{\kappa}{4}\right)^2} = \sin\left(\tfrac{8\pi}{\kappa}\sqrt{\tfrac12 \kappa \xi +\left(1-\tfrac{\kappa}{4}\right)^2}\right) \quad ,\quad 0<\xi <1-\tfrac{2}{\kappa} </math> Depending on <math>\kappa</math>, this equation has one or two solutions. If there are two solutions, one of them is <math>1-\tfrac{\kappa}{8}</math>, and the exponent is the other one. === CFT variables === The parameter <math>\kappa</math> is related to the central charge <math>c</math> and the contractible loop weight <math>n</math> by :<math> c= 13-6\beta^2 -6\beta^{-2} \quad , \quad \beta =\frac{2}{\sqrt{\kappa}} \quad , \quad \frac12<\beta^2 <1 \quad , \quad n = -2\cos(\pi\beta^2) </math> The exponent is a conformal dimension, i.e. twice a chiral conformal dimension. The corresponding momentum <math>P</math> and loop weight <math>w</math> are defined by :<math> \xi = \frac{c-1}{12} + 2P^2 \quad \iff \quad P = \sqrt{\tfrac{\xi}{2} + \kappa^{-1}\left(1-\tfrac{\kappa}{4}\right)^2} \quad , \quad w = 2\cos(2\pi \beta P) </math> The momentum is a natural variable in CFT, as it allows many formulas to simplify, including the fusion rules of degenerate fields, and the DOZZ formula for the three-point structure constants of Liouville theory. Special values of the momentums are :<math> P_{(r,s)} = \frac{r}{2}\beta - \frac{s}{2}\beta^{-1} </math> where <math>r,s</math> are Kac table indices, which are integer for degenerate representations of the Virasoro algebra. Many known critical exponents correspond to momentums <math>P_{(r,s)}</math> with rational indices. In terms of the CFT variables <math>\beta, P</math>, the equation that determines the backbone exponent reads :<math> \varphi\left(4\pi\beta P\right) = \varphi\left(4\pi \beta P_{(1,0)}\right) \quad \text{with} \quad \varphi(x)=\frac{\sin(x)}{x} \quad \text{and} \quad \beta^{-1}-\beta < 2P < \beta^{-1} </math> === Derivation === The derivation involves coupling SLE<math>_\kappa</math> to what is called Liouville theory with parameter <math>\beta=\frac{\gamma}{2}</math> and central charge <math>c^f=1+6(\beta+\beta^{-1})^2</math> such that <math>c^f+c=26</math> where <math>c</math> is the loop model's central charge. However, the bulk cosmological constant vanishes, so this is a [[w:Massless_free_scalar_bosons_in_two_dimensions|free bosonic CFT]] with conformal boundaries i.e. boundary conditions that break the affine symmetry. (Such boundary conditions are well-known in the case of compactified free bosons with <math>c=1</math>.<ref name="jan01"/>) The coupling between the loop model and the free boson involves fields with total dimension <math>\Delta+\Delta^f=1</math>. We write <math>\Delta=\frac14(\beta-\beta^{-1})^2 +P^2</math> and <math>\Delta^f = \frac14(\beta+\beta^{-1})^2 -P^2</math>, then the momentum <math>P</math> is related to the "Liouville" bulk or boundary momentum <math>\alpha</math> by <math>\alpha = Q+2P</math> with <math>Q=\beta+\beta^{-1}</math>. In terms of Kac table indices <math>r,s</math> we have <math>P^f_{(r,s)} = \frac{r}{2}\beta + \frac{s}{2}\beta^{-1}= P_{(r,-s)}</math> so that <math>\Delta_{(r,s)}+\Delta^f_{(r,-s)}=1</math>. The fundamental quantity is the joint moment (4.8), which comes from conformal welding (3.15), and reads in our notations :<math> g(P, p) = \nu_\kappa\left[|\psi'(i)|^{-2\Delta_P} |\varphi'(1)|^{\Delta_{(3,1)}-\Delta_p}\right] </math> where <math>\nu_\kappa</math> is the SLE measure, <math>\psi,\varphi</math> are conformal transformations, and <math>P,p</math> are a bulk and a boundary momentum. It turns out that up to simple factors, this quantity is an integral transform of a certain boundary 3-point structure constant, with respect to a boundary cosmological constant. This is written in Eq. (4.9), which has the form :<math> g(P, p) \left\langle V_P B_p\right\rangle \propto \int_0^\infty d\mu \frac{\mu^{-2\beta^{-1}P}}{\mu(\mu+1)} \left\langle B_{P_{(1,0)}}B_{P_{(1,0)}}B_p\right\rangle_{\mu,1,1} </math> where <math>V_P</math> and <math>B_p,B_{P_{(1,0)}}</math> are bulk and boundary fields in "Liouville" theory. The monochromatic momentum is a solution of (1.6), which (up to a detail) reads :<math> g(P, P_{(3,1)}) = 1 </math> And <math>g(P, P_{(3,1)})</math> can be computed by performing the integeral. The appearance of boundary fields of momentum <math>P_{(1,0)}</math> is somewhat natural, since the exponent is related to bulk fields of the same momentum. The appearance of the boundary momentum <math>P_{(3,1)}=P^f_{(3,-1)}</math> is more mysterious. Apparently <math>2\Delta_{(3,1)}</math> is the dimension of the set of touching points of the boundary touching loop. == Other exponents and probabilities == === Monochromatic exponents === Monochromatic exponents are labelled by integers <math>k\in\mathbb{N}^*</math>. They govern probabilities of existence of connections with <math>k</math> disjoint paths. Let <math> P^\text{mono}_k</math> be the corresponding momentums. In particular, the backbone exponent corresponds to <math>k=2</math>. Numerical estimates for the monochromatic exponents for percolation:<ref name="jz02"/> :<math> \xi_2 = 0.3569 \pm 0.0006\quad , \quad \xi_3 = 0.77 \pm 0.02 \quad , \quad \xi_4 = 1.33\pm 0.03 \quad , \quad \xi_5= 2.1\pm 0.2 \quad , \quad \xi_6 = 3\pm 0.3 </math> For <math>\beta^2=\frac12</math>, clusters are trees, and the monochromatic exponents correspond to <math>P_k^\text{mono}= P_{(k,0)}= P_{(0,\frac{k}{2})}</math> so that <math>\xi_k = \frac{k^2-1}{4}</math>. Exact (but very complicated) determination of the monochromatic exponents for percolation:<ref name="cai26"/> === Polychromatic exponents === Polychromatic exponents <math> P^\text{poly}_k</math> are defined similarly to the monochromatic exponents, however the disjoint paths need not all be on clusters, they can also be on dual clusters. The exponents do not change if we assume that the clusters and dual clusters alternate. Clusters and dual clusters are separated by loops, so the polychromatic exponents have an interpretation in loop models. The corresponding momentums are<ref name="ada99"/> :<math> P^\text{poly}_k \underset{k\geq 2}{=} P_{(\frac{k}{2},0)} = \frac{k}{4}\beta </math> In critical loop models, these momentums correspond to the conformal dimensions of the so-called watermelon operators, which create <math>k</math> loops. The case <math>k=1</math> is special, with :<math> P^\text{poly}_1 = P^\text{mono}_1 = P_{(0,\frac12)} </math> In the case of percolation <math>\kappa=6</math> i.e. <math>\beta^2=\frac23</math>, we have the Beffara-Nolin inequalities<ref name="bn09"/> :<math> P^\text{poly}_k < P^\text{mono}_k < P^\text{poly}_{k+1} </math> === Boundary touching probabilities === The expectation value of a certain conformal radius <math>R</math> to a power <math>\lambda = -\xi</math> is<ref name="asyz24"/> :<math> P^2<P_{(0,\frac12)}^2 \implies \mathbb{E}(R^{-\xi}) = -\frac{\cos(\pi \beta^2)}{\cos(2\pi \beta P)} = \frac{n}{w} </math> where the momentum <math>P</math> corresponds to the conformal dimension <math>\xi</math>. This results is then refined: we have <math>\mathbb{E}(R^{-\xi})= \mathbb{E}_1(R^{-\xi})+\mathbb{E}_2(R^{-\xi})</math> depending on whether a certain loop touches the boundary of the unit disk or not, with (Theorem 1.2) :<math> P^2<P_{(1,0)}^2 \implies \mathbb{E}_1(R^{-\xi}) = \frac{2\cos(\pi \beta^2)\sin\left(2\pi (\beta-\beta^{-1})P\right)}{\sin(2\pi \beta^{-1}P)} </math> :<math> P^2<P_{(0,\frac12)}^2 \implies \mathbb{E}_2(R^{-\xi}) = \frac{\cos(\pi \beta^2)\sin\left(2\pi (\beta^{-1}-2\beta)P\right)}{\cos(2\pi \beta P)\sin(2\pi \beta^{-1}P)} </math> A further refinement, corresponding to removing boundary-touching loops, is (Theorem 1.3) :<math> P^2<P_{(1,0)}^2 \implies \widetilde{\mathbb{E}}_2(R^{-\xi}) = \frac{\sin\left(2\pi (2\beta-\beta^{-1})P\right)}{\sin(2\pi \beta^{-1}P)} </math> i.e. <math>\mathbb{E}_2(R^{-\xi}) =\mathbb{E}(R^{-\xi})\widetilde{\mathbb{E}}_2(R^{-\xi})</math>. Then given a parameter <math>a</math> (a sort of loop weight), the corresponding nested path exponent <math>\xi_{NP}</math> is a solution of <math>\widetilde{\mathbb{E}}_2(R^{-\xi_{NP}})=a</math>. The nested loop exponent <math>\xi_{NL}</math> is a solution of <math>\mathbb{E}(R^{-\xi_{NL}})=a</math> i.e. it is the dimension associated to the loop weight <math>w=\frac{n}{a}</math>. === Bulk one-arm exponent === Assume that CLE clusters are randomly colored, with two colors of probabilities <math>r, 1-r</math>. Percolation along clusters of the second color gives rise to a critical exponent <math>\xi=\frac{c-1}{12}+2P^2</math> called the one-arm exponent. To write this exponent it is convenient to introduce :<math> \rho' = \frac{\rho+4}{2} = \frac{1}{\pi}\arctan\left(\frac{\sin(2\pi\beta^{-2})}{\cos(2\pi\beta^{-2})+\frac{r}{r-1}}\right) \in (1,2\beta^{-2}) \quad \iff \quad \frac{r}{r-1}=\frac{\sin\pi(2\beta^{-2}-\rho')}{\sin\pi \rho'} </math> Then the momentum <math>P</math> is a solution of<ref name="lsyz24"/> :<math> \frac{\sin\pi(\beta^{-2}+\rho')}{\sin\pi(\beta^{-2}-\rho')} = \frac{\sin 2\pi(\beta^{-1}+\beta \rho')P}{\sin 2\pi(\beta^{-1}-\beta \rho')P} </math> This equation can be rewritten as :<math> \sum_{\begin{array}{c} \epsilon_i = \pm 1 \\ \epsilon_1\epsilon_2\epsilon_3\epsilon_4 = -1\end{array}} \epsilon_2\epsilon_3 e^{i\pi \sum_{i=1}^4\epsilon_i x_i} = 0 \quad \text{with}\quad (x_1,x_2,x_3,x_4) = \Big(\beta^{-2},\rho',2\beta^{-1}P,2\beta\rho'P\Big) </math> == Speculations == In the case of percolation, the backbone exponent is the leading eigenvalue of a differential operator.<ref name="lsw01"/> === A guess for higher monochromatic exponents in percolation=== We assume that the difference between the monochromatic and polychromatic momentums decreases as <math>\frac{1}{k}</math>. This leads to the formula :<math> P^\text{mono}_k = \frac{k}{4}\beta + \frac{2P^\text{mono}_2-\beta}{k} = \frac{k^2-4}{4k}\beta +\frac{2}{k} P_2^\text{mono} </math> This leads to the numerical values :<math> \xi_2 = 0.3567 \quad , \quad \xi_3 = 0.7692 \quad , \quad \xi_4 = 1.3511 \quad , \quad \xi_5= 2.100 \quad , \quad \xi_6 = 3.017 </math> where we take the exact value of <math>\xi_2</math>, and deduce the other exponents. These are very close to the numerical estimates.<ref name="jz02"/> But this works only for the case of percolation <math>\beta^2=\frac23</math>, in particular this fails for <math>\beta^2=\frac12</math>. === Relation with modular kernels === The modular S-matrix is :<math> S_{P,P'} = \cos(4\pi PP') </math> In rational CFTs, this matrix is relevant to the computation of the annulus partition function. However, it is not quite what appears in the computation of the backbone exponent. A quantity that is more closely related to that computation is the modular kernel :<math> S_{P,P'}[\Delta=1] = \frac{\sin(4\pi PP')}{P'} </math> This kernel describes the modular properties of 1-point conformal blocks on a torus with a primary field of conformal dimension 1. (Why would such a field appear?) If <math>P'</math> is the momentum related to the backbone exponent, then <math>P=P_{(2,0)}</math> is the momentum of a watermelon operator that creates 4 loops. === Relation with monodromies of BPZ equations === The function <math>\varphi(x)</math> that defines the backbone exponent can be rewritten in terms of Euler's Gamma function. The Gamma function appears in the monodromy of solutions of BPZ differential equations, which are obeyed by correlation functions that involve degenerate fields. This suggests that the backbone exponent might be determined from a simple condition on the monodromy of a degenerate field around the field that creates the two paths. To be more specific, we have <math>\varphi(\pi x)= \frac{1}{\Gamma(1-x)\Gamma(1+x)}</math>, so the equation that defines the backbone exponent can be rewritten as :<math> \prod_\pm \frac{\Gamma(1\pm 4\beta P)}{\Gamma(1\pm 2\beta^2)} = 1 </math> The BPZ equations for a four-point functions of the type <math>\left\langle V_{\langle r,1\rangle}V_{P_1}V_{P_2}V_{P_3}\right\rangle</math> with <math>r=2,3,\dots</math> have monodromies that involve Gamma functions of combinations of <math>\beta P_i</math>. === Fate of interchiral symmetry === In CFT, does the field whose dimension is the monochromatic exponent break interchiral symmetry? That symmetry is associated to the existence of degenerate fields, and it constrains non-diagonal fields to have fractional Kac indices. Breaking that symmetry would make the CFT harder to approach with known methods. On the other hand, having exact results in a situation with no interchiral symmetry would be new and interesting. == References == {{Reflist|refs= <ref name="cai26">{{cite web | last=Cai | first=Gefei | title=Monochromatic arm exponents for critical planar percolation | website=arXiv.org | date=2026-11-03 | url=https://arxiv.org/abs/2611.05101 | access-date=2026-11-06}}</ref> <ref name="lsw01">{{cite web | last1=Lawler | first1=Gregory F. | last2=Schramm | first2=Oded | last3=Werner | first3=Wendelin | title=One-arm exponent for critical 2D percolation | website=arXiv.org | date=2001-08-30 | url=https://arxiv.org/abs/math/0108211 | access-date=2026-06-04}}</ref> <ref name="lsyz24">{{cite web | last=Liu | first=Haoyu | last2=Sun | first2=Xin | last3=Yu | first3=Pu | last4=Zhuang | first4=Zijie | title=The bulk one-arm exponent for the CLE$_{κ'}$ percolations | website=arXiv.org | date=2024-10-16 | url=https://arxiv.org/abs/2410.12724 | access-date=2025-03-14}}</ref> <ref name="asyz24">{{cite journal | last=Ang | first=Morris | last2=Sun | first2=Xin | last3=Yu | first3=Pu | last4=Zhuang | first4=Zijie | title=Boundary touching probability and nested-path exponent for non-simple CLE | publisher=arXiv | date=2024-01-29 | url=https://arxiv.org/abs/2401.15904 | access-date=2024-02-04}}</ref> <ref name="jz02">{{cite journal | last=Jacobsen | first=Jesper Lykke | last2=Zinn-Justin | first2=Paul | title=Monochromatic path crossing exponents and graph connectivity in 2D percolation | publisher=arXiv | doi=10.48550/ARXIV.COND-MAT/0207063 | url=https://arxiv.org/abs/cond-mat/0207063 }}</ref> <ref name="bn09">{{cite journal | last=Beffara | first=Vincent | last2=Nolin | first2=Pierre | title=On monochromatic arm exponents for 2D critical percolation | journal=arXiv | publisher=arXiv | doi=10.48550/ARXIV.0906.3570 | url=https://arxiv.org/abs/0906.3570 | }}</ref> <ref name="ada99">{{cite journal | last=Aizenman | first=Michael | last2=Duplantier | first2=Bertrand | last3=Aharony | first3=Amnon | title=Path Crossing Exponents and the External Perimeter in 2D Percolation | journal=arXiv | publisher=arXiv | doi=10.48550/ARXIV.COND-MAT/9901018 | url=https://arxiv.org/abs/cond-mat/9901018 |}}</ref> <ref name="nqsz23">{{cite journal | last=Nolin | first=Pierre | last2=Qian | first2=Wei | last3=Sun | first3=Xin | last4=Zhuang | first4=Zijie | title=Backbone exponent for two-dimensional percolation | publisher=arXiv | doi=10.48550/ARXIV.2309.05050 | url=https://arxiv.org/abs/2309.05050}}</ref> <ref name="jan01">{{cite journal | last=Janik | first=Romuald A. | title=Exceptional boundary states at c=1 | journal=Nuclear Physics B | date=2001-09-04 | volume=618 | issue=3 | pages=675–688 | doi=10.1016/S0550-3213(01)00486-2 | arxiv=hep-th/0109021 | s2cid=9079750 | url = https://arxiv.org/abs/hep-th/0109021}}</ref> }} [[Category:Research]] [[Category:Conformal field theory]] qqrphb6knmrhxhrs7cf3q3xuvn2ie9d 0 328204 2844631 2844607 2026-11-05T16:03:42Z Unitfreak 695864 /* The Sumerian Origins of Genesis */ 2844631 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 = The early history in Genesis shares deep structural, thematic, and chronological ties with ancient Mesopotamian writing traditions. There is a close parallel between the biblical figures listed in Genesis 5, who lived before the flood, and those listed in the '''[[w:Sumerian King List|Sumerian King List (SKL)]]'''. This ancient text details a succession of long-lived rulers who reigned right up until a cataclysmic flood destroyed their civilization. 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. st2l8flxy6my22iwreyjtbk5n1tw5jo 0 330494 2844633 2844601 2026-11-05T16:27:09Z JSINST 3110286 2844633 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 developing 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 the academic literature can be utilized to identify the high risk group by requesting authors of the publications to re-analyse the 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== n4hl1dqhi42734joc19zf2g1ijrrwg6 2844634 2844633 2026-11-05T16:28:13Z JSINST 3110286 2844634 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, pre-hypertension, hypertension, and being overweight before the age of 30, often without any symptoms.[1] These conditions constitute the background for developing type 2 diabetes mellitus, cardiovascular diseases (CVDs), which are the leading cause of death and disability worldwide. Early diagnosis and prevention is possible, and a cost-effective preventive strategy can improve long-term health outcomes. '''Prevalence studies''' are used to identify high-risk groups among specific occupational groups, with repeated studies conducted in schools and universities—targeting, for example, seafarers, nurses, and medical students. Alternatively, data from the academic literature can be utilized to identify the high risk group by requesting authors of the publications to re-analyse the data.</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== ctv4bgi2656ho7ejxth6t3pldhz1s6a 2844635 2844634 2026-11-05T16:29:26Z JSINST 3110286 2844635 wikitext text/x-wiki <big>The John Snow prediabetes Institute is an international research network focused on early identification of prediabetes, prehypertension and other types of metabolic syndrome in the early ages for lifestyle health promotion. (https://w.wiki/Skm7)</big> <big>'''Background:''' Millions of people are at an increased risk of developing metabolic syndrome, starting with atherosclerosis, pre-diabetes, pre-hypertension, hypertension, and being overweight before the age of 30, often without any symptoms.[1] These conditions constitute the background for developing type 2 diabetes mellitus, cardiovascular diseases (CVDs), which are the leading cause of death and disability worldwide. Early diagnosis and prevention is possible, and a cost-effective preventive strategy can improve long-term health outcomes. '''Prevalence studies''' are used to identify high-risk groups among specific occupational groups, with repeated studies conducted in schools and universities—targeting, for example, seafarers, nurses, and medical students. Alternatively, data from the academic literature can be utilized to identify the high risk group by requesting authors of the publications to re-analyse the data.</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== 649ymq5xnz9akooqkiqgwagavb5mrmv 0 331111 2844608 2844573 2026-11-05T12:10:21Z U3143751 3106785 /* Affective empathy */ Updated 2844608 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 empathy === Affective empathy is the "how" and focuses on the feeling aspect of empathy. For example, empathising with another person and physically experiencing their emotions. Research links it to emotional contagion (Panksepp and Lahvis, 2011), a theory that suggests emotions are contagious and spread from one person to another. Hatfield (1993) coined the term and suggested that emotional contagion is an automatic [[wikipedia:Mimicry|mimicry]] process. Elfenbein (2014) has expanded on this explanation by introducing affective process theory. As shown in table 1., this original mimicry idea, referred to as complementary linkage, is just one of the three processes of emotional contagion. Insert table here === Cognitive empathy === While the affective component is the essence of empathy, the cognitive component is the process through which that essence comes into being (Bošnjaković & Radionov, 2018). 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) * Storytelling is a way to promote position emotions (Wei et al., 2024). '''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]] iwt3k0zi5lmwtwhmzhb9b74nxxzq1lj 2844609 2844608 2026-11-05T12:19:43Z U3143751 3106785 Included table in Affective empathy section based on Elfenbein's APT 2844609 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 empathy === Affective empathy is the "how" and focuses on the feeling aspect of empathy. For example, empathising with another person and physically experiencing their emotions. Research links it to emotional contagion (Panksepp and Lahvis, 2011), a theory that suggests emotions are contagious and spread from one person to another. Hatfield (1993) coined the term and suggested that emotional contagion is an automatic [[wikipedia:Mimicry|mimicry]] process. Elfenbein (2014) has expanded on this explanation by introducing affective process theory. As shown in table 1., this original mimicry idea, referred to as complementary linkage, is just one of the three processes of emotional contagion. '''Table 1. The three elements of emotional contagion (Elfenbein et al., 2014)''' {| class="wikitable" style="margin: auto; |- ! Type !! Description |- | '''Convergent''' || People sharing an interpretation of an emotional event. |- | '''Divergent''' || People interpreting an emotionial event differently |- | '''Complementary''' || One person acts as the source of an emotional event. |- |} === Cognitive empathy === While the affective component is the essence of empathy, the cognitive component is the process through which that essence comes into being (Bošnjaković & Radionov, 2018). 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) * Storytelling is a way to promote position emotions (Wei et al., 2024). '''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]] 1f88yo4js8mwgg53tcwqe8ns88fxkrc 2844610 2844609 2026-11-05T12:27:21Z U3143751 3106785 /* Affective empathy */ Finished affective empathy 2844610 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 empathy === Affective empathy is the "how" and focuses on the feeling aspect of empathy. Research links it to emotional contagion (Panksepp and Lahvis, 2011), a theory that suggests emotions are contagious and spread from one person to another. Hatfield (1993) coined the term and suggested that emotional contagion is an automatic [[wikipedia:Mimicry|mimicry]] process. Elfenbein (2014) has expanded on this explanation by introducing affective process theory. As shown in Table 1., this original mimicry idea, referred to as complementary linkage, is just one of the three processes of emotional contagion. The other processes describe how people can experience the same event but potentially interpret it differently, highlighting how individual differences play an important role in affective empathy. '''Table 1. Types of Affective Linkage in the Affective Process Theory (Elfenbein et al., 2014)''' {| class="wikitable" style="margin: auto; |- ! Type !! Description |- | '''Convergent''' || People sharing an interpretation of an emotional event. |- | '''Divergent''' || People interpreting an emotionial event differently |- | '''Complementary''' || One person acts as the source of an emotional event. |} === 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). === Empathy and 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) * Storytelling is a way to promote position emotions (Wei et al., 2024). '''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= 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]] rqcsp5iwd2gerlzc6qftj9jpr4194cs 2844620 2844610 2026-11-05T13:21:39Z U3143751 3106785 /* Affective empathy */ adjusted information from a dual model to tripartite model of empathy 2844620 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 == At its core, [[Empathy]] is the 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 drains energy and resources. Further, there are distinct types of empathy that each have different energy costs. Research is varied when it comes to the number of constructs that empathy has. A dual model of affective and cognitive empathy (Yu and Chou, 2018) appears to be the simplest, where affective empathy occurs as an emotional reaction to someone else's mental state (Shao et al., 2024), and cognitive empathy occurs when a person understands someone else's feelings or emotions (Cui et al., 2022). However, a tripartite model will be the primary focus in this chapter. === Cognitive empathy === Cognitive empathy is the "what" and focuses on a person tangibly understanding what another person feels. It is associated with activation in the dorsomedial prefrontal cortex and the right inferior parietal cortex (Ruby & Decety, 2004) and, considered as a top-down attention process (Shao et al., 2024), requires a conscious, sustained focus. === Affective empathy, compassion, and emotional contagion === Affective empathy is the "how" and focuses on the feeling aspect of empathy. It is associated with mirror neuron activity inferior frontal cortex, the premotor areas and the insula (Yu and Chou, 2018). Through the tripartite model, affective empathy is further categorised as either other or self focused (Mantinago et al. 2026). When the affective response is self-oriented, the boundaries between a person and the target of empathy blur, leading to [[wikipedia:Emotional_contagion|emotional contagion]] (Panksepp and Lahvis, 2011) - a theory that suggests emotions are contagious and spread from one person to another through an automatic [[wikipedia:Mimicry|mimicry]] process (Hatfield, 1993). However, when the affective response is other-oriented, observers are aware that they are perceiving and reacting to someone else’s emotional experience. (Mantinago et al. 2026) === Empathy and 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. Compassion negatively correlated with secondary traumatic stress and contagion is positively associated with secondary traumatic stress (Hunt and Gooney, 2019). 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''' In the terms of the JD-R and CORT models, Compassion and Cognitive empathy function as job resources, thus helping to buffer individuals from burnout, whereas contagion functions as a job demand. 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 * {{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) * Storytelling is a way to promote position emotions (Wei et al., 2024). '''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= 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]] o2xbafjw12zopmnzy9k6x1ovrhqq5lx 2844621 2844620 2026-11-05T13:32:06Z U3143751 3106785 /* Empathy fatigue */ updated compassion 2844621 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 == At its core, [[empathy]] is the ability to imagine or understand another person's feelings and/or emotions (Riess, 2017; Shao et al., 2024). Research around empathy is diverse, which is especially apparent when looking at theories considering the constructs of empathy. A dual model consisting of affective and cognitive empathy (Yu and Chou, 2018) appears to be the simplest, where affective empathy occurs as an emotional reaction to someone else's mental state (Shao et al., 2024), and cognitive empathy occurs when a person understands someone else's feelings or emotions (Cui et al., 2022). However, a tripartite model will be the primary focus in this chapter. === Cognitive empathy === Cognitive empathy is the "what" and focuses on a person tangibly understanding what another person feels. It is associated with activation in the dorsomedial prefrontal cortex and the right inferior parietal cortex (Ruby & Decety, 2004) and, considered as a top-down attention process (Shao et al., 2024), requires a conscious, sustained focus. In relation to the JD-R and CORT models, cognitive empathy has been labelled as a job resource, such that engaging in cognitive empathy is negatively associated with emotional exhaustion (reference!). === Affective empathy, compassion, and emotional contagion === Affective empathy is the "how" and focuses on the feeling aspect of empathy. It is associated with mirror neuron activity inferior frontal cortex, the premotor areas and the insula (Yu and Chou, 2018). Through the tripartite model, affective empathy is further categorised as either other or self focused (Mantinago et al. 2026). When the affective response is self-oriented, the boundaries between a person and the target of empathy blur, leading to [[wikipedia:Emotional_contagion|emotional contagion]] (Panksepp and Lahvis, 2011) - a theory that suggests emotions are contagious and spread from one person to another through an automatic [[wikipedia:Mimicry|mimicry]] process (Hatfield, 1993). However, when the affective response is other-oriented, observers are aware that they are perceiving and reacting to someone else’s emotional experience (Mantinago et al. 2026), leading to compassion. Compassion is blah blah, it's motivating, and much like cognitive empathy, is labelled as a job resource, so is negatively associated with emotional exhaustion. === Empathy and 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. Compassion negatively correlated with secondary traumatic stress and contagion is positively associated with secondary traumatic stress (Hunt and Gooney, 2019). 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''' In the terms of the JD-R and CORT models, Compassion and Cognitive empathy function as job resources, thus helping to buffer individuals from burnout, whereas contagion functions as a job demand. 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 * {{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) * Storytelling is a way to promote position emotions (Wei et al., 2024). '''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= 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]] 0863xevi93gcrhstir7ol3zum4gud0m 2844622 2844621 2026-11-05T13:46:20Z U3143751 3106785 /* Empathy and compassion fatigue */ 2844622 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 == At its core, [[empathy]] is the ability to imagine or understand another person's feelings and/or emotions (Riess, 2017; Shao et al., 2024). Research around empathy is diverse, which is especially apparent when looking at theories considering the constructs of empathy. A dual model consisting of affective and cognitive empathy (Yu and Chou, 2018) appears to be the simplest, where affective empathy occurs as an emotional reaction to someone else's mental state (Shao et al., 2024), and cognitive empathy occurs when a person understands someone else's feelings or emotions (Cui et al., 2022). However, a tripartite model will be the primary focus in this chapter. === Cognitive empathy === Cognitive empathy is the "what" and focuses on a person tangibly understanding what another person feels. It is associated with activation in the dorsomedial prefrontal cortex and the right inferior parietal cortex (Ruby & Decety, 2004) and, considered as a top-down attention process (Shao et al., 2024), requires a conscious, sustained focus. In relation to the JD-R and CORT models, cognitive empathy has been labelled as a job resource, such that engaging in cognitive empathy is negatively associated with emotional exhaustion (reference!). === Affective empathy, compassion, and emotional contagion === Affective empathy is the "how" and focuses on the feeling aspect of empathy. It is associated with mirror neuron activity inferior frontal cortex, the premotor areas and the insula (Yu and Chou, 2018). Through the tripartite model, affective empathy is further categorised as either other or self focused (Mantinago et al. 2026). When the affective response is self-oriented, the boundaries between a person and the target of empathy blur, leading to [[wikipedia:Emotional_contagion|emotional contagion]] (Panksepp and Lahvis, 2011) - a theory that suggests emotions are contagious and spread from one person to another through an automatic [[wikipedia:Mimicry|mimicry]] process (Hatfield, 1993). However, when the affective response is other-oriented, observers are aware that they are perceiving and reacting to someone else’s emotional experience (Mantinago et al. 2026), leading to compassion. Compassion is blah blah, it's motivating, and much like cognitive empathy, is labelled as a job resource, so is negatively associated with emotional exhaustion. === Empathy and compassion fatigue === Empathy fatigue occurs when... (sustained empathic response! Job demands over a long period of time! It's all about this lol) 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, [[wikipedia:Secondary_trauma|secondary traumatic stress]]. Compassion negatively correlated with secondary traumatic stress and contagion is positively associated with secondary traumatic stress (Hunt and Gooney, 2019). 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''' In the terms of the JD-R and CORT models, Compassion and Cognitive empathy function as job resources, thus helping to buffer individuals from burnout, whereas contagion functions as a job demand. 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 * {{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) * Storytelling is a way to promote position emotions (Wei et al., 2024). '''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= 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]] 800o0jz7i804tw340h8aobs2njmun3q 2844657 2844622 2026-11-05T18:32:36Z U3143751 3106785 /* Cognitive empathy */ 2844657 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 == At its core, [[empathy]] is the ability to imagine or understand another person's feelings and/or emotions (Riess, 2017; Shao et al., 2024). Research around empathy is diverse, which is especially apparent when looking at theories considering the constructs of empathy. A dual model consisting of affective and cognitive empathy (Yu and Chou, 2018) appears to be the simplest, where affective empathy occurs as an emotional reaction to someone else's mental state (Shao et al., 2024), and cognitive empathy occurs when a person understands someone else's feelings or emotions (Cui et al., 2022). However, a tripartite model will be the primary focus in this chapter. === Cognitive empathy === Cognitive empathy is the "what" and focuses on a person tangibly understanding what another person feels or, simply, "understanding the mental states of others" (Martinago et al., 2026 p. 802). It is associated with activation in the dorsomedial prefrontal cortex and the right inferior parietal cortex (Ruby & Decety, 2004) and, considered as a [[wikipedia:Attention#Orienting|top-down attention]] process, requires a conscious, sustained focus (Shao et al., 2024). In relation to the JD-R and CORT models, engaging in cognitive empathy is negatively associated with emotional exhaustion and may be labelled as a job resource (Ferguson et al., 2021). === Affective empathy, compassion, and emotional contagion === Affective empathy is the "how" and focuses on the feeling aspect of empathy. It is associated with mirror neuron activity inferior frontal cortex, the premotor areas and the insula (Yu and Chou, 2018). Through the tripartite model, affective empathy is further categorised as either other or self focused (Mantinago et al. 2026). When the affective response is self-oriented, the boundaries between a person and the target of empathy blur, leading to [[wikipedia:Emotional_contagion|emotional contagion]] - a theory that suggests emotions are contagious and spread from one person to another through an automatic [[wikipedia:Mimicry|mimicry]] process (Panksepp and Lahvis, 2011; Hatfield, 1993). However, when the affective response is other-oriented and Observers understand that the emotions they are reacting to belong to another person, rather than themselves, it leads to empathic concern in the form of sympathy or compassion (Mantinago et al. 2026). Much like cognitive empathy, empathic concern, primarily the compassion aspect, is negatively associated with emotional exhaustion and is considered a job resource. Table 1. outlines the primary differences between the different types of empathy. A meta-analysis published this year (Martingano et al., 2026) looked at the data === Empathy and compassion fatigue === Empathy fatigue occurs when... (sustained empathic response! Job demands over a long period of time! It's all about this lol) Usually found in healthcare professionals (Figley, 2002), secondary traumatic stress occurs when ...Although compassion fatigue can be considered synonymous with empathy fatigue, it is its own concept. Also known as [[wikipedia:Secondary_trauma|secondary traumatic stress]], it is linked with emotional contagion. Where compassion negatively correlated with secondary traumatic stress, contagion was positively associated (Hunt and Gooney, 2019). This lends to the results from Martingano et al. (2026). {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' In the terms of the JD-R and CORT models, Compassion and Cognitive empathy function as job resources, thus helping to buffer individuals from burnout, whereas contagion functions as a job demand. 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 * {{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) * Storytelling is a way to promote position emotions (Wei et al., 2024). '''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= 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]] ecxp43bbor904j6gscr5qn4kur4ga0c 2844658 2844657 2026-11-05T18:45:42Z U3143751 3106785 /* Affective empathy, compassion, and emotional contagion */ 2844658 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 == At its core, [[empathy]] is the ability to imagine or understand another person's feelings and/or emotions (Riess, 2017; Shao et al., 2024). Research around empathy is diverse, which is especially apparent when looking at theories considering the constructs of empathy. A dual model consisting of affective and cognitive empathy (Yu and Chou, 2018) appears to be the simplest, where affective empathy occurs as an emotional reaction to someone else's mental state (Shao et al., 2024), and cognitive empathy occurs when a person understands someone else's feelings or emotions (Cui et al., 2022). However, a tripartite model will be the primary focus in this chapter. === Cognitive empathy === Cognitive empathy is the "what" and focuses on a person tangibly understanding what another person feels or, simply, "understanding the mental states of others" (Martinago et al., 2026 p. 802). It is associated with activation in the dorsomedial prefrontal cortex and the right inferior parietal cortex (Ruby & Decety, 2004) and, considered as a [[wikipedia:Attention#Orienting|top-down attention]] process, requires a conscious, sustained focus (Shao et al., 2024). In relation to the JD-R and CORT models, engaging in cognitive empathy is negatively associated with emotional exhaustion and may be labelled as a job resource (Ferguson et al., 2021). === Affective empathy, compassion, and emotional contagion === Affective empathy is the "how" and focuses on the feeling aspect of empathy. It is associated with mirror neuron activity inferior frontal cortex, the premotor areas and the insula (Yu and Chou, 2018). Through the tripartite model, affective empathy is further categorised as either other or self focused (Mantinago et al. 2026). When the affective response is self-oriented, the boundaries between a person and the target of empathy blur, leading to [[wikipedia:Emotional_contagion|emotional contagion]] - a theory that suggests emotions are contagious and spread from one person to another through an automatic [[wikipedia:Mimicry|mimicry]] process (Panksepp and Lahvis, 2011; Hatfield, 1993). However, when the affective response is other-oriented and Observers understand that the emotions they are reacting to belong to another person, rather than themselves, it leads to empathic concern in the form of sympathy or compassion (Mantinago et al. 2026). Much like cognitive empathy, empathic concern, primarily the compassion aspect, is negatively associated with emotional exhaustion and is considered a job resource. Table 1. outlines the primary differences between the different types of empathy. A meta-analysis published this year (Martingano et al., 2026) looked at the data === Empathy and compassion fatigue === Empathy fatigue occurs when... (sustained empathic response! Job demands over a long period of time! It's all about this lol) Usually found in healthcare professionals (Figley, 2002), secondary traumatic stress occurs when ...Although compassion fatigue can be considered synonymous with empathy fatigue, it is its own concept. Also known as [[wikipedia:Secondary_trauma|secondary traumatic stress]], it is linked with emotional contagion. Where compassion negatively correlated with secondary traumatic stress, contagion was positively associated (Hunt and Gooney, 2019). This lends to the results from Martingano et al. (2026). {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' In the terms of the JD-R and CORT models, Compassion and Cognitive empathy function as job resources, thus helping to buffer individuals from burnout, whereas contagion functions as a job demand. 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 * {{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) * Storytelling is a way to promote position emotions (Wei et al., 2024). '''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= 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]] rikuw50enoi1vj1lqanepbyng3glk53 2844659 2844658 2026-11-05T18:52:47Z U3143751 3106785 /* Affective empathy, compassion, and emotional contagion */ 2844659 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 == At its core, [[empathy]] is the ability to imagine or understand another person's feelings and/or emotions (Riess, 2017; Shao et al., 2024). Research around empathy is diverse, which is especially apparent when looking at theories considering the constructs of empathy. A dual model consisting of affective and cognitive empathy (Yu and Chou, 2018) appears to be the simplest, where affective empathy occurs as an emotional reaction to someone else's mental state (Shao et al., 2024), and cognitive empathy occurs when a person understands someone else's feelings or emotions (Cui et al., 2022). However, a tripartite model will be the primary focus in this chapter. === Cognitive empathy === Cognitive empathy is the "what" and focuses on a person tangibly understanding what another person feels or, simply, "understanding the mental states of others" (Martinago et al., 2026 p. 802). It is associated with activation in the dorsomedial prefrontal cortex and the right inferior parietal cortex (Ruby & Decety, 2004) and, considered as a [[wikipedia:Attention#Orienting|top-down attention]] process, requires a conscious, sustained focus (Shao et al., 2024). In relation to the JD-R and CORT models, engaging in cognitive empathy is negatively associated with emotional exhaustion and may be labelled as a job resource (Ferguson et al., 2021). === Affective empathy, compassion, and emotional contagion === Affective empathy is the "how" and focuses on the feeling aspect of empathy. It is associated with mirror neuron activity inferior frontal cortex, the premotor areas and the insula (Yu and Chou, 2018). Through the tripartite model, affective empathy is further categorised as either other- or self-focused (Mantinago et al. 2026). When the affective response is self-oriented, the boundaries between a person and the target of empathy blur, leading to [[wikipedia:Emotional_contagion|emotional contagion]] - a theory that suggests emotions are contagious and spread from one person to another through an automatic [[wikipedia:Mimicry|mimicry]] process (Panksepp and Lahvis, 2011; Hatfield, 1993). Contagion is negatively associated with emotional exhaustion and may be labelled as a job demand. However, when the affective response is other-oriented and Observers understand that the emotions they are reacting to belong to another person, rather than themselves, it leads to empathic concern in the form of sympathy or compassion (Mantinago et al. 2026). Much like cognitive empathy, empathic concern, primarily the compassion aspect, is negatively associated with emotional exhaustion and is considered a job resource. Table 1. outlines the primary differences between the different types of empathy. A meta-analysis published this year (Martingano et al., 2026) looked at the data === Empathy and compassion fatigue === Empathy fatigue occurs when... (sustained empathic response! Job demands over a long period of time! It's all about this lol) Usually found in healthcare professionals (Figley, 2002), secondary traumatic stress occurs when ...Although compassion fatigue can be considered synonymous with empathy fatigue, it is its own concept. Also known as [[wikipedia:Secondary_trauma|secondary traumatic stress]], it is linked with emotional contagion. Where compassion negatively correlated with secondary traumatic stress, contagion was positively associated (Hunt and Gooney, 2019). This lends to the results from Martingano et al. (2026). {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' In the terms of the JD-R and CORT models, Compassion and Cognitive empathy function as job resources, thus helping to buffer individuals from burnout, whereas contagion functions as a job demand. 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 * {{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" emotional labor might be considered the emotional regulation of empathy, such that it is viewed externally. emotionally === 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) * Storytelling is a way to promote position emotions (Wei et al., 2024). '''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= 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]] 41pvh7qnnxt6jzr0hb0xdrnxp3crmes 2844660 2844659 2026-11-05T19:15:32Z U3143751 3106785 /* Empathy and compassion fatigue */ 2844660 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 == At its core, [[empathy]] is the ability to imagine or understand another person's feelings and/or emotions (Riess, 2017; Shao et al., 2024). Research around empathy is diverse, which is especially apparent when looking at theories considering the constructs of empathy. A dual model consisting of affective and cognitive empathy (Yu and Chou, 2018) appears to be the simplest, where affective empathy occurs as an emotional reaction to someone else's mental state (Shao et al., 2024), and cognitive empathy occurs when a person understands someone else's feelings or emotions (Cui et al., 2022). However, a tripartite model will be the primary focus in this chapter. === Cognitive empathy === Cognitive empathy is the "what" and focuses on a person tangibly understanding what another person feels or, simply, "understanding the mental states of others" (Martinago et al., 2026 p. 802). It is associated with activation in the dorsomedial prefrontal cortex and the right inferior parietal cortex (Ruby & Decety, 2004) and, considered as a [[wikipedia:Attention#Orienting|top-down attention]] process, requires a conscious, sustained focus (Shao et al., 2024). In relation to the JD-R and CORT models, engaging in cognitive empathy is negatively associated with emotional exhaustion and may be labelled as a job resource (Ferguson et al., 2021). === Affective empathy, compassion, and emotional contagion === Affective empathy is the "how" and focuses on the feeling aspect of empathy. It is associated with mirror neuron activity inferior frontal cortex, the premotor areas and the insula (Yu and Chou, 2018). Through the tripartite model, affective empathy is further categorised as either other- or self-focused (Mantinago et al. 2026). When the affective response is self-oriented, the boundaries between a person and the target of empathy blur, leading to [[wikipedia:Emotional_contagion|emotional contagion]] - a theory that suggests emotions are contagious and spread from one person to another through an automatic [[wikipedia:Mimicry|mimicry]] process (Panksepp and Lahvis, 2011; Hatfield, 1993). Contagion is negatively associated with emotional exhaustion and may be labelled as a job demand (Hunt and Gooney, 2019). However, when the affective response is other-oriented and Observers understand that the emotions they are reacting to belong to another person, rather than themselves, it leads to empathic concern in the form of sympathy or compassion (Mantinago et al. 2026). Much like cognitive empathy, empathic concern, primarily the compassion aspect, is negatively associated with emotional exhaustion and is considered a job resource. Table 1. outlines the primary differences between the different types of empathy. A meta-analysis published this year (Martingano et al., 2026) looked at the data === Empathy and compassion fatigue === [[wikipedia:Compassion_fatigue|Compassion fatigue]], also known as [[wikipedia:Secondary_trauma|secondary traumatic stress]], is a term introduced by Figley (2002) to describe the negative experience affecting healthcare professionals when they are constantly exposed to the trauma of their patients and is associated with burnout. However, looking at more modern literature, it is understood that compassion does not lead to emotional exhaustion. As such, it will be referred to as empathy fatigue - an alternative term to support clarity in its antecedents (Cigrand et al., 2025). {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' In the terms of the JD-R and CORT models, Compassion and Cognitive empathy function as job resources, thus helping to buffer individuals from burnout, whereas contagion functions as a job demand. 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 * {{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" emotional labor might be considered the emotional regulation of empathy, such that it is viewed externally. emotionally === 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) * Storytelling is a way to promote position emotions (Wei et al., 2024). '''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= 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]] 1gio5ltz9wcaemys0nga3v3f7r5srwa 2844661 2844660 2026-11-05T19:29:31Z U3143751 3106785 /* Affective empathy, compassion, and emotional contagion */ 2844661 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 == At its core, [[empathy]] is the ability to imagine or understand another person's feelings and/or emotions (Riess, 2017; Shao et al., 2024). Research around empathy is diverse, which is especially apparent when looking at theories considering the constructs of empathy. A dual model consisting of affective and cognitive empathy (Yu and Chou, 2018) appears to be the simplest, where affective empathy occurs as an emotional reaction to someone else's mental state (Shao et al., 2024), and cognitive empathy occurs when a person understands someone else's feelings or emotions (Cui et al., 2022). However, a tripartite model will be the primary focus in this chapter. === Cognitive empathy === Cognitive empathy is the "what" and focuses on a person tangibly understanding what another person feels or, simply, "understanding the mental states of others" (Martinago et al., 2026 p. 802). It is associated with activation in the dorsomedial prefrontal cortex and the right inferior parietal cortex (Ruby & Decety, 2004) and, considered as a [[wikipedia:Attention#Orienting|top-down attention]] process, requires a conscious, sustained focus (Shao et al., 2024). In relation to the JD-R and CORT models, engaging in cognitive empathy is negatively associated with emotional exhaustion and may be labelled as a job resource (Ferguson et al., 2021). === Affective empathy, compassion, and emotional contagion === Affective empathy is the "how" and focuses on the feeling aspect of empathy. It is associated with mirror neuron activity inferior frontal cortex, the premotor areas and the insula (Yu and Chou, 2018). Through the tripartite model, affective empathy is further categorised as either other- or self-focused (Mantinago et al. 2026). When the affective response is self-oriented, the boundaries between a person and the target of empathy blur, leading to [[wikipedia:Emotional_contagion|emotional contagion]] - a theory that suggests emotions are contagious and spread from one person to another through an automatic [[wikipedia:Mimicry|mimicry]] process (Panksepp and Lahvis, 2011; Hatfield, 1993). Contagion is negatively associated with emotional exhaustion and may be labelled as a job demand (Hunt and Gooney, 2019). However, when the affective response is other-oriented and Observers understand that the emotions they are reacting to belong to another person, rather than themselves, it leads to empathic concern in the form of sympathy or compassion (Mantinago et al. 2026). Much like cognitive empathy, empathic concern, primarily the compassion aspect, is negatively associated with emotional exhaustion and is considered a job resource. Table 1. outlines the primary differences between the different types of empathy. === Empathy and compassion fatigue === [[wikipedia:Compassion_fatigue|Compassion fatigue]], also known as [[wikipedia:Secondary_trauma|secondary traumatic stress]], is a term introduced by Figley (2002) to describe the negative experience affecting healthcare professionals when they are constantly exposed to the trauma of their patients and is associated with burnout. However, looking at more modern literature, it is understood that compassion does not lead to emotional exhaustion. As such, it will be referred to as empathy fatigue - an alternative term to support clarity in its antecedents (Cigrand et al., 2025). {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' *In the terms of the JD-R and CORT models, Compassion and Cognitive empathy function as job resources, thus helping to buffer individuals from burnout, whereas self-focused affective empathy (contagion) functions as a job demand. *Empathy fatigue occurs because of contagion depleting energy faster than resources can restore or mitigate demands. Aelin has engaged in contagion with their clients and is now experiencing empathy fatigue, contributing to their emotional exhaustion. * {{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" emotional labor might be considered the emotional regulation of empathy, such that it is viewed externally. emotionally === 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) * Storytelling is a way to promote position emotions (Wei et al., 2024). '''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= 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]] qzng7lyv7d588t6jnuee1wzj6k0g399 2844662 2844661 2026-11-05T19:35:53Z U3143751 3106785 /* Emotional labor as a mediating factor between empathy fatigue and emotional exhaustion */ Used ChatGPT to guide updated introduction for this section: https://chatgpt.com/share/6aecda6e-e02c-83ec-87e0-3f7fe6c3fba7 2844662 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 == At its core, [[empathy]] is the ability to imagine or understand another person's feelings and/or emotions (Riess, 2017; Shao et al., 2024). Research around empathy is diverse, which is especially apparent when looking at theories considering the constructs of empathy. A dual model consisting of affective and cognitive empathy (Yu and Chou, 2018) appears to be the simplest, where affective empathy occurs as an emotional reaction to someone else's mental state (Shao et al., 2024), and cognitive empathy occurs when a person understands someone else's feelings or emotions (Cui et al., 2022). However, a tripartite model will be the primary focus in this chapter. === Cognitive empathy === Cognitive empathy is the "what" and focuses on a person tangibly understanding what another person feels or, simply, "understanding the mental states of others" (Martinago et al., 2026 p. 802). It is associated with activation in the dorsomedial prefrontal cortex and the right inferior parietal cortex (Ruby & Decety, 2004) and, considered as a [[wikipedia:Attention#Orienting|top-down attention]] process, requires a conscious, sustained focus (Shao et al., 2024). In relation to the JD-R and CORT models, engaging in cognitive empathy is negatively associated with emotional exhaustion and may be labelled as a job resource (Ferguson et al., 2021). === Affective empathy, compassion, and emotional contagion === Affective empathy is the "how" and focuses on the feeling aspect of empathy. It is associated with mirror neuron activity inferior frontal cortex, the premotor areas and the insula (Yu and Chou, 2018). Through the tripartite model, affective empathy is further categorised as either other- or self-focused (Mantinago et al. 2026). When the affective response is self-oriented, the boundaries between a person and the target of empathy blur, leading to [[wikipedia:Emotional_contagion|emotional contagion]] - a theory that suggests emotions are contagious and spread from one person to another through an automatic [[wikipedia:Mimicry|mimicry]] process (Panksepp and Lahvis, 2011; Hatfield, 1993). Contagion is negatively associated with emotional exhaustion and may be labelled as a job demand (Hunt and Gooney, 2019). However, when the affective response is other-oriented and Observers understand that the emotions they are reacting to belong to another person, rather than themselves, it leads to empathic concern in the form of sympathy or compassion (Mantinago et al. 2026). Much like cognitive empathy, empathic concern, primarily the compassion aspect, is negatively associated with emotional exhaustion and is considered a job resource. Table 1. outlines the primary differences between the different types of empathy. === Empathy and compassion fatigue === [[wikipedia:Compassion_fatigue|Compassion fatigue]], also known as [[wikipedia:Secondary_trauma|secondary traumatic stress]], is a term introduced by Figley (2002) to describe the negative experience affecting healthcare professionals when they are constantly exposed to the trauma of their patients and is associated with burnout. However, looking at more modern literature, it is understood that compassion does not lead to emotional exhaustion. As such, it will be referred to as empathy fatigue - an alternative term to support clarity in its antecedents (Cigrand et al., 2025). {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' *In the terms of the JD-R and CORT models, Compassion and Cognitive empathy function as job resources, thus helping to buffer individuals from burnout, whereas self-focused affective empathy (contagion) functions as a job demand. *Empathy fatigue occurs because of contagion depleting energy faster than resources can restore or mitigate demands. Aelin has engaged in contagion with their clients and is now experiencing empathy fatigue, contributing to their emotional exhaustion. * {{RoundBoxBottom}} == Emotional labor as a mediating factor between empathy fatigue and emotional exhaustion == [[wikipedia:Social_norm|Social norms]] provide guidelines for socially acceptable emotional expression. As social beings, humans rely on these shared expectations to regulate their emotional responses and maintain appropriate social interactions. . 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" emotional labor might be considered the emotional regulation of empathy, such that it is viewed externally. emotionally === 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) * Storytelling is a way to promote position emotions (Wei et al., 2024). '''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= 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]] ough3zaaytsgk6vwr7qkhvbts9cc2om 2844665 2844662 2026-11-05T20:00:46Z U3143751 3106785 /* Emotional labor as a mediating factor between empathy fatigue and emotional exhaustion */ 2844665 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 == At its core, [[empathy]] is the ability to imagine or understand another person's feelings and/or emotions (Riess, 2017; Shao et al., 2024). Research around empathy is diverse, which is especially apparent when looking at theories considering the constructs of empathy. A dual model consisting of affective and cognitive empathy (Yu and Chou, 2018) appears to be the simplest, where affective empathy occurs as an emotional reaction to someone else's mental state (Shao et al., 2024), and cognitive empathy occurs when a person understands someone else's feelings or emotions (Cui et al., 2022). However, a tripartite model will be the primary focus in this chapter. === Cognitive empathy === Cognitive empathy is the "what" and focuses on a person tangibly understanding what another person feels or, simply, "understanding the mental states of others" (Martinago et al., 2026 p. 802). It is associated with activation in the dorsomedial prefrontal cortex and the right inferior parietal cortex (Ruby & Decety, 2004) and, considered as a [[wikipedia:Attention#Orienting|top-down attention]] process, requires a conscious, sustained focus (Shao et al., 2024). In relation to the JD-R and CORT models, engaging in cognitive empathy is negatively associated with emotional exhaustion and may be labelled as a job resource (Ferguson et al., 2021). === Affective empathy, compassion, and emotional contagion === Affective empathy is the "how" and focuses on the feeling aspect of empathy. It is associated with mirror neuron activity inferior frontal cortex, the premotor areas and the insula (Yu and Chou, 2018). Through the tripartite model, affective empathy is further categorised as either other- or self-focused (Mantinago et al. 2026). When the affective response is self-oriented, the boundaries between a person and the target of empathy blur, leading to [[wikipedia:Emotional_contagion|emotional contagion]] - a theory that suggests emotions are contagious and spread from one person to another through an automatic [[wikipedia:Mimicry|mimicry]] process (Panksepp and Lahvis, 2011; Hatfield, 1993). Contagion is negatively associated with emotional exhaustion and may be labelled as a job demand (Hunt and Gooney, 2019). However, when the affective response is other-oriented and Observers understand that the emotions they are reacting to belong to another person, rather than themselves, it leads to empathic concern in the form of sympathy or compassion (Mantinago et al. 2026). Much like cognitive empathy, empathic concern, primarily the compassion aspect, is negatively associated with emotional exhaustion and is considered a job resource. Table 1. outlines the primary differences between the different types of empathy. === Empathy and compassion fatigue === [[wikipedia:Compassion_fatigue|Compassion fatigue]], also known as [[wikipedia:Secondary_trauma|secondary traumatic stress]], is a term introduced by Figley (2002) to describe the negative experience affecting healthcare professionals when they are constantly exposed to the trauma of their patients and is associated with burnout. However, looking at more modern literature, it is understood that compassion does not lead to emotional exhaustion. As such, it will be referred to as empathy fatigue - an alternative term to support clarity in its antecedents (Cigrand et al., 2025). {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' *In the terms of the JD-R and CORT models, Compassion and Cognitive empathy function as job resources, thus helping to buffer individuals from burnout, whereas self-focused affective empathy (contagion) functions as a job demand. *Empathy fatigue occurs because of contagion depleting energy faster than resources can restore or mitigate demands. Aelin has engaged in contagion with their clients and is now experiencing empathy fatigue, contributing to their emotional exhaustion. * {{RoundBoxBottom}} == Emotional labour as a mediating factor between empathy fatigue and emotional exhaustion == [[wikipedia:Social_norm|Social norms]] provide guidelines for socially acceptable emotional expression. As social beings, humans rely on these shared expectations to regulate their emotional responses and maintain appropriate social interactions. This is the essence of [[Emotion labour|emotional labour]]: "the regulation of felt and expressed emotions (whether effortful or not) to match these emotional display rules with the objective of fulfilling organisational goals" (Vial & Cowgill, 2022 p. 2). Emotional labour is typically displayed through one of two ways. === Types of acting === Surface acting involves changing how felt emotions are externally expressed, such as adjusting facial expressions and body language (Vial & Cowgill, 2022). People who surface act are more likely to experience emotional exhaustion through emotional dissonance. A meta-analysis from Kenworthy et al. (2014) looked at the relationship between emotional dissonance and burnout. Across 57 articles published between 1998 and 2011, they found a medium effect size (''r'' =.34) across a large sample (n=16165). Demographics were not specifically reported on, however they considered moderating factors and found a significant slope after regressing for gender (''Z'' = 2.24, ''p'' = .03). The second type, deep acting, involves genuinely adjusting the way a person feels to match the feelings they have with the feelings that are appropriate to display in a given situation) often report positive relationships with emotional labor * 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 ** ** Actual language - in this situation, this is what I ''actually'' feel === The mediating effect of emotional labor === 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" emotional labor might be considered the emotional regulation of empathy, such that it is viewed externally. emotionally 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) * Storytelling is a way to promote position emotions (Wei et al., 2024). '''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= 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]] 70tlwvzswz9vqa86z3qukpkfjlnylp8 2844667 2844665 2026-11-05T20:52:40Z U3143751 3106785 /* Types of acting */ Utilised chatgpt to synthesis research and provide a foundation to update the mediating effect of emotional labour: https://chatgpt.com/share/6aecec58-3cc0-83ec-ac8f-85be62e897e6 2844667 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 == At its core, [[empathy]] is the ability to imagine or understand another person's feelings and/or emotions (Riess, 2017; Shao et al., 2024). Research around empathy is diverse, which is especially apparent when looking at theories considering the constructs of empathy. A dual model consisting of affective and cognitive empathy (Yu and Chou, 2018) appears to be the simplest, where affective empathy occurs as an emotional reaction to someone else's mental state (Shao et al., 2024), and cognitive empathy occurs when a person understands someone else's feelings or emotions (Cui et al., 2022). However, a tripartite model will be the primary focus in this chapter. === Cognitive empathy === Cognitive empathy is the "what" and focuses on a person tangibly understanding what another person feels or, simply, "understanding the mental states of others" (Martinago et al., 2026 p. 802). It is associated with activation in the dorsomedial prefrontal cortex and the right inferior parietal cortex (Ruby & Decety, 2004) and, considered as a [[wikipedia:Attention#Orienting|top-down attention]] process, requires a conscious, sustained focus (Shao et al., 2024). In relation to the JD-R and CORT models, engaging in cognitive empathy is negatively associated with emotional exhaustion and may be labelled as a job resource (Ferguson et al., 2021). === Affective empathy, compassion, and emotional contagion === Affective empathy is the "how" and focuses on the feeling aspect of empathy. It is associated with mirror neuron activity inferior frontal cortex, the premotor areas and the insula (Yu and Chou, 2018). Through the tripartite model, affective empathy is further categorised as either other- or self-focused (Mantinago et al. 2026). When the affective response is self-oriented, the boundaries between a person and the target of empathy blur, leading to [[wikipedia:Emotional_contagion|emotional contagion]] - a theory that suggests emotions are contagious and spread from one person to another through an automatic [[wikipedia:Mimicry|mimicry]] process (Panksepp and Lahvis, 2011; Hatfield, 1993). Contagion is negatively associated with emotional exhaustion and may be labelled as a job demand (Hunt and Gooney, 2019). However, when the affective response is other-oriented and Observers understand that the emotions they are reacting to belong to another person, rather than themselves, it leads to empathic concern in the form of sympathy or compassion (Mantinago et al. 2026). Much like cognitive empathy, empathic concern, primarily the compassion aspect, is negatively associated with emotional exhaustion and is considered a job resource. Table 1. outlines the primary differences between the different types of empathy. === Empathy and compassion fatigue === [[wikipedia:Compassion_fatigue|Compassion fatigue]], also known as [[wikipedia:Secondary_trauma|secondary traumatic stress]], is a term introduced by Figley (2002) to describe the negative experience affecting healthcare professionals when they are constantly exposed to the trauma of their patients and is associated with burnout. However, looking at more modern literature, it is understood that compassion does not lead to emotional exhaustion. As such, it will be referred to as empathy fatigue - an alternative term to support clarity in its antecedents (Cigrand et al., 2025). {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' *In the terms of the JD-R and CORT models, Compassion and Cognitive empathy function as job resources, thus helping to buffer individuals from burnout, whereas self-focused affective empathy (contagion) functions as a job demand. *Empathy fatigue occurs because of contagion depleting energy faster than resources can restore or mitigate demands. Aelin has engaged in contagion with their clients and is now experiencing empathy fatigue, contributing to their emotional exhaustion. * {{RoundBoxBottom}} == Emotional labour as a mediating factor between empathy fatigue and emotional exhaustion == [[wikipedia:Social_norm|Social norms]] provide guidelines for socially acceptable emotional expression. As social beings, humans rely on these shared expectations to regulate their emotional responses and maintain appropriate social interactions. This is the essence of [[Emotion labour|emotional labour]]: "the regulation of felt and expressed emotions (whether effortful or not) to match these emotional display rules with the objective of fulfilling organisational goals" (Vial & Cowgill, 2022 p. 2). Emotional labour is typically displayed through one of two ways. === Types of acting === Surface acting involves changing how felt emotions are externally expressed, such as adjusting facial expressions and body language to 'fake' the appropriate emotion (Vial & Cowgill, 2022). This creates an internal conflict known as emotional dissonance, where what someone is feeling and what they are expressing is misaligned (Middleton, 1989, as cited in Kenworthy et al., 2014). A meta-analysis from Kenworthy et al. (2014) looked at the relationship between emotional dissonance and burnout. Across 57 articles published between 1998 and 2011, they found a medium effect size (''r'' =.34) across a large sample (n=16165). Demographics were not specifically reported on, however they considered moderating factors and found a significant slope after regressing for gender (''Z'' = 2.24, ''p'' = .03). Under the JD-R, surface acting may be considered as a job demand. Surface and deep acting, the second type of emotional labour, are both [[emotional regulation]] strategies, but differ in what they are regulating. Surface acting regulates the expression of emotion, and deep acting regulates the internal feeling of the emotion through strategies such as [[wikipedia:Cognitive_appraisal|cognitive reappraisal]] (Vial & Cowgill, 2022). It is important to note that some literature discusses a third emotional labour, expression of naturally felt emotions, which just describes that a person's genuine feelings are the appropriate to express in the moment and does not require additional emotional regulation (Zhu et al., 2025). === The mediating effect of emotional labour === Pulling together the researhc, emotional labour is one of the primary bridges between emotional exhaustion and empathy fatigue. When performing surface acting, people become emotionally exhausted. As such, individuals have fewer resources available to manage their emotions leading to emotional dysregulation (Nangle et al., 2019). This creates a job demand spiral, where they are still required to display appropriate emotional responses (Vial & Cowgill, 2022). Under the CORT model, this further depletes already limited emotional resources, intensifying emotional exhaustion (Hobfoll, 1989). Consequently, while this emotional labour is sustained, it reduces an individual's capacity to engage in cognitive and other-oriented affective empathy. Surface acting may exacerbate this effect by increasing emotional demands, while repeated exposure to others' emotions may further contribute to emotional strain through emotional contagion (Hunt & Gooney, 2019). Therefore, emotional labour, particularly surface acting, acts as the mechanism through which empathy fatigue contributes to emotional exhaustion. {{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 == (Zhu et al., 2025) DA and ENFE are conducive to the conservation and gain of resources, while SA leads to resource consumption 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) * Storytelling is a way to promote position emotions (Wei et al., 2024). '''Table 3. 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= 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]] 8r7kwywsr9pefc6isj4pw8q9iam8vgd 2844668 2844667 2026-11-05T20:58:52Z U3143751 3106785 /* The mediating effect of emotional labour */ Updated scenario box 2844668 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 == At its core, [[empathy]] is the ability to imagine or understand another person's feelings and/or emotions (Riess, 2017; Shao et al., 2024). Research around empathy is diverse, which is especially apparent when looking at theories considering the constructs of empathy. A dual model consisting of affective and cognitive empathy (Yu and Chou, 2018) appears to be the simplest, where affective empathy occurs as an emotional reaction to someone else's mental state (Shao et al., 2024), and cognitive empathy occurs when a person understands someone else's feelings or emotions (Cui et al., 2022). However, a tripartite model will be the primary focus in this chapter. === Cognitive empathy === Cognitive empathy is the "what" and focuses on a person tangibly understanding what another person feels or, simply, "understanding the mental states of others" (Martinago et al., 2026 p. 802). It is associated with activation in the dorsomedial prefrontal cortex and the right inferior parietal cortex (Ruby & Decety, 2004) and, considered as a [[wikipedia:Attention#Orienting|top-down attention]] process, requires a conscious, sustained focus (Shao et al., 2024). In relation to the JD-R and CORT models, engaging in cognitive empathy is negatively associated with emotional exhaustion and may be labelled as a job resource (Ferguson et al., 2021). === Affective empathy, compassion, and emotional contagion === Affective empathy is the "how" and focuses on the feeling aspect of empathy. It is associated with mirror neuron activity inferior frontal cortex, the premotor areas and the insula (Yu and Chou, 2018). Through the tripartite model, affective empathy is further categorised as either other- or self-focused (Mantinago et al. 2026). When the affective response is self-oriented, the boundaries between a person and the target of empathy blur, leading to [[wikipedia:Emotional_contagion|emotional contagion]] - a theory that suggests emotions are contagious and spread from one person to another through an automatic [[wikipedia:Mimicry|mimicry]] process (Panksepp and Lahvis, 2011; Hatfield, 1993). Contagion is negatively associated with emotional exhaustion and may be labelled as a job demand (Hunt and Gooney, 2019). However, when the affective response is other-oriented and Observers understand that the emotions they are reacting to belong to another person, rather than themselves, it leads to empathic concern in the form of sympathy or compassion (Mantinago et al. 2026). Much like cognitive empathy, empathic concern, primarily the compassion aspect, is negatively associated with emotional exhaustion and is considered a job resource. Table 1. outlines the primary differences between the different types of empathy. === Empathy and compassion fatigue === [[wikipedia:Compassion_fatigue|Compassion fatigue]], also known as [[wikipedia:Secondary_trauma|secondary traumatic stress]], is a term introduced by Figley (2002) to describe the negative experience affecting healthcare professionals when they are constantly exposed to the trauma of their patients and is associated with burnout. However, looking at more modern literature, it is understood that compassion does not lead to emotional exhaustion. As such, it will be referred to as empathy fatigue - an alternative term to support clarity in its antecedents (Cigrand et al., 2025). {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' *In the terms of the JD-R and CORT models, Compassion and Cognitive empathy function as job resources, thus helping to buffer individuals from burnout, whereas self-focused affective empathy (contagion) functions as a job demand. *Empathy fatigue occurs because of contagion depleting energy faster than resources can restore or mitigate demands. Aelin has engaged in contagion with their clients and is now experiencing empathy fatigue, contributing to their emotional exhaustion. * {{RoundBoxBottom}} == Emotional labour as a mediating factor between empathy fatigue and emotional exhaustion == [[wikipedia:Social_norm|Social norms]] provide guidelines for socially acceptable emotional expression. As social beings, humans rely on these shared expectations to regulate their emotional responses and maintain appropriate social interactions. This is the essence of [[Emotion labour|emotional labour]]: "the regulation of felt and expressed emotions (whether effortful or not) to match these emotional display rules with the objective of fulfilling organisational goals" (Vial & Cowgill, 2022 p. 2). Emotional labour is typically displayed through one of two ways. === Types of acting === Surface acting involves changing how felt emotions are externally expressed, such as adjusting facial expressions and body language to 'fake' the appropriate emotion (Vial & Cowgill, 2022). This creates an internal conflict known as emotional dissonance, where what someone is feeling and what they are expressing is misaligned (Middleton, 1989, as cited in Kenworthy et al., 2014). A meta-analysis from Kenworthy et al. (2014) looked at the relationship between emotional dissonance and burnout. Across 57 articles published between 1998 and 2011, they found a medium effect size (''r'' =.34) across a large sample (n=16165). Demographics were not specifically reported on, however they considered moderating factors and found a significant slope after regressing for gender (''Z'' = 2.24, ''p'' = .03). Under the JD-R, surface acting may be considered as a job demand. Surface and deep acting, the second type of emotional labour, are both [[emotional regulation]] strategies, but differ in what they are regulating. Surface acting regulates the expression of emotion, and deep acting regulates the internal feeling of the emotion through strategies such as [[wikipedia:Cognitive_appraisal|cognitive reappraisal]] (Vial & Cowgill, 2022). It is important to note that some literature discusses a third emotional labour, expression of naturally felt emotions, which just describes that a person's genuine feelings are the appropriate to express in the moment and does not require additional emotional regulation (Zhu et al., 2025). === The mediating effect of emotional labour === Pulling together the researhc, emotional labour is one of the primary bridges between emotional exhaustion and empathy fatigue. When performing surface acting, people become emotionally exhausted. As such, individuals have fewer resources available to manage their emotions leading to emotional dysregulation (Nangle et al., 2019). This creates a job demand spiral, where they are still required to display appropriate emotional responses (Vial & Cowgill, 2022). Under the CORT model, this further depletes already limited emotional resources, intensifying emotional exhaustion (Hobfoll, 1989). Consequently, while this emotional labour is sustained, it reduces an individual's capacity to engage in cognitive and other-oriented affective empathy. Surface acting may exacerbate this effect by increasing emotional demands, while repeated exposure to others' emotions may further contribute to emotional strain through emotional contagion (Hunt & Gooney, 2019). Therefore, emotional labour, particularly surface acting, acts as the mechanism through which empathy fatigue contributes to emotional exhaustion. {{RoundBoxTop|theme=3}} '''What role does emotional labor play in emotional exhaustion and empathy fatigue''' * In relation to the CORT and JD-R theories, surface acting expends energy as a job demand. When energy levels are exhausted, people are more likely to engage in contagion due to emotional dysregulation, leading to a job demands spiral that exacerbates emotional exhaustion. * Aelin has exhausted their energy supplies by surface acting and is stuck in a job demands spiral due to contagion. '''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 {Deep acting is negatively associated with burnout: |type="()"} + True - False </quiz> {{RoundBoxBottom}} == Strategies for combating empathy fatigue and emotional exhaustion == (Zhu et al., 2025) DA and ENFE are conducive to the conservation and gain of resources, while SA leads to resource consumption 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) * Storytelling is a way to promote position emotions (Wei et al., 2024). '''Table 3. 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= 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]] brnw1imtkfekvbhk4tuqwamjpg80473 2844669 2844668 2026-11-05T21:00:01Z U3143751 3106785 2844669 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 == At its core, [[empathy]] is the ability to imagine or understand another person's feelings and/or emotions (Riess, 2017; Shao et al., 2024). Research around empathy is diverse, which is especially apparent when looking at theories considering the constructs of empathy. A dual model consisting of affective and cognitive empathy (Yu and Chou, 2018) appears to be the simplest, where affective empathy occurs as an emotional reaction to someone else's mental state (Shao et al., 2024), and cognitive empathy occurs when a person understands someone else's feelings or emotions (Cui et al., 2022). However, a tripartite model will be the primary focus in this chapter. === Cognitive empathy === Cognitive empathy is the "what" and focuses on a person tangibly understanding what another person feels or, simply, "understanding the mental states of others" (Martinago et al., 2026 p. 802). It is associated with activation in the dorsomedial prefrontal cortex and the right inferior parietal cortex (Ruby & Decety, 2004) and, considered as a [[wikipedia:Attention#Orienting|top-down attention]] process, requires a conscious, sustained focus (Shao et al., 2024). In relation to the JD-R and CORT models, engaging in cognitive empathy is negatively associated with emotional exhaustion and may be labelled as a job resource (Ferguson et al., 2021). === Affective empathy, compassion, and emotional contagion === Affective empathy is the "how" and focuses on the feeling aspect of empathy. It is associated with mirror neuron activity inferior frontal cortex, the premotor areas and the insula (Yu and Chou, 2018). Through the tripartite model, affective empathy is further categorised as either other- or self-focused (Mantinago et al. 2026). When the affective response is self-oriented, the boundaries between a person and the target of empathy blur, leading to [[wikipedia:Emotional_contagion|emotional contagion]] - a theory that suggests emotions are contagious and spread from one person to another through an automatic [[wikipedia:Mimicry|mimicry]] process (Panksepp and Lahvis, 2011; Hatfield, 1993). Contagion is negatively associated with emotional exhaustion and may be labelled as a job demand (Hunt and Gooney, 2019). However, when the affective response is other-oriented and Observers understand that the emotions they are reacting to belong to another person, rather than themselves, it leads to empathic concern in the form of sympathy or compassion (Mantinago et al. 2026). Much like cognitive empathy, empathic concern, primarily the compassion aspect, is negatively associated with emotional exhaustion and is considered a job resource. Table 1. outlines the primary differences between the different types of empathy. === Empathy and compassion fatigue === [[wikipedia:Compassion_fatigue|Compassion fatigue]], also known as [[wikipedia:Secondary_trauma|secondary traumatic stress]], is a term introduced by Figley (2002) to describe the negative experience affecting healthcare professionals when they are constantly exposed to the trauma of their patients and is associated with burnout. However, looking at more modern literature, it is understood that compassion does not lead to emotional exhaustion. As such, it will be referred to as empathy fatigue - an alternative term to support clarity in its antecedents (Cigrand et al., 2025). {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' *In the terms of the JD-R and CORT models, Compassion and Cognitive empathy function as job resources, thus helping to buffer individuals from burnout, whereas self-focused affective empathy (contagion) functions as a job demand. *Empathy fatigue occurs because of contagion depleting energy faster than resources can restore or mitigate demands. Aelin has engaged in contagion with their clients and is now experiencing empathy fatigue, contributing to their emotional exhaustion. * {{RoundBoxBottom}} == Emotional labour as a mediating factor between empathy fatigue and emotional exhaustion == [[wikipedia:Social_norm|Social norms]] provide guidelines for socially acceptable emotional expression. As social beings, humans rely on these shared expectations to regulate their emotional responses and maintain appropriate social interactions. This is the essence of [[Emotion labour|emotional labour]]: "the regulation of felt and expressed emotions (whether effortful or not) to match these emotional display rules with the objective of fulfilling organisational goals" (Vial & Cowgill, 2022 p. 2). Emotional labour is typically displayed through one of two ways. === Types of acting === Surface acting involves changing how felt emotions are externally expressed, such as adjusting facial expressions and body language to 'fake' the appropriate emotion (Vial & Cowgill, 2022). This creates an internal conflict known as emotional dissonance, where what someone is feeling and what they are expressing is misaligned (Middleton, 1989, as cited in Kenworthy et al., 2014). A meta-analysis from Kenworthy et al. (2014) looked at the relationship between emotional dissonance and burnout. Across 57 articles published between 1998 and 2011, they found a medium effect size (''r'' =.34) across a large sample (n=16165). Demographics were not specifically reported on, however they considered moderating factors and found a significant slope after regressing for gender (''Z'' = 2.24, ''p'' = .03). Under the JD-R, surface acting may be considered as a job demand. Surface and deep acting, the second type of emotional labour, are both [[emotional regulation]] strategies, but differ in what they are regulating. Surface acting regulates the expression of emotion, and deep acting regulates the internal feeling of the emotion through strategies such as [[wikipedia:Cognitive_appraisal|cognitive reappraisal]] (Vial & Cowgill, 2022). It is important to note that some literature discusses a third emotional labour, expression of naturally felt emotions, which just describes that a person's genuine feelings are the appropriate to express in the moment and does not require additional emotional regulation (Zhu et al., 2025). === The mediating effect of emotional labour === Pulling together the researhc, emotional labour is one of the primary bridges between emotional exhaustion and empathy fatigue. When performing surface acting, people become emotionally exhausted. As such, individuals have fewer resources available to manage their emotions leading to emotional dysregulation (Nangle et al., 2019). This creates a job demand spiral, where they are still required to display appropriate emotional responses (Vial & Cowgill, 2022). Under the CORT model, this further depletes already limited emotional resources, intensifying emotional exhaustion (Hobfoll, 1989). Consequently, while this emotional labour is sustained, it reduces an individual's capacity to engage in cognitive and other-oriented affective empathy. Surface acting may exacerbate this effect by increasing emotional demands, while repeated exposure to others' emotions may further contribute to emotional strain through emotional contagion (Hunt & Gooney, 2019). Therefore, emotional labour, particularly surface acting, acts as the mechanism through which empathy fatigue contributes to emotional exhaustion. '''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 {Deep acting is negatively associated with burnout: |type="()"} + True - False </quiz> {{RoundBoxTop|theme=3}} '''What role does emotional labor play in emotional exhaustion and empathy fatigue''' * In relation to the CORT and JD-R theories, surface acting expends energy as a job demand. When energy levels are exhausted, people are more likely to engage in contagion due to emotional dysregulation, leading to a job demands spiral that exacerbates emotional exhaustion. * Aelin has exhausted their energy supplies by surface acting and is stuck in a job demands spiral due to contagion. {{RoundBoxBottom}} == Strategies for combating empathy fatigue and emotional exhaustion == (Zhu et al., 2025) DA and ENFE are conducive to the conservation and gain of resources, while SA leads to resource consumption 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) * Storytelling is a way to promote position emotions (Wei et al., 2024). '''Table 3. 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= 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]] 039ti3vmwcb8zccbl4n5zkwvmhscpce 2844670 2844669 2026-11-05T21:08:13Z U3143751 3106785 /* Strategies for combating empathy fatigue and emotional exhaustion */ 2844670 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 == At its core, [[empathy]] is the ability to imagine or understand another person's feelings and/or emotions (Riess, 2017; Shao et al., 2024). Research around empathy is diverse, which is especially apparent when looking at theories considering the constructs of empathy. A dual model consisting of affective and cognitive empathy (Yu and Chou, 2018) appears to be the simplest, where affective empathy occurs as an emotional reaction to someone else's mental state (Shao et al., 2024), and cognitive empathy occurs when a person understands someone else's feelings or emotions (Cui et al., 2022). However, a tripartite model will be the primary focus in this chapter. === Cognitive empathy === Cognitive empathy is the "what" and focuses on a person tangibly understanding what another person feels or, simply, "understanding the mental states of others" (Martinago et al., 2026 p. 802). It is associated with activation in the dorsomedial prefrontal cortex and the right inferior parietal cortex (Ruby & Decety, 2004) and, considered as a [[wikipedia:Attention#Orienting|top-down attention]] process, requires a conscious, sustained focus (Shao et al., 2024). In relation to the JD-R and CORT models, engaging in cognitive empathy is negatively associated with emotional exhaustion and may be labelled as a job resource (Ferguson et al., 2021). === Affective empathy, compassion, and emotional contagion === Affective empathy is the "how" and focuses on the feeling aspect of empathy. It is associated with mirror neuron activity inferior frontal cortex, the premotor areas and the insula (Yu and Chou, 2018). Through the tripartite model, affective empathy is further categorised as either other- or self-focused (Mantinago et al. 2026). When the affective response is self-oriented, the boundaries between a person and the target of empathy blur, leading to [[wikipedia:Emotional_contagion|emotional contagion]] - a theory that suggests emotions are contagious and spread from one person to another through an automatic [[wikipedia:Mimicry|mimicry]] process (Panksepp and Lahvis, 2011; Hatfield, 1993). Contagion is negatively associated with emotional exhaustion and may be labelled as a job demand (Hunt and Gooney, 2019). However, when the affective response is other-oriented and Observers understand that the emotions they are reacting to belong to another person, rather than themselves, it leads to empathic concern in the form of sympathy or compassion (Mantinago et al. 2026). Much like cognitive empathy, empathic concern, primarily the compassion aspect, is negatively associated with emotional exhaustion and is considered a job resource. Table 1. outlines the primary differences between the different types of empathy. === Empathy and compassion fatigue === [[wikipedia:Compassion_fatigue|Compassion fatigue]], also known as [[wikipedia:Secondary_trauma|secondary traumatic stress]], is a term introduced by Figley (2002) to describe the negative experience affecting healthcare professionals when they are constantly exposed to the trauma of their patients and is associated with burnout. However, looking at more modern literature, it is understood that compassion does not lead to emotional exhaustion. As such, it will be referred to as empathy fatigue - an alternative term to support clarity in its antecedents (Cigrand et al., 2025). {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' *In the terms of the JD-R and CORT models, Compassion and Cognitive empathy function as job resources, thus helping to buffer individuals from burnout, whereas self-focused affective empathy (contagion) functions as a job demand. *Empathy fatigue occurs because of contagion depleting energy faster than resources can restore or mitigate demands. Aelin has engaged in contagion with their clients and is now experiencing empathy fatigue, contributing to their emotional exhaustion. * {{RoundBoxBottom}} == Emotional labour as a mediating factor between empathy fatigue and emotional exhaustion == [[wikipedia:Social_norm|Social norms]] provide guidelines for socially acceptable emotional expression. As social beings, humans rely on these shared expectations to regulate their emotional responses and maintain appropriate social interactions. This is the essence of [[Emotion labour|emotional labour]]: "the regulation of felt and expressed emotions (whether effortful or not) to match these emotional display rules with the objective of fulfilling organisational goals" (Vial & Cowgill, 2022 p. 2). Emotional labour is typically displayed through one of two ways. === Types of acting === Surface acting involves changing how felt emotions are externally expressed, such as adjusting facial expressions and body language to 'fake' the appropriate emotion (Vial & Cowgill, 2022). This creates an internal conflict known as emotional dissonance, where what someone is feeling and what they are expressing is misaligned (Middleton, 1989, as cited in Kenworthy et al., 2014). A meta-analysis from Kenworthy et al. (2014) looked at the relationship between emotional dissonance and burnout. Across 57 articles published between 1998 and 2011, they found a medium effect size (''r'' =.34) across a large sample (n=16165). Demographics were not specifically reported on, however they considered moderating factors and found a significant slope after regressing for gender (''Z'' = 2.24, ''p'' = .03). Under the JD-R, surface acting may be considered as a job demand. Surface and deep acting, the second type of emotional labour, are both [[emotional regulation]] strategies, but differ in what they are regulating. Surface acting regulates the expression of emotion, and deep acting regulates the internal feeling of the emotion through strategies such as [[wikipedia:Cognitive_appraisal|cognitive reappraisal]] (Vial & Cowgill, 2022). It is important to note that some literature discusses a third emotional labour, expression of naturally felt emotions, which just describes that a person's genuine feelings are the appropriate to express in the moment and does not require additional emotional regulation (Zhu et al., 2025). === The mediating effect of emotional labour === Pulling together the researhc, emotional labour is one of the primary bridges between emotional exhaustion and empathy fatigue. When performing surface acting, people become emotionally exhausted. As such, individuals have fewer resources available to manage their emotions leading to emotional dysregulation (Nangle et al., 2019). This creates a job demand spiral, where they are still required to display appropriate emotional responses (Vial & Cowgill, 2022). Under the CORT model, this further depletes already limited emotional resources, intensifying emotional exhaustion (Hobfoll, 1989). Consequently, while this emotional labour is sustained, it reduces an individual's capacity to engage in cognitive and other-oriented affective empathy. Surface acting may exacerbate this effect by increasing emotional demands, while repeated exposure to others' emotions may further contribute to emotional strain through emotional contagion (Hunt & Gooney, 2019). Therefore, emotional labour, particularly surface acting, acts as the mechanism through which empathy fatigue contributes to emotional exhaustion. '''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 {Deep acting is negatively associated with burnout: |type="()"} + True - False </quiz> {{RoundBoxTop|theme=3}} '''What role does emotional labor play in emotional exhaustion and empathy fatigue''' * In relation to the CORT and JD-R theories, surface acting expends energy as a job demand. When energy levels are exhausted, people are more likely to engage in contagion due to emotional dysregulation, leading to a job demands spiral that exacerbates emotional exhaustion. * Aelin has exhausted their energy supplies by surface acting and is stuck in a job demands spiral due to contagion. {{RoundBoxBottom}} == Strategies for combating emotional exhaustion == Fortunately, burnout is not a permanent condition and treatment is available for dealing with emotional exhaustion. Table 2. has been adapated from van Dam (2021) to provide an overview of goals and interventions that are available. Someone who is experiencing burnout would begin at phase 1, whereas someone who is not would begin at phase 3. (Zhu et al., 2025) DA and ENFE are conducive to the conservation and gain of resources, while SA leads to resource consumption * 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) * Storytelling is a way to promote position emotions (Wei et al., 2024). '''Table 3. 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. These interventions are primarily focused on reducing job demands. || 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. These interventions are primarily focused on increasing job resources. || 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. These interventions are focused on reducing job demands and increasing job resources. || 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. These interventions are focused on reducing job demands and increasing job resources. || Setting priorities regarding quality of life and interpersonal relationships. |} ==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= 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]] 4w3cb7sa6b5y9iavaxmdxoo3ec3pqvs 2844671 2844670 2026-11-05T21:20:09Z U3143751 3106785 /* References */ updating reference list 2844671 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]] and the [[wikipedia:Job_demands-resources_model|Job Demands-Resources model]]. === Conservation of resources stress model === The Conservation of resources stress model (CORT; Hobfoll, 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 == At its core, [[empathy]] is the ability to imagine or understand another person's feelings and/or emotions (Riess, 2017; Shao et al., 2024). Research around empathy is diverse, which is especially apparent when looking at theories considering the constructs of empathy. A dual model consisting of affective and cognitive empathy (Yu and Chou, 2018) appears to be the simplest, where affective empathy occurs as an emotional reaction to someone else's mental state (Shao et al., 2024), and cognitive empathy occurs when a person understands someone else's feelings or emotions (Cui et al., 2022). However, a tripartite model will be the primary focus in this chapter. === Cognitive empathy === Cognitive empathy is the "what" and focuses on a person tangibly understanding what another person feels or, simply, "understanding the mental states of others" (Martinago et al., 2026 p. 802). It is associated with activation in the dorsomedial prefrontal cortex and the right inferior parietal cortex (Ruby & Decety, 2004) and, considered as a [[wikipedia:Attention#Orienting|top-down attention]] process, requires a conscious, sustained focus (Shao et al., 2024). In relation to the JD-R and CORT models, engaging in cognitive empathy is negatively associated with emotional exhaustion and may be labelled as a job resource (Ferguson et al., 2021). === Affective empathy, compassion, and emotional contagion === Affective empathy is the "how" and focuses on the feeling aspect of empathy. It is associated with mirror neuron activity inferior frontal cortex, the premotor areas and the insula (Yu and Chou, 2018). Through the tripartite model, affective empathy is further categorised as either other- or self-focused (Mantinago et al. 2026). When the affective response is self-oriented, the boundaries between a person and the target of empathy blur, leading to [[wikipedia:Emotional_contagion|emotional contagion]] - a theory that suggests emotions are contagious and spread from one person to another through an automatic [[wikipedia:Mimicry|mimicry]] process (Panksepp and Lahvis, 2011). Contagion is negatively associated with emotional exhaustion and may be labelled as a job demand (Hunt and Gooney, 2019). However, when the affective response is other-oriented and Observers understand that the emotions they are reacting to belong to another person, rather than themselves, it leads to empathic concern in the form of sympathy or compassion (Mantinago et al. 2026). Much like cognitive empathy, empathic concern, primarily the compassion aspect, is negatively associated with emotional exhaustion and is considered a job resource. Table 1. outlines the primary differences between the different types of empathy. === Empathy and compassion fatigue === [[wikipedia:Compassion_fatigue|Compassion fatigue]], also known as [[wikipedia:Secondary_trauma|secondary traumatic stress]], is a term introduced by Figley (2002) to describe the negative experience affecting healthcare professionals when they are constantly exposed to the trauma of their patients and is associated with burnout. However, looking at more modern literature, it is understood that compassion does not lead to emotional exhaustion. As such, it will be referred to as empathy fatigue - an alternative term to support clarity in its antecedents (Cigrand et al., 2025). {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' *In the terms of the JD-R and CORT models, Compassion and Cognitive empathy function as job resources, thus helping to buffer individuals from burnout, whereas self-focused affective empathy (contagion) functions as a job demand. *Empathy fatigue occurs because of contagion depleting energy faster than resources can restore or mitigate demands. Aelin has engaged in contagion with their clients and is now experiencing empathy fatigue, contributing to their emotional exhaustion. * {{RoundBoxBottom}} == Emotional labour as a mediating factor between empathy fatigue and emotional exhaustion == [[wikipedia:Social_norm|Social norms]] provide guidelines for socially acceptable emotional expression. As social beings, humans rely on these shared expectations to regulate their emotional responses and maintain appropriate social interactions. This is the essence of [[Emotion labour|emotional labour]]: "the regulation of felt and expressed emotions (whether effortful or not) to match these emotional display rules with the objective of fulfilling organisational goals" (Vial & Cowgill, 2022 p. 2). Emotional labour is typically displayed through one of two ways. === Types of acting === Surface acting involves changing how felt emotions are externally expressed, such as adjusting facial expressions and body language to 'fake' the appropriate emotion (Vial & Cowgill, 2022). This creates an internal conflict known as emotional dissonance, where what someone is feeling and what they are expressing is misaligned (Middleton, 1989, as cited in Kenworthy et al., 2014). A meta-analysis from Kenworthy et al. (2014) looked at the relationship between emotional dissonance and burnout. Across 57 articles published between 1998 and 2011, they found a medium effect size (''r'' =.34) across a large sample (n=16165). Demographics were not specifically reported on, however they considered moderating factors and found a significant slope after regressing for gender (''Z'' = 2.24, ''p'' = .03). Under the JD-R, surface acting may be considered as a job demand. Surface and deep acting, the second type of emotional labour, are both [[emotional regulation]] strategies, but differ in what they are regulating. Surface acting regulates the expression of emotion, and deep acting regulates the internal feeling of the emotion through strategies such as [[wikipedia:Cognitive_appraisal|cognitive reappraisal]] (Vial & Cowgill, 2022). It is important to note that some literature discusses a third emotional labour, expression of naturally felt emotions (ENFE), which just describes that a person's genuine feelings are the appropriate to express in the moment and does not require additional emotional regulation (Zhu et al., 2025). === The mediating effect of emotional labour === Pulling together the researhc, emotional labour is one of the primary bridges between emotional exhaustion and empathy fatigue. When performing surface acting, people become emotionally exhausted. As such, individuals have fewer resources available to manage their emotions leading to emotional dysregulation (Nangle et al., 2019). This creates a job demand spiral, where they are still required to display appropriate emotional responses (Vial & Cowgill, 2022). Under the CORT model, this further depletes already limited emotional resources, intensifying emotional exhaustion (Hobfoll, 1989). Consequently, while this emotional labour is sustained, it reduces an individual's capacity to engage in cognitive and other-oriented affective empathy. Surface acting may exacerbate this effect by increasing emotional demands, while repeated exposure to others' emotions may further contribute to emotional strain through emotional contagion (Hunt & Gooney, 2019). Therefore, emotional labour, particularly surface acting, acts as the mechanism through which empathy fatigue contributes to emotional exhaustion. '''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 {Deep acting is negatively associated with burnout: |type="()"} + True - False </quiz> {{RoundBoxTop|theme=3}} '''What role does emotional labor play in emotional exhaustion and empathy fatigue''' * In relation to the CORT and JD-R theories, surface acting expends energy as a job demand. When energy levels are exhausted, people are more likely to engage in contagion due to emotional dysregulation, leading to a job demands spiral that exacerbates emotional exhaustion. * Aelin has exhausted their energy supplies by surface acting and is stuck in a job demands spiral due to contagion. {{RoundBoxBottom}} == Strategies for combating emotional exhaustion == Fortunately, burnout is not a permanent condition and treatment is available for dealing with emotional exhaustion. Table 2. has been adapated from van Dam (2021) to provide an overview of goals and interventions that are available. Someone who is experiencing burnout would begin at phase 1, whereas someone who is not would begin at phase 3. Importantly, when it comes to psychoeducation, it is critical to understand how to reduce the amount of surface acting someone is engaging in. Suggestions include knowledge around communication and boundaries (Jeung et al., 2018), promotion of positive emotions through storytelling (Wei et al., 2024) and a transition to deep acting or ENFE as these are associated with the conservation and gain of resources instead of increasing job demands (Zhu et al., 2025). '''Table 3. Treatment phases of clinical burnout (based on Table 2 from van Dam, A. 2021)''' {| class="wikitable" style="margin: auto; |- ! Phase !! Goal !! Interventions |- | '''Phase 1: Crisis''' || Recognition of the patient that the problems are serious, and that serious action is required. These interventions are primarily focused on reducing job demands. || 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. These interventions are primarily focused on increasing job resources. || 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. These interventions are focused on reducing job demands and increasing job resources. || 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. These interventions are focused on reducing job demands and increasing job resources. || Setting priorities regarding quality of life and interpersonal relationships. |} ==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= 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 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 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 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]] dmkbdmg1r63vho047nxazc0scpaewx3 2844672 2844671 2026-11-05T21:22:16Z U3143751 3106785 /* References */ 2844672 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 (Lloyd 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]] and the [[wikipedia:Job_demands-resources_model|Job Demands-Resources model]]. === Conservation of resources stress model === The Conservation of resources stress model (CORT; Hobfoll, 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 == At its core, [[empathy]] is the ability to imagine or understand another person's feelings and/or emotions (Riess, 2017; Shao et al., 2024). Research around empathy is diverse, which is especially apparent when looking at theories considering the constructs of empathy. A dual model consisting of affective and cognitive empathy (Yu and Chou, 2018) appears to be the simplest, where affective empathy occurs as an emotional reaction to someone else's mental state (Shao et al., 2024), and cognitive empathy occurs when a person understands someone else's feelings or emotions (Cui et al., 2022). However, a tripartite model will be the primary focus in this chapter. === Cognitive empathy === Cognitive empathy is the "what" and focuses on a person tangibly understanding what another person feels or, simply, "understanding the mental states of others" (Martingano et al., 2026 p. 802). It is associated with activation in the dorsomedial prefrontal cortex and the right inferior parietal cortex (Ruby & Decety, 2004) and, considered as a [[wikipedia:Attention#Orienting|top-down attention]] process, requires a conscious, sustained focus (Shao et al., 2024). In relation to the JD-R and CORT models, engaging in cognitive empathy is negatively associated with emotional exhaustion and may be labelled as a job resource (Ferguson et al., 2021). === Affective empathy, compassion, and emotional contagion === Affective empathy is the "how" and focuses on the feeling aspect of empathy. It is associated with mirror neuron activity inferior frontal cortex, the premotor areas and the insula (Yu and Chou, 2018). Through the tripartite model, affective empathy is further categorised as either other- or self-focused (Mantinago et al. 2026). When the affective response is self-oriented, the boundaries between a person and the target of empathy blur, leading to [[wikipedia:Emotional_contagion|emotional contagion]] - a theory that suggests emotions are contagious and spread from one person to another through an automatic [[wikipedia:Mimicry|mimicry]] process (Panksepp and Lahvis, 2011). Contagion is negatively associated with emotional exhaustion and may be labelled as a job demand (Hunt and Gooney, 2019). However, when the affective response is other-oriented and Observers understand that the emotions they are reacting to belong to another person, rather than themselves, it leads to empathic concern in the form of sympathy or compassion (Mantinago et al. 2026). Much like cognitive empathy, empathic concern, primarily the compassion aspect, is negatively associated with emotional exhaustion and is considered a job resource. Table 1. outlines the primary differences between the different types of empathy. === Empathy and compassion fatigue === [[wikipedia:Compassion_fatigue|Compassion fatigue]], also known as [[wikipedia:Secondary_trauma|secondary traumatic stress]], is a term introduced by Figley (2002) to describe the negative experience affecting healthcare professionals when they are constantly exposed to the trauma of their patients and is associated with burnout. However, looking at more modern literature, it is understood that compassion does not lead to emotional exhaustion. As such, it will be referred to as empathy fatigue - an alternative term to support clarity in its antecedents (Cigrand et al., 2025). {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' *In the terms of the JD-R and CORT models, Compassion and Cognitive empathy function as job resources, thus helping to buffer individuals from burnout, whereas self-focused affective empathy (contagion) functions as a job demand. *Empathy fatigue occurs because of contagion depleting energy faster than resources can restore or mitigate demands. Aelin has engaged in contagion with their clients and is now experiencing empathy fatigue, contributing to their emotional exhaustion. * {{RoundBoxBottom}} == Emotional labour as a mediating factor between empathy fatigue and emotional exhaustion == [[wikipedia:Social_norm|Social norms]] provide guidelines for socially acceptable emotional expression. As social beings, humans rely on these shared expectations to regulate their emotional responses and maintain appropriate social interactions. This is the essence of [[Emotion labour|emotional labour]]: "the regulation of felt and expressed emotions (whether effortful or not) to match these emotional display rules with the objective of fulfilling organisational goals" (Vial & Cowgill, 2022 p. 2). Emotional labour is typically displayed through one of two ways. === Types of acting === Surface acting involves changing how felt emotions are externally expressed, such as adjusting facial expressions and body language to 'fake' the appropriate emotion (Vial & Cowgill, 2022). This creates an internal conflict known as emotional dissonance, where what someone is feeling and what they are expressing is misaligned (Middleton, 1989, as cited in Kenworthy et al., 2014). A meta-analysis from Kenworthy et al. (2014) looked at the relationship between emotional dissonance and burnout. Across 57 articles published between 1998 and 2011, they found a medium effect size (''r'' =.34) across a large sample (n=16165). Demographics were not specifically reported on, however they considered moderating factors and found a significant slope after regressing for gender (''Z'' = 2.24, ''p'' = .03). Under the JD-R, surface acting may be considered as a job demand. Surface and deep acting, the second type of emotional labour, are both [[emotional regulation]] strategies, but differ in what they are regulating. Surface acting regulates the expression of emotion, and deep acting regulates the internal feeling of the emotion through strategies such as [[wikipedia:Cognitive_appraisal|cognitive reappraisal]] (Vial & Cowgill, 2022). It is important to note that some literature discusses a third emotional labour, expression of naturally felt emotions (ENFE), which just describes that a person's genuine feelings are the appropriate to express in the moment and does not require additional emotional regulation (Zhu et al., 2025). === The mediating effect of emotional labour === Pulling together the researhc, emotional labour is one of the primary bridges between emotional exhaustion and empathy fatigue. When performing surface acting, people become emotionally exhausted. As such, individuals have fewer resources available to manage their emotions leading to emotional dysregulation (Nangle et al., 2019). This creates a job demand spiral, where they are still required to display appropriate emotional responses (Vial & Cowgill, 2022). Under the CORT model, this further depletes already limited emotional resources, intensifying emotional exhaustion (Hobfoll, 1989). Consequently, while this emotional labour is sustained, it reduces an individual's capacity to engage in cognitive and other-oriented affective empathy. Surface acting may exacerbate this effect by increasing emotional demands, while repeated exposure to others' emotions may further contribute to emotional strain through emotional contagion (Hunt & Gooney, 2019). Therefore, emotional labour, particularly surface acting, acts as the mechanism through which empathy fatigue contributes to emotional exhaustion. '''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 {Deep acting is negatively associated with burnout: |type="()"} + True - False </quiz> {{RoundBoxTop|theme=3}} '''What role does emotional labor play in emotional exhaustion and empathy fatigue''' * In relation to the CORT and JD-R theories, surface acting expends energy as a job demand. When energy levels are exhausted, people are more likely to engage in contagion due to emotional dysregulation, leading to a job demands spiral that exacerbates emotional exhaustion. * Aelin has exhausted their energy supplies by surface acting and is stuck in a job demands spiral due to contagion. {{RoundBoxBottom}} == Strategies for combating emotional exhaustion == Fortunately, burnout is not a permanent condition and treatment is available for dealing with emotional exhaustion. Table 2. has been adapated from van Dam (2021) to provide an overview of goals and interventions that are available. Someone who is experiencing burnout would begin at phase 1, whereas someone who is not would begin at phase 3. Importantly, when it comes to psychoeducation, it is critical to understand how to reduce the amount of surface acting someone is engaging in. Suggestions include knowledge around communication and boundaries (Jeung et al., 2018), promotion of positive emotions through storytelling (Wei et al., 2024) and a transition to deep acting or ENFE as these are associated with the conservation and gain of resources instead of increasing job demands (Zhu et al., 2025). '''Table 3. Treatment phases of clinical burnout (based on Table 2 from van Dam, A. 2021)''' {| class="wikitable" style="margin: auto; |- ! Phase !! Goal !! Interventions |- | '''Phase 1: Crisis''' || Recognition of the patient that the problems are serious, and that serious action is required. These interventions are primarily focused on reducing job demands. || 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. These interventions are primarily focused on increasing job resources. || 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. These interventions are focused on reducing job demands and increasing job resources. || 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. These interventions are focused on reducing job demands and increasing job resources. || Setting priorities regarding quality of life and interpersonal relationships. |} ==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= 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 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 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 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 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 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 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]] 32u2zmfjbrkbsgyqbrkohnbfyqvzwf1 2844673 2844672 2026-11-05T21:28:38Z U3143751 3106785 /* References */ 2844673 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 (Lloyd 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; 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]] and the [[wikipedia:Job_demands-resources_model|Job Demands-Resources model]]. === Conservation of resources stress model === The Conservation of resources stress model (CORT; Hobfoll, 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 == At its core, [[empathy]] is the ability to imagine or understand another person's feelings and/or emotions (Shao et al., 2024). Research around empathy is diverse, which is especially apparent when looking at theories considering the constructs of empathy. A dual model consisting of affective and cognitive empathy (Yu and Chou, 2018) appears to be the simplest, where affective empathy occurs as an emotional reaction to someone else's mental state (Shao et al., 2024), and cognitive empathy occurs when a person understands someone else's feelings or emotions (Cui et al., 2022). However, a tripartite model will be the primary focus in this chapter. === Cognitive empathy === Cognitive empathy is the "what" and focuses on a person tangibly understanding what another person feels or, simply, "understanding the mental states of others" (Martingano et al., 2026 p. 802). It is associated with activation in the dorsomedial prefrontal cortex and the right inferior parietal cortex (Ruby & Decety, 2004) and, considered as a [[wikipedia:Attention#Orienting|top-down attention]] process, requires a conscious, sustained focus (Shao et al., 2024). In relation to the JD-R and CORT models, engaging in cognitive empathy is negatively associated with emotional exhaustion and may be labelled as a job resource (Ferguson et al., 2021). === Affective empathy, compassion, and emotional contagion === Affective empathy is the "how" and focuses on the feeling aspect of empathy. It is associated with mirror neuron activity inferior frontal cortex, the premotor areas and the insula (Yu and Chou, 2018). Through the tripartite model, affective empathy is further categorised as either other- or self-focused (Mantinago et al. 2026). When the affective response is self-oriented, the boundaries between a person and the target of empathy blur, leading to [[wikipedia:Emotional_contagion|emotional contagion]] - a theory that suggests emotions are contagious and spread from one person to another through an automatic [[wikipedia:Mimicry|mimicry]] process (Panksepp and Lahvis, 2011). Contagion is negatively associated with emotional exhaustion and may be labelled as a job demand (Hunt and Gooney, 2019). However, when the affective response is other-oriented and Observers understand that the emotions they are reacting to belong to another person, rather than themselves, it leads to empathic concern in the form of sympathy or compassion (Mantinago et al. 2026). Much like cognitive empathy, empathic concern, primarily the compassion aspect, is negatively associated with emotional exhaustion and is considered a job resource. Table 1. outlines the primary differences between the different types of empathy. === Empathy and compassion fatigue === [[wikipedia:Compassion_fatigue|Compassion fatigue]], also known as [[wikipedia:Secondary_trauma|secondary traumatic stress]], is a term introduced by Figley (2002) to describe the negative experience affecting healthcare professionals when they are constantly exposed to the trauma of their patients and is associated with burnout. However, looking at more modern literature, it is understood that compassion does not lead to emotional exhaustion. As such, it will be referred to as empathy fatigue - an alternative term to support clarity in its antecedents (Cigrand et al., 2025). {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' *In the terms of the JD-R and CORT models, Compassion and Cognitive empathy function as job resources, thus helping to buffer individuals from burnout, whereas self-focused affective empathy (contagion) functions as a job demand. *Empathy fatigue occurs because of contagion depleting energy faster than resources can restore or mitigate demands. Aelin has engaged in contagion with their clients and is now experiencing empathy fatigue, contributing to their emotional exhaustion. * {{RoundBoxBottom}} == Emotional labour as a mediating factor between empathy fatigue and emotional exhaustion == [[wikipedia:Social_norm|Social norms]] provide guidelines for socially acceptable emotional expression. As social beings, humans rely on these shared expectations to regulate their emotional responses and maintain appropriate social interactions. This is the essence of [[Emotion labour|emotional labour]]: "the regulation of felt and expressed emotions (whether effortful or not) to match these emotional display rules with the objective of fulfilling organisational goals" (Vial & Cowgill, 2022 p. 2). Emotional labour is typically displayed through one of two ways. === Types of acting === Surface acting involves changing how felt emotions are externally expressed, such as adjusting facial expressions and body language to 'fake' the appropriate emotion (Vial & Cowgill, 2022). This creates an internal conflict known as emotional dissonance, where what someone is feeling and what they are expressing is misaligned (Middleton, 1989, as cited in Kenworthy et al., 2014). A meta-analysis from Kenworthy et al. (2014) looked at the relationship between emotional dissonance and burnout. Across 57 articles published between 1998 and 2011, they found a medium effect size (''r'' =.34) across a large sample (n=16165). Demographics were not specifically reported on, however they considered moderating factors and found a significant slope after regressing for gender (''Z'' = 2.24, ''p'' = .03). Under the JD-R, surface acting may be considered as a job demand. Surface and deep acting, the second type of emotional labour, are both [[emotional regulation]] strategies, but differ in what they are regulating. Surface acting regulates the expression of emotion, and deep acting regulates the internal feeling of the emotion through strategies such as [[wikipedia:Cognitive_appraisal|cognitive reappraisal]] (Vial & Cowgill, 2022). It is important to note that some literature discusses a third emotional labour, expression of naturally felt emotions (ENFE), which just describes that a person's genuine feelings are the appropriate to express in the moment and does not require additional emotional regulation (Zhu et al., 2025). === The mediating effect of emotional labour === Pulling together the research, emotional labour is one of the primary bridges between emotional exhaustion and empathy fatigue. When performing surface acting, people become emotionally exhausted. As such, individuals have fewer resources available to manage their emotions leading to emotional dysregulation (Nangle et al., 2019). This creates a job demand spiral, where they are still required to display appropriate emotional responses (Vial & Cowgill, 2022). Under the CORT model, this further depletes already limited emotional resources, intensifying emotional exhaustion (Hobfoll, 1989). Consequently, while this emotional labour is sustained, it reduces an individual's capacity to engage in cognitive and other-oriented affective empathy. Surface acting may exacerbate this effect by increasing emotional demands, while repeated exposure to others' emotions may further contribute to emotional strain through emotional contagion (Hunt & Gooney, 2019). Therefore, emotional labour, particularly surface acting, acts as the mechanism through which empathy fatigue contributes to emotional exhaustion. '''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 {Deep acting is negatively associated with burnout: |type="()"} + True - False </quiz> {{RoundBoxTop|theme=3}} '''What role does emotional labor play in emotional exhaustion and empathy fatigue''' * In relation to the CORT and JD-R theories, surface acting expends energy as a job demand. When energy levels are exhausted, people are more likely to engage in contagion due to emotional dysregulation, leading to a job demands spiral that exacerbates emotional exhaustion. * Aelin has exhausted their energy supplies by surface acting and is stuck in a job demands spiral due to contagion. {{RoundBoxBottom}} == Strategies for combating emotional exhaustion == Fortunately, burnout is not a permanent condition and treatment is available for dealing with emotional exhaustion. Table 2. has been adapated from van Dam (2021) to provide an overview of goals and interventions that are available. Someone who is experiencing burnout would begin at phase 1, whereas someone who is not would begin at phase 3. Importantly, when it comes to psychoeducation, it is critical to understand how to reduce the amount of surface acting someone is engaging in. Suggestions include knowledge around communication and boundaries (Jeung et al., 2018), promotion of positive emotions through storytelling (Wei et al., 2024) and a transition to deep acting or ENFE as these are associated with the conservation and gain of resources instead of increasing job demands (Zhu et al., 2025). '''Table 3. Treatment phases of clinical burnout (based on Table 2 from van Dam, A. 2021)''' {| class="wikitable" style="margin: auto; |- ! Phase !! Goal !! Interventions |- | '''Phase 1: Crisis''' || Recognition of the patient that the problems are serious, and that serious action is required. These interventions are primarily focused on reducing job demands. || 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. These interventions are primarily focused on increasing job resources. || 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. These interventions are focused on reducing job demands and increasing job resources. || 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. These interventions are focused on reducing job demands and increasing job resources. || Setting priorities regarding quality of life and interpersonal relationships. |} ==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= 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 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 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 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 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 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 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 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 Nangle, M. R., Henry, J. D., von Hippel, C., & Kjelsaas, K. (2019). An empirical study of how emotion dysregulation and social cognition relate to occupational burnout in dentistry. British Dental Journal, 227(4), 285-290. https://doi.org/10.1038/s41415-019-0620-0 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 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 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]] qqcydkau3hzxves9ptf0nhkmnkyglaz 2844674 2844673 2026-11-05T21:30:45Z U3143751 3106785 /* Strategies for combating emotional exhaustion */ 2844674 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 (Lloyd 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; 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]] and the [[wikipedia:Job_demands-resources_model|Job Demands-Resources model]]. === Conservation of resources stress model === The Conservation of resources stress model (CORT; Hobfoll, 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 == At its core, [[empathy]] is the ability to imagine or understand another person's feelings and/or emotions (Shao et al., 2024). Research around empathy is diverse, which is especially apparent when looking at theories considering the constructs of empathy. A dual model consisting of affective and cognitive empathy (Yu and Chou, 2018) appears to be the simplest, where affective empathy occurs as an emotional reaction to someone else's mental state (Shao et al., 2024), and cognitive empathy occurs when a person understands someone else's feelings or emotions (Cui et al., 2022). However, a tripartite model will be the primary focus in this chapter. === Cognitive empathy === Cognitive empathy is the "what" and focuses on a person tangibly understanding what another person feels or, simply, "understanding the mental states of others" (Martingano et al., 2026 p. 802). It is associated with activation in the dorsomedial prefrontal cortex and the right inferior parietal cortex (Ruby & Decety, 2004) and, considered as a [[wikipedia:Attention#Orienting|top-down attention]] process, requires a conscious, sustained focus (Shao et al., 2024). In relation to the JD-R and CORT models, engaging in cognitive empathy is negatively associated with emotional exhaustion and may be labelled as a job resource (Ferguson et al., 2021). === Affective empathy, compassion, and emotional contagion === Affective empathy is the "how" and focuses on the feeling aspect of empathy. It is associated with mirror neuron activity inferior frontal cortex, the premotor areas and the insula (Yu and Chou, 2018). Through the tripartite model, affective empathy is further categorised as either other- or self-focused (Mantinago et al. 2026). When the affective response is self-oriented, the boundaries between a person and the target of empathy blur, leading to [[wikipedia:Emotional_contagion|emotional contagion]] - a theory that suggests emotions are contagious and spread from one person to another through an automatic [[wikipedia:Mimicry|mimicry]] process (Panksepp and Lahvis, 2011). Contagion is negatively associated with emotional exhaustion and may be labelled as a job demand (Hunt and Gooney, 2019). However, when the affective response is other-oriented and Observers understand that the emotions they are reacting to belong to another person, rather than themselves, it leads to empathic concern in the form of sympathy or compassion (Mantinago et al. 2026). Much like cognitive empathy, empathic concern, primarily the compassion aspect, is negatively associated with emotional exhaustion and is considered a job resource. Table 1. outlines the primary differences between the different types of empathy. === Empathy and compassion fatigue === [[wikipedia:Compassion_fatigue|Compassion fatigue]], also known as [[wikipedia:Secondary_trauma|secondary traumatic stress]], is a term introduced by Figley (2002) to describe the negative experience affecting healthcare professionals when they are constantly exposed to the trauma of their patients and is associated with burnout. However, looking at more modern literature, it is understood that compassion does not lead to emotional exhaustion. As such, it will be referred to as empathy fatigue - an alternative term to support clarity in its antecedents (Cigrand et al., 2025). {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' *In the terms of the JD-R and CORT models, Compassion and Cognitive empathy function as job resources, thus helping to buffer individuals from burnout, whereas self-focused affective empathy (contagion) functions as a job demand. *Empathy fatigue occurs because of contagion depleting energy faster than resources can restore or mitigate demands. Aelin has engaged in contagion with their clients and is now experiencing empathy fatigue, contributing to their emotional exhaustion. * {{RoundBoxBottom}} == Emotional labour as a mediating factor between empathy fatigue and emotional exhaustion == [[wikipedia:Social_norm|Social norms]] provide guidelines for socially acceptable emotional expression. As social beings, humans rely on these shared expectations to regulate their emotional responses and maintain appropriate social interactions. This is the essence of [[Emotion labour|emotional labour]]: "the regulation of felt and expressed emotions (whether effortful or not) to match these emotional display rules with the objective of fulfilling organisational goals" (Vial & Cowgill, 2022 p. 2). Emotional labour is typically displayed through one of two ways. === Types of acting === Surface acting involves changing how felt emotions are externally expressed, such as adjusting facial expressions and body language to 'fake' the appropriate emotion (Vial & Cowgill, 2022). This creates an internal conflict known as emotional dissonance, where what someone is feeling and what they are expressing is misaligned (Middleton, 1989, as cited in Kenworthy et al., 2014). A meta-analysis from Kenworthy et al. (2014) looked at the relationship between emotional dissonance and burnout. Across 57 articles published between 1998 and 2011, they found a medium effect size (''r'' =.34) across a large sample (n=16165). Demographics were not specifically reported on, however they considered moderating factors and found a significant slope after regressing for gender (''Z'' = 2.24, ''p'' = .03). Under the JD-R, surface acting may be considered as a job demand. Surface and deep acting, the second type of emotional labour, are both [[emotional regulation]] strategies, but differ in what they are regulating. Surface acting regulates the expression of emotion, and deep acting regulates the internal feeling of the emotion through strategies such as [[wikipedia:Cognitive_appraisal|cognitive reappraisal]] (Vial & Cowgill, 2022). It is important to note that some literature discusses a third emotional labour, expression of naturally felt emotions (ENFE), which just describes that a person's genuine feelings are the appropriate to express in the moment and does not require additional emotional regulation (Zhu et al., 2025). === The mediating effect of emotional labour === Pulling together the research, emotional labour is one of the primary bridges between emotional exhaustion and empathy fatigue. When performing surface acting, people become emotionally exhausted. As such, individuals have fewer resources available to manage their emotions leading to emotional dysregulation (Nangle et al., 2019). This creates a job demand spiral, where they are still required to display appropriate emotional responses (Vial & Cowgill, 2022). Under the CORT model, this further depletes already limited emotional resources, intensifying emotional exhaustion (Hobfoll, 1989). Consequently, while this emotional labour is sustained, it reduces an individual's capacity to engage in cognitive and other-oriented affective empathy. Surface acting may exacerbate this effect by increasing emotional demands, while repeated exposure to others' emotions may further contribute to emotional strain through emotional contagion (Hunt & Gooney, 2019). Therefore, emotional labour, particularly surface acting, acts as the mechanism through which empathy fatigue contributes to emotional exhaustion. '''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 {Deep acting is negatively associated with burnout: |type="()"} + True - False </quiz> {{RoundBoxTop|theme=3}} '''What role does emotional labor play in emotional exhaustion and empathy fatigue''' * In relation to the CORT and JD-R theories, surface acting expends energy as a job demand. When energy levels are exhausted, people are more likely to engage in contagion due to emotional dysregulation, leading to a job demands spiral that exacerbates emotional exhaustion. * Aelin has exhausted their energy supplies by surface acting and is stuck in a job demands spiral due to contagion. {{RoundBoxBottom}} == Strategies for combating emotional exhaustion == Fortunately, burnout is not a permanent condition and treatment is available for dealing with emotional exhaustion. Table 2. has been adapated from van Dam (2021) to provide an overview of goals and interventions that are available. Someone who is experiencing burnout would begin at phase 1, whereas someone who is not would begin at phase 3. Importantly, when it comes to psychoeducation, it is critical to understand how to reduce the amount of surface acting someone is engaging in. Suggestions include knowledge around communication and boundaries (Bakker et al., 2021), promotion of positive emotions through storytelling (Wei et al., 2024) and a transition to deep acting or ENFE as these are associated with the conservation and gain of resources instead of increasing job demands (Zhu et al., 2025). '''Table 3. Treatment phases of clinical burnout (based on Table 2 from van Dam, A. 2021)''' {| class="wikitable" style="margin: auto; |- ! Phase !! Goal !! Interventions |- | '''Phase 1: Crisis''' || Recognition of the patient that the problems are serious, and that serious action is required. These interventions are primarily focused on reducing job demands. || 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. These interventions are primarily focused on increasing job resources. || 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. These interventions are focused on reducing job demands and increasing job resources. || 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. These interventions are focused on reducing job demands and increasing job resources. || Setting priorities regarding quality of life and interpersonal relationships. |} ==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= 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 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 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 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 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 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 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 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 Nangle, M. R., Henry, J. D., von Hippel, C., & Kjelsaas, K. (2019). An empirical study of how emotion dysregulation and social cognition relate to occupational burnout in dentistry. British Dental Journal, 227(4), 285-290. https://doi.org/10.1038/s41415-019-0620-0 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 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 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]] kj1dcqsk8o2qdt63zj3cg5pkt1vewxf 2844675 2844674 2026-11-05T21:33:24Z U3143751 3106785 /* Conservation of resources stress model */ 2844675 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 (Lloyd 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; 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]] and the [[wikipedia:Job_demands-resources_model|Job Demands-Resources model]]. === Conservation of resources stress model === The Conservation of resources stress model (CORT; Hobfoll, 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 & 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 & 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 == At its core, [[empathy]] is the ability to imagine or understand another person's feelings and/or emotions (Shao et al., 2024). Research around empathy is diverse, which is especially apparent when looking at theories considering the constructs of empathy. A dual model consisting of affective and cognitive empathy (Yu and Chou, 2018) appears to be the simplest, where affective empathy occurs as an emotional reaction to someone else's mental state (Shao et al., 2024), and cognitive empathy occurs when a person understands someone else's feelings or emotions (Cui et al., 2022). However, a tripartite model will be the primary focus in this chapter. === Cognitive empathy === Cognitive empathy is the "what" and focuses on a person tangibly understanding what another person feels or, simply, "understanding the mental states of others" (Martingano et al., 2026 p. 802). It is associated with activation in the dorsomedial prefrontal cortex and the right inferior parietal cortex (Ruby & Decety, 2004) and, considered as a [[wikipedia:Attention#Orienting|top-down attention]] process, requires a conscious, sustained focus (Shao et al., 2024). In relation to the JD-R and CORT models, engaging in cognitive empathy is negatively associated with emotional exhaustion and may be labelled as a job resource (Ferguson et al., 2021). === Affective empathy, compassion, and emotional contagion === Affective empathy is the "how" and focuses on the feeling aspect of empathy. It is associated with mirror neuron activity inferior frontal cortex, the premotor areas and the insula (Yu and Chou, 2018). Through the tripartite model, affective empathy is further categorised as either other- or self-focused (Mantinago et al. 2026). When the affective response is self-oriented, the boundaries between a person and the target of empathy blur, leading to [[wikipedia:Emotional_contagion|emotional contagion]] - a theory that suggests emotions are contagious and spread from one person to another through an automatic [[wikipedia:Mimicry|mimicry]] process (Panksepp and Lahvis, 2011). Contagion is negatively associated with emotional exhaustion and may be labelled as a job demand (Hunt and Gooney, 2019). However, when the affective response is other-oriented and Observers understand that the emotions they are reacting to belong to another person, rather than themselves, it leads to empathic concern in the form of sympathy or compassion (Mantinago et al. 2026). Much like cognitive empathy, empathic concern, primarily the compassion aspect, is negatively associated with emotional exhaustion and is considered a job resource. Table 1. outlines the primary differences between the different types of empathy. === Empathy and compassion fatigue === [[wikipedia:Compassion_fatigue|Compassion fatigue]], also known as [[wikipedia:Secondary_trauma|secondary traumatic stress]], is a term introduced by Figley (2002) to describe the negative experience affecting healthcare professionals when they are constantly exposed to the trauma of their patients and is associated with burnout. However, looking at more modern literature, it is understood that compassion does not lead to emotional exhaustion. As such, it will be referred to as empathy fatigue - an alternative term to support clarity in its antecedents (Cigrand et al., 2025). {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' *In the terms of the JD-R and CORT models, Compassion and Cognitive empathy function as job resources, thus helping to buffer individuals from burnout, whereas self-focused affective empathy (contagion) functions as a job demand. *Empathy fatigue occurs because of contagion depleting energy faster than resources can restore or mitigate demands. Aelin has engaged in contagion with their clients and is now experiencing empathy fatigue, contributing to their emotional exhaustion. * {{RoundBoxBottom}} == Emotional labour as a mediating factor between empathy fatigue and emotional exhaustion == [[wikipedia:Social_norm|Social norms]] provide guidelines for socially acceptable emotional expression. As social beings, humans rely on these shared expectations to regulate their emotional responses and maintain appropriate social interactions. This is the essence of [[Emotion labour|emotional labour]]: "the regulation of felt and expressed emotions (whether effortful or not) to match these emotional display rules with the objective of fulfilling organisational goals" (Vial & Cowgill, 2022 p. 2). Emotional labour is typically displayed through one of two ways. === Types of acting === Surface acting involves changing how felt emotions are externally expressed, such as adjusting facial expressions and body language to 'fake' the appropriate emotion (Vial & Cowgill, 2022). This creates an internal conflict known as emotional dissonance, where what someone is feeling and what they are expressing is misaligned (Middleton, 1989, as cited in Kenworthy et al., 2014). A meta-analysis from Kenworthy et al. (2014) looked at the relationship between emotional dissonance and burnout. Across 57 articles published between 1998 and 2011, they found a medium effect size (''r'' =.34) across a large sample (n=16165). Demographics were not specifically reported on, however they considered moderating factors and found a significant slope after regressing for gender (''Z'' = 2.24, ''p'' = .03). Under the JD-R, surface acting may be considered as a job demand. Surface and deep acting, the second type of emotional labour, are both [[emotional regulation]] strategies, but differ in what they are regulating. Surface acting regulates the expression of emotion, and deep acting regulates the internal feeling of the emotion through strategies such as [[wikipedia:Cognitive_appraisal|cognitive reappraisal]] (Vial & Cowgill, 2022). It is important to note that some literature discusses a third emotional labour, expression of naturally felt emotions (ENFE), which just describes that a person's genuine feelings are the appropriate to express in the moment and does not require additional emotional regulation (Zhu et al., 2025). === The mediating effect of emotional labour === Pulling together the research, emotional labour is one of the primary bridges between emotional exhaustion and empathy fatigue. When performing surface acting, people become emotionally exhausted. As such, individuals have fewer resources available to manage their emotions leading to emotional dysregulation (Nangle et al., 2019). This creates a job demand spiral, where they are still required to display appropriate emotional responses (Vial & Cowgill, 2022). Under the CORT model, this further depletes already limited emotional resources, intensifying emotional exhaustion (Hobfoll, 1989). Consequently, while this emotional labour is sustained, it reduces an individual's capacity to engage in cognitive and other-oriented affective empathy. Surface acting may exacerbate this effect by increasing emotional demands, while repeated exposure to others' emotions may further contribute to emotional strain through emotional contagion (Hunt & Gooney, 2019). Therefore, emotional labour, particularly surface acting, acts as the mechanism through which empathy fatigue contributes to emotional exhaustion. '''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 {Deep acting is negatively associated with burnout: |type="()"} + True - False </quiz> {{RoundBoxTop|theme=3}} '''What role does emotional labor play in emotional exhaustion and empathy fatigue''' * In relation to the CORT and JD-R theories, surface acting expends energy as a job demand. When energy levels are exhausted, people are more likely to engage in contagion due to emotional dysregulation, leading to a job demands spiral that exacerbates emotional exhaustion. * Aelin has exhausted their energy supplies by surface acting and is stuck in a job demands spiral due to contagion. {{RoundBoxBottom}} == Strategies for combating emotional exhaustion == Fortunately, burnout is not a permanent condition and treatment is available for dealing with emotional exhaustion. Table 2. has been adapated from van Dam (2021) to provide an overview of goals and interventions that are available. Someone who is experiencing burnout would begin at phase 1, whereas someone who is not would begin at phase 3. Importantly, when it comes to psychoeducation, it is critical to understand how to reduce the amount of surface acting someone is engaging in. Suggestions include knowledge around communication and boundaries (Bakker et al., 2021), promotion of positive emotions through storytelling (Wei et al., 2024) and a transition to deep acting or ENFE as these are associated with the conservation and gain of resources instead of increasing job demands (Zhu et al., 2025). '''Table 3. Treatment phases of clinical burnout (based on Table 2 from van Dam, A. 2021)''' {| class="wikitable" style="margin: auto; |- ! Phase !! Goal !! Interventions |- | '''Phase 1: Crisis''' || Recognition of the patient that the problems are serious, and that serious action is required. These interventions are primarily focused on reducing job demands. || 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. These interventions are primarily focused on increasing job resources. || 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. These interventions are focused on reducing job demands and increasing job resources. || 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. These interventions are focused on reducing job demands and increasing job resources. || Setting priorities regarding quality of life and interpersonal relationships. |} ==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= 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 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 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 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 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 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 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 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 Nangle, M. R., Henry, J. D., von Hippel, C., & Kjelsaas, K. (2019). An empirical study of how emotion dysregulation and social cognition relate to occupational burnout in dentistry. British Dental Journal, 227(4), 285-290. https://doi.org/10.1038/s41415-019-0620-0 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 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 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]] 61nbq4zu87mvadjo7v0zvz5of24jjtn 2844676 2844675 2026-11-05T21:40:07Z U3143751 3106785 2844676 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 (Lloyd 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; 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]] and the [[wikipedia:Job_demands-resources_model|Job Demands-Resources model]]. === Conservation of resources stress model === The Conservation of resources stress model (CORT; Hobfoll, 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 & 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 & 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 == At its core, [[empathy]] is the ability to imagine or understand another person's feelings and/or emotions (Shao et al., 2024). Research around empathy is diverse, which is especially apparent when looking at theories considering the constructs of empathy. A dual model consisting of affective and cognitive empathy (Yu and Chou, 2018) appears to be the simplest, where affective empathy occurs as an emotional reaction to someone else's mental state (Shao et al., 2024), and cognitive empathy occurs when a person understands someone else's feelings or emotions (Cui et al., 2022). However, a tripartite model will be the primary focus in this chapter. === Cognitive empathy === Cognitive empathy is the "what" and focuses on a person tangibly understanding what another person feels or, simply, "understanding the mental states of others" (Martingano et al., 2026 p. 802). It is associated with activation in the dorsomedial prefrontal cortex and the right inferior parietal cortex (Ruby & Decety, 2004) and, considered as a [[wikipedia:Attention#Orienting|top-down attention]] process, requires a conscious, sustained focus (Shao et al., 2024). In relation to the JD-R and CORT models, engaging in cognitive empathy is negatively associated with emotional exhaustion and may be labelled as a job resource (Ferguson et al., 2021). === Affective empathy, compassion, and emotional contagion === Affective empathy is the "how" and focuses on the feeling aspect of empathy. It is associated with mirror neuron activity inferior frontal cortex, the premotor areas and the insula (Yu and Chou, 2018). Through the tripartite model, affective empathy is further categorised as either other- or self-focused (Mantinago et al. 2026). When the affective response is self-oriented, the boundaries between a person and the target of empathy blur, leading to [[wikipedia:Emotional_contagion|emotional contagion]] - a theory that suggests emotions are contagious and spread from one person to another through an automatic [[wikipedia:Mimicry|mimicry]] process (Panksepp and Lahvis, 2011). Contagion is negatively associated with emotional exhaustion and may be labelled as a job demand (Hunt and Gooney, 2019). However, when the affective response is other-oriented and Observers understand that the emotions they are reacting to belong to another person, rather than themselves, it leads to empathic concern in the form of sympathy or compassion (Mantinago et al. 2026). Much like cognitive empathy, empathic concern, primarily the compassion aspect, is negatively associated with emotional exhaustion and is considered a job resource. Table 1. outlines the primary differences between the different types of empathy. {| class="wikitable" |+ '''Table 1. Differences Between Empathy Types''' ! Type of empathy ! Description ! What this looks like |- | '''Cognitive empathy''' | Empathy that focuses on understanding and recognising another person's thoughts, feelings and perspective without necessarily sharing their emotions. | Understanding why a friend feels upset after receiving bad news. |- | '''Affective empathy''' | The ability to emotionally respond to and share in another person's emotional experience. | Feeling sad when you see a friend crying. |- | '''Compassion''' (affective-other) | Affective empathy which motivates a desire to take action to make change | Feeling concerned about a struggling friend and offering support. |- | '''Sympathy''' (affective-other) | Affective empathy that does '''not''' motivate a desire to take action. | Feeling sorry for a friend who has lost their job. |- | '''Emotional contagion''' (affective-self) | Mirroring another person's emotional state. | Feeling anxious after spending time with someone who is visibly anxious. |} === Empathy and compassion fatigue === [[wikipedia:Compassion_fatigue|Compassion fatigue]], also known as [[wikipedia:Secondary_trauma|secondary traumatic stress]], is a term introduced by Figley (2002) to describe the negative experience affecting healthcare professionals when they are constantly exposed to the trauma of their patients and is associated with burnout. However, looking at more modern literature, it is understood that compassion does not lead to emotional exhaustion. As such, it will be referred to as empathy fatigue - an alternative term to support clarity in its antecedents (Cigrand et al., 2025). {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' *In the terms of the JD-R and CORT models, Compassion and Cognitive empathy function as job resources, thus helping to buffer individuals from burnout, whereas self-focused affective empathy (contagion) functions as a job demand. *Empathy fatigue occurs because of contagion depleting energy faster than resources can restore or mitigate demands. Aelin has engaged in contagion with their clients and is now experiencing empathy fatigue, contributing to their emotional exhaustion. * {{RoundBoxBottom}} == Emotional labour as a mediating factor between empathy fatigue and emotional exhaustion == [[wikipedia:Social_norm|Social norms]] provide guidelines for socially acceptable emotional expression. As social beings, humans rely on these shared expectations to regulate their emotional responses and maintain appropriate social interactions. This is the essence of [[Emotion labour|emotional labour]]: "the regulation of felt and expressed emotions (whether effortful or not) to match these emotional display rules with the objective of fulfilling organisational goals" (Vial & Cowgill, 2022 p. 2). Emotional labour is typically displayed through one of two ways. === Types of acting === Surface acting involves changing how felt emotions are externally expressed, such as adjusting facial expressions and body language to 'fake' the appropriate emotion (Vial & Cowgill, 2022). This creates an internal conflict known as emotional dissonance, where what someone is feeling and what they are expressing is misaligned (Middleton, 1989, as cited in Kenworthy et al., 2014). A meta-analysis from Kenworthy et al. (2014) looked at the relationship between emotional dissonance and burnout. Across 57 articles published between 1998 and 2011, they found a medium effect size (''r'' =.34) across a large sample (n=16165). Demographics were not specifically reported on, however they considered moderating factors and found a significant slope after regressing for gender (''Z'' = 2.24, ''p'' = .03). Under the JD-R, surface acting may be considered as a job demand. Surface and deep acting, the second type of emotional labour, are both [[emotional regulation]] strategies, but differ in what they are regulating. Surface acting regulates the expression of emotion, and deep acting regulates the internal feeling of the emotion through strategies such as [[wikipedia:Cognitive_appraisal|cognitive reappraisal]] (Vial & Cowgill, 2022). It is important to note that some literature discusses a third emotional labour, expression of naturally felt emotions (ENFE), which just describes that a person's genuine feelings are the appropriate to express in the moment and does not require additional emotional regulation (Zhu et al., 2025). === The mediating effect of emotional labour === Pulling together the research, emotional labour is one of the primary bridges between emotional exhaustion and empathy fatigue. When performing surface acting, people become emotionally exhausted. As such, individuals have fewer resources available to manage their emotions leading to emotional dysregulation (Nangle et al., 2019). This creates a job demand spiral, where they are still required to display appropriate emotional responses (Vial & Cowgill, 2022). Under the CORT model, this further depletes already limited emotional resources, intensifying emotional exhaustion (Hobfoll, 1989). Consequently, while this emotional labour is sustained, it reduces an individual's capacity to engage in cognitive and other-oriented affective empathy. Surface acting may exacerbate this effect by increasing emotional demands, while repeated exposure to others' emotions may further contribute to emotional strain through emotional contagion (Hunt & Gooney, 2019). Therefore, emotional labour, particularly surface acting, acts as the mechanism through which empathy fatigue contributes to emotional exhaustion. '''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 {Deep acting is negatively associated with burnout: |type="()"} + True - False </quiz> {{RoundBoxTop|theme=3}} '''What role does emotional labor play in emotional exhaustion and empathy fatigue''' * In relation to the CORT and JD-R theories, surface acting expends energy as a job demand. When energy levels are exhausted, people are more likely to engage in contagion due to emotional dysregulation, leading to a job demands spiral that exacerbates emotional exhaustion. * Aelin has exhausted their energy supplies by surface acting and is stuck in a job demands spiral due to contagion. {{RoundBoxBottom}} == Strategies for combating emotional exhaustion == Fortunately, burnout is not a permanent condition and treatment is available for dealing with emotional exhaustion. Table 2. has been adapated from van Dam (2021) to provide an overview of goals and interventions that are available. Someone who is experiencing burnout would begin at phase 1, whereas someone who is not would begin at phase 3. Importantly, when it comes to psychoeducation, it is critical to understand how to reduce the amount of surface acting someone is engaging in. Suggestions include knowledge around communication and boundaries (Bakker et al., 2021), promotion of positive emotions through storytelling (Wei et al., 2024) and a transition to deep acting or ENFE as these are associated with the conservation and gain of resources instead of increasing job demands (Zhu et al., 2025). '''Table 3. Treatment phases of clinical burnout (based on Table 2 from van Dam, A. 2021)''' {| class="wikitable" style="margin: auto; |- ! Phase !! Goal !! Interventions |- | '''Phase 1: Crisis''' || Recognition of the patient that the problems are serious, and that serious action is required. These interventions are primarily focused on reducing job demands. || 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. These interventions are primarily focused on increasing job resources. || 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. These interventions are focused on reducing job demands and increasing job resources. || 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. These interventions are focused on reducing job demands and increasing job resources. || Setting priorities regarding quality of life and interpersonal relationships. |} ==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= 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 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 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 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 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 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 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 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 Nangle, M. R., Henry, J. D., von Hippel, C., & Kjelsaas, K. (2019). An empirical study of how emotion dysregulation and social cognition relate to occupational burnout in dentistry. British Dental Journal, 227(4), 285-290. https://doi.org/10.1038/s41415-019-0620-0 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 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 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]] 23xitvqpty45ao2vfxhhh0ug3vqyg3q 2844677 2844676 2026-11-05T21:44:15Z U3143751 3106785 /* References */ Used ChatGPT to format reference list: https://chatgpt.com/share/6aecf893-7fe8-83ec-8d17-745d1c86c8af 2844677 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 (Lloyd 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; 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]] and the [[wikipedia:Job_demands-resources_model|Job Demands-Resources model]]. === Conservation of resources stress model === The Conservation of resources stress model (CORT; Hobfoll, 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 & 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 & 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 == At its core, [[empathy]] is the ability to imagine or understand another person's feelings and/or emotions (Shao et al., 2024). Research around empathy is diverse, which is especially apparent when looking at theories considering the constructs of empathy. A dual model consisting of affective and cognitive empathy (Yu and Chou, 2018) appears to be the simplest, where affective empathy occurs as an emotional reaction to someone else's mental state (Shao et al., 2024), and cognitive empathy occurs when a person understands someone else's feelings or emotions (Cui et al., 2022). However, a tripartite model will be the primary focus in this chapter. === Cognitive empathy === Cognitive empathy is the "what" and focuses on a person tangibly understanding what another person feels or, simply, "understanding the mental states of others" (Martingano et al., 2026 p. 802). It is associated with activation in the dorsomedial prefrontal cortex and the right inferior parietal cortex (Ruby & Decety, 2004) and, considered as a [[wikipedia:Attention#Orienting|top-down attention]] process, requires a conscious, sustained focus (Shao et al., 2024). In relation to the JD-R and CORT models, engaging in cognitive empathy is negatively associated with emotional exhaustion and may be labelled as a job resource (Ferguson et al., 2021). === Affective empathy, compassion, and emotional contagion === Affective empathy is the "how" and focuses on the feeling aspect of empathy. It is associated with mirror neuron activity inferior frontal cortex, the premotor areas and the insula (Yu and Chou, 2018). Through the tripartite model, affective empathy is further categorised as either other- or self-focused (Mantinago et al. 2026). When the affective response is self-oriented, the boundaries between a person and the target of empathy blur, leading to [[wikipedia:Emotional_contagion|emotional contagion]] - a theory that suggests emotions are contagious and spread from one person to another through an automatic [[wikipedia:Mimicry|mimicry]] process (Panksepp and Lahvis, 2011). Contagion is negatively associated with emotional exhaustion and may be labelled as a job demand (Hunt and Gooney, 2019). However, when the affective response is other-oriented and Observers understand that the emotions they are reacting to belong to another person, rather than themselves, it leads to empathic concern in the form of sympathy or compassion (Mantinago et al. 2026). Much like cognitive empathy, empathic concern, primarily the compassion aspect, is negatively associated with emotional exhaustion and is considered a job resource. Table 1. outlines the primary differences between the different types of empathy. {| class="wikitable" |+ '''Table 1. Differences Between Empathy Types''' ! Type of empathy ! Description ! What this looks like |- | '''Cognitive empathy''' | Empathy that focuses on understanding and recognising another person's thoughts, feelings and perspective without necessarily sharing their emotions. | Understanding why a friend feels upset after receiving bad news. |- | '''Affective empathy''' | The ability to emotionally respond to and share in another person's emotional experience. | Feeling sad when you see a friend crying. |- | '''Compassion''' (affective-other) | Affective empathy which motivates a desire to take action to make change | Feeling concerned about a struggling friend and offering support. |- | '''Sympathy''' (affective-other) | Affective empathy that does '''not''' motivate a desire to take action. | Feeling sorry for a friend who has lost their job. |- | '''Emotional contagion''' (affective-self) | Mirroring another person's emotional state. | Feeling anxious after spending time with someone who is visibly anxious. |} === Empathy and compassion fatigue === [[wikipedia:Compassion_fatigue|Compassion fatigue]], also known as [[wikipedia:Secondary_trauma|secondary traumatic stress]], is a term introduced by Figley (2002) to describe the negative experience affecting healthcare professionals when they are constantly exposed to the trauma of their patients and is associated with burnout. However, looking at more modern literature, it is understood that compassion does not lead to emotional exhaustion. As such, it will be referred to as empathy fatigue - an alternative term to support clarity in its antecedents (Cigrand et al., 2025). {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' *In the terms of the JD-R and CORT models, Compassion and Cognitive empathy function as job resources, thus helping to buffer individuals from burnout, whereas self-focused affective empathy (contagion) functions as a job demand. *Empathy fatigue occurs because of contagion depleting energy faster than resources can restore or mitigate demands. Aelin has engaged in contagion with their clients and is now experiencing empathy fatigue, contributing to their emotional exhaustion. * {{RoundBoxBottom}} == Emotional labour as a mediating factor between empathy fatigue and emotional exhaustion == [[wikipedia:Social_norm|Social norms]] provide guidelines for socially acceptable emotional expression. As social beings, humans rely on these shared expectations to regulate their emotional responses and maintain appropriate social interactions. This is the essence of [[Emotion labour|emotional labour]]: "the regulation of felt and expressed emotions (whether effortful or not) to match these emotional display rules with the objective of fulfilling organisational goals" (Vial & Cowgill, 2022 p. 2). Emotional labour is typically displayed through one of two ways. === Types of acting === Surface acting involves changing how felt emotions are externally expressed, such as adjusting facial expressions and body language to 'fake' the appropriate emotion (Vial & Cowgill, 2022). This creates an internal conflict known as emotional dissonance, where what someone is feeling and what they are expressing is misaligned (Middleton, 1989, as cited in Kenworthy et al., 2014). A meta-analysis from Kenworthy et al. (2014) looked at the relationship between emotional dissonance and burnout. Across 57 articles published between 1998 and 2011, they found a medium effect size (''r'' =.34) across a large sample (n=16165). Demographics were not specifically reported on, however they considered moderating factors and found a significant slope after regressing for gender (''Z'' = 2.24, ''p'' = .03). Under the JD-R, surface acting may be considered as a job demand. Surface and deep acting, the second type of emotional labour, are both [[emotional regulation]] strategies, but differ in what they are regulating. Surface acting regulates the expression of emotion, and deep acting regulates the internal feeling of the emotion through strategies such as [[wikipedia:Cognitive_appraisal|cognitive reappraisal]] (Vial & Cowgill, 2022). It is important to note that some literature discusses a third emotional labour, expression of naturally felt emotions (ENFE), which just describes that a person's genuine feelings are the appropriate to express in the moment and does not require additional emotional regulation (Zhu et al., 2025). === The mediating effect of emotional labour === Pulling together the research, emotional labour is one of the primary bridges between emotional exhaustion and empathy fatigue. When performing surface acting, people become emotionally exhausted. As such, individuals have fewer resources available to manage their emotions leading to emotional dysregulation (Nangle et al., 2019). This creates a job demand spiral, where they are still required to display appropriate emotional responses (Vial & Cowgill, 2022). Under the CORT model, this further depletes already limited emotional resources, intensifying emotional exhaustion (Hobfoll, 1989). Consequently, while this emotional labour is sustained, it reduces an individual's capacity to engage in cognitive and other-oriented affective empathy. Surface acting may exacerbate this effect by increasing emotional demands, while repeated exposure to others' emotions may further contribute to emotional strain through emotional contagion (Hunt & Gooney, 2019). Therefore, emotional labour, particularly surface acting, acts as the mechanism through which empathy fatigue contributes to emotional exhaustion. '''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 {Deep acting is negatively associated with burnout: |type="()"} + True - False </quiz> {{RoundBoxTop|theme=3}} '''What role does emotional labor play in emotional exhaustion and empathy fatigue''' * In relation to the CORT and JD-R theories, surface acting expends energy as a job demand. When energy levels are exhausted, people are more likely to engage in contagion due to emotional dysregulation, leading to a job demands spiral that exacerbates emotional exhaustion. * Aelin has exhausted their energy supplies by surface acting and is stuck in a job demands spiral due to contagion. {{RoundBoxBottom}} == Strategies for combating emotional exhaustion == Fortunately, burnout is not a permanent condition and treatment is available for dealing with emotional exhaustion. Table 2. has been adapated from van Dam (2021) to provide an overview of goals and interventions that are available. Someone who is experiencing burnout would begin at phase 1, whereas someone who is not would begin at phase 3. Importantly, when it comes to psychoeducation, it is critical to understand how to reduce the amount of surface acting someone is engaging in. Suggestions include knowledge around communication and boundaries (Bakker et al., 2021), promotion of positive emotions through storytelling (Wei et al., 2024) and a transition to deep acting or ENFE as these are associated with the conservation and gain of resources instead of increasing job demands (Zhu et al., 2025). '''Table 3. Treatment phases of clinical burnout (based on Table 2 from van Dam, A. 2021)''' {| class="wikitable" style="margin: auto; |- ! Phase !! Goal !! Interventions |- | '''Phase 1: Crisis''' || Recognition of the patient that the problems are serious, and that serious action is required. These interventions are primarily focused on reducing job demands. || 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. These interventions are primarily focused on increasing job resources. || 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. These interventions are focused on reducing job demands and increasing job resources. || 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. These interventions are focused on reducing job demands and increasing job resources. || Setting priorities regarding quality of life and interpersonal relationships. |} ==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= == References == Alarcon, G. M. (2011). A meta-analysis of burnout with job demands, resources, and attitudes. ''Journal of Vocational Behavior, 79''(2), 549–562. https://doi.org/10.1016/j.jvb.2011.03.007 Bakker, A. B., & de Vries, J. D. (2021). Job demands–resources theory and self-regulation: New explanations and remedies for job burnout. ''Anxiety, Stress, & Coping, 34''(1), 1–21. https://doi.org/10.1080/10615806.2020.1797695 Bakker, A. B., Demerouti, E., & Sanz-Vergel, A. I. (2014). Burnout and work engagement: The JD–R approach. ''Annual Review of Organizational Psychology and Organizational Behavior, 1'', 389–411. https://doi.org/10.1146/annurev-orgpsych-031413-091235 Chan, F. M. (2026). ''The state of workplace burnout 2026: The workplace divide''. Infinite Potential. https://www.ahri.com.au/wp-content/uploads/The-State-of-Workplace-Burnout-2026-Final.pdf Cigrand, C., Quaglia, J. T., Lawlor, S., & Bailey, C. (2025). Healing the healers: The role of contemplative practice in mitigating counselor burnout and empathy fatigue. ''Journal of Mental Health Counseling, 47''(2), 241–261. https://doi.org/10.17744/mehc.47.3.04 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, 32''(23), 5330–5342. https://doi.org/10.1093/cercor/bhac017 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 Ferguson, A. M., Cameron, C. D., & Inzlicht, M. (2021). When does empathy feel good? ''Current Opinion in Behavioral Sciences, 39'', 125–129. https://doi.org/10.1016/j.cobeha.2021.03.011 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 Gynning, B. E., Karlsson, E., Teoh, K., Gustavsson, P., Christiansen, F., & Brulin, E. (2025). Do job resources buffer the harmful effects of job demands on burnout complaints? A 1-year cohort study of Swedish healthcare professionals. ''International Journal of Nursing Studies Advances, 9'', Article 100397. https://doi.org/10.1016/j.ijnsa.2025.100397 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, 2''(3), 96–99. https://doi.org/10.1111/1467-8721.ep10770953 Hobfoll, S. E. (1989). Conservation of resources: A new attempt at conceptualizing stress. ''American Psychologist, 44''(3), 513–524. https://doi.org/10.1037/0003-066X.44.3.513 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 Hunt, P., Denieffe, S., & Gooney, M. (2019). Running on empathy: Relationship of empathy to compassion satisfaction and compassion fatigue in cancer healthcare professionals. ''European Journal of Cancer Care, 28''(4), Article e13124. https://doi.org/10.1111/ecc.13124 Ito, J. K., & Brotheridge, C. M. (2003). Resources, coping strategies, and emotional exhaustion: A conservation of resources perspective. ''Journal of Vocational Behavior, 63''(3), 490–509. https://doi.org/10.1016/S0001-8791(02)00033-7 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/10.1111/jasp.12211 Lesener, T., Gusy, B., & Wolter, C. (2019). The job demands–resources model: A meta-analytic review of longitudinal studies. ''Work & Stress, 33''(1), 76–103. https://doi.org/10.1080/02678373.2018.1529065 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/10.1007/s00127-004-0808-7 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 Nangle, M. R., Henry, J. D., von Hippel, C., & Kjelsaas, K. (2019). An empirical study of how emotion dysregulation and social cognition relate to occupational burnout in dentistry. ''British Dental Journal, 227''(4), 285–290. https://doi.org/10.1038/s41415-019-0620-0 Panksepp, J. B., & Lahvis, G. P. (2011). Rodent empathy and affective neuroscience. ''Neuroscience & 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), Article 04023060. https://doi.org/10.1061/JCEMD4.COENG-13339 Ruby, P., & Decety, J. (2004). How would you feel versus how do you think she would feel? A neuroimaging study of perspective-taking with social emotions. ''Journal of Cognitive Neuroscience, 16''(6), 988–999. 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'', 607–628. 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'', Article 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, 34''(1), Article bhad441. https://doi.org/10.1093/cercor/bhad441 Thaler, R. H. (2000). From homo economicus to homo sapiens. ''The Journal of Economic Perspectives, 14''(1), 133–141. https://www.proquest.com/scholarly-journals/homo-economicus-sapiens/docview/212067365 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'', Article 849566. https://doi.org/10.3389/fpsyg.2022.849566 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 Yu, C. L., & Chou, T. L. (2018). A dual route model of empathy: A neurobiological prospective. ''Frontiers in Psychology, 9'', Article 2212. https://doi.org/10.3389/fpsyg.2018.02212 Zhu, X., Tian, G., Liu, R., & Li, Y. (2025). Emotional leadership and teachers’ psychological well-being: The mediating role of emotional labour and moderating roles of cultural values in JD-R theory. ''European Journal of Education, 60''(1), Article e70006. https://doi.org/10.1111/ejed.70006 }} ==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]] sl4w5z6pdnjh6b4kaeprycuz98otdcn 2844678 2844677 2026-11-05T21:46:24Z U3143751 3106785 /* Overview */ Used ChatGPT to make scenario more concise: https://chatgpt.com/share/6aecf920-29a0-83ec-963f-2f691049be6e 2844678 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 at a non-profit organisation providing mental health services to young people aged 18–25, many of whom have experienced challenging living conditions. They support approximately 20 clients each week. Initially, Aelin is highly motivated, demonstrating strong active listening skills, empathy and an ability to create a safe, welcoming environment for clients. They remember personal details and adapt their approach to respond sensitively to clients' emotional needs. However, as their caseload grows and clients present with increasingly complex needs and trauma histories, Aelin becomes increasingly invested in their work. They begin taking work home, with work-related thoughts gradually extending into weekends. Over time, they experience persistent exhaustion, even before the working day begins. Several months later, Aelin notices a shift in their interactions with others, including clients. They feel increasingly detached, frustrated and less empathetic, rather than engaged and attentive. Work leaves them emotionally drained, with little energy for friends or family. As their irritability increases, Aelin begins questioning whether they can 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 (Lloyd 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; 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]] and the [[wikipedia:Job_demands-resources_model|Job Demands-Resources model]]. === Conservation of resources stress model === The Conservation of resources stress model (CORT; Hobfoll, 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 & 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 & 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 == At its core, [[empathy]] is the ability to imagine or understand another person's feelings and/or emotions (Shao et al., 2024). Research around empathy is diverse, which is especially apparent when looking at theories considering the constructs of empathy. A dual model consisting of affective and cognitive empathy (Yu and Chou, 2018) appears to be the simplest, where affective empathy occurs as an emotional reaction to someone else's mental state (Shao et al., 2024), and cognitive empathy occurs when a person understands someone else's feelings or emotions (Cui et al., 2022). However, a tripartite model will be the primary focus in this chapter. === Cognitive empathy === Cognitive empathy is the "what" and focuses on a person tangibly understanding what another person feels or, simply, "understanding the mental states of others" (Martingano et al., 2026 p. 802). It is associated with activation in the dorsomedial prefrontal cortex and the right inferior parietal cortex (Ruby & Decety, 2004) and, considered as a [[wikipedia:Attention#Orienting|top-down attention]] process, requires a conscious, sustained focus (Shao et al., 2024). In relation to the JD-R and CORT models, engaging in cognitive empathy is negatively associated with emotional exhaustion and may be labelled as a job resource (Ferguson et al., 2021). === Affective empathy, compassion, and emotional contagion === Affective empathy is the "how" and focuses on the feeling aspect of empathy. It is associated with mirror neuron activity inferior frontal cortex, the premotor areas and the insula (Yu and Chou, 2018). Through the tripartite model, affective empathy is further categorised as either other- or self-focused (Mantinago et al. 2026). When the affective response is self-oriented, the boundaries between a person and the target of empathy blur, leading to [[wikipedia:Emotional_contagion|emotional contagion]] - a theory that suggests emotions are contagious and spread from one person to another through an automatic [[wikipedia:Mimicry|mimicry]] process (Panksepp and Lahvis, 2011). Contagion is negatively associated with emotional exhaustion and may be labelled as a job demand (Hunt and Gooney, 2019). However, when the affective response is other-oriented and Observers understand that the emotions they are reacting to belong to another person, rather than themselves, it leads to empathic concern in the form of sympathy or compassion (Mantinago et al. 2026). Much like cognitive empathy, empathic concern, primarily the compassion aspect, is negatively associated with emotional exhaustion and is considered a job resource. Table 1. outlines the primary differences between the different types of empathy. {| class="wikitable" |+ '''Table 1. Differences Between Empathy Types''' ! Type of empathy ! Description ! What this looks like |- | '''Cognitive empathy''' | Empathy that focuses on understanding and recognising another person's thoughts, feelings and perspective without necessarily sharing their emotions. | Understanding why a friend feels upset after receiving bad news. |- | '''Affective empathy''' | The ability to emotionally respond to and share in another person's emotional experience. | Feeling sad when you see a friend crying. |- | '''Compassion''' (affective-other) | Affective empathy which motivates a desire to take action to make change | Feeling concerned about a struggling friend and offering support. |- | '''Sympathy''' (affective-other) | Affective empathy that does '''not''' motivate a desire to take action. | Feeling sorry for a friend who has lost their job. |- | '''Emotional contagion''' (affective-self) | Mirroring another person's emotional state. | Feeling anxious after spending time with someone who is visibly anxious. |} === Empathy and compassion fatigue === [[wikipedia:Compassion_fatigue|Compassion fatigue]], also known as [[wikipedia:Secondary_trauma|secondary traumatic stress]], is a term introduced by Figley (2002) to describe the negative experience affecting healthcare professionals when they are constantly exposed to the trauma of their patients and is associated with burnout. However, looking at more modern literature, it is understood that compassion does not lead to emotional exhaustion. As such, it will be referred to as empathy fatigue - an alternative term to support clarity in its antecedents (Cigrand et al., 2025). {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' *In the terms of the JD-R and CORT models, Compassion and Cognitive empathy function as job resources, thus helping to buffer individuals from burnout, whereas self-focused affective empathy (contagion) functions as a job demand. *Empathy fatigue occurs because of contagion depleting energy faster than resources can restore or mitigate demands. Aelin has engaged in contagion with their clients and is now experiencing empathy fatigue, contributing to their emotional exhaustion. * {{RoundBoxBottom}} == Emotional labour as a mediating factor between empathy fatigue and emotional exhaustion == [[wikipedia:Social_norm|Social norms]] provide guidelines for socially acceptable emotional expression. As social beings, humans rely on these shared expectations to regulate their emotional responses and maintain appropriate social interactions. This is the essence of [[Emotion labour|emotional labour]]: "the regulation of felt and expressed emotions (whether effortful or not) to match these emotional display rules with the objective of fulfilling organisational goals" (Vial & Cowgill, 2022 p. 2). Emotional labour is typically displayed through one of two ways. === Types of acting === Surface acting involves changing how felt emotions are externally expressed, such as adjusting facial expressions and body language to 'fake' the appropriate emotion (Vial & Cowgill, 2022). This creates an internal conflict known as emotional dissonance, where what someone is feeling and what they are expressing is misaligned (Middleton, 1989, as cited in Kenworthy et al., 2014). A meta-analysis from Kenworthy et al. (2014) looked at the relationship between emotional dissonance and burnout. Across 57 articles published between 1998 and 2011, they found a medium effect size (''r'' =.34) across a large sample (n=16165). Demographics were not specifically reported on, however they considered moderating factors and found a significant slope after regressing for gender (''Z'' = 2.24, ''p'' = .03). Under the JD-R, surface acting may be considered as a job demand. Surface and deep acting, the second type of emotional labour, are both [[emotional regulation]] strategies, but differ in what they are regulating. Surface acting regulates the expression of emotion, and deep acting regulates the internal feeling of the emotion through strategies such as [[wikipedia:Cognitive_appraisal|cognitive reappraisal]] (Vial & Cowgill, 2022). It is important to note that some literature discusses a third emotional labour, expression of naturally felt emotions (ENFE), which just describes that a person's genuine feelings are the appropriate to express in the moment and does not require additional emotional regulation (Zhu et al., 2025). === The mediating effect of emotional labour === Pulling together the research, emotional labour is one of the primary bridges between emotional exhaustion and empathy fatigue. When performing surface acting, people become emotionally exhausted. As such, individuals have fewer resources available to manage their emotions leading to emotional dysregulation (Nangle et al., 2019). This creates a job demand spiral, where they are still required to display appropriate emotional responses (Vial & Cowgill, 2022). Under the CORT model, this further depletes already limited emotional resources, intensifying emotional exhaustion (Hobfoll, 1989). Consequently, while this emotional labour is sustained, it reduces an individual's capacity to engage in cognitive and other-oriented affective empathy. Surface acting may exacerbate this effect by increasing emotional demands, while repeated exposure to others' emotions may further contribute to emotional strain through emotional contagion (Hunt & Gooney, 2019). Therefore, emotional labour, particularly surface acting, acts as the mechanism through which empathy fatigue contributes to emotional exhaustion. '''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 {Deep acting is negatively associated with burnout: |type="()"} + True - False </quiz> {{RoundBoxTop|theme=3}} '''What role does emotional labor play in emotional exhaustion and empathy fatigue''' * In relation to the CORT and JD-R theories, surface acting expends energy as a job demand. When energy levels are exhausted, people are more likely to engage in contagion due to emotional dysregulation, leading to a job demands spiral that exacerbates emotional exhaustion. * Aelin has exhausted their energy supplies by surface acting and is stuck in a job demands spiral due to contagion. {{RoundBoxBottom}} == Strategies for combating emotional exhaustion == Fortunately, burnout is not a permanent condition and treatment is available for dealing with emotional exhaustion. Table 2. has been adapated from van Dam (2021) to provide an overview of goals and interventions that are available. Someone who is experiencing burnout would begin at phase 1, whereas someone who is not would begin at phase 3. Importantly, when it comes to psychoeducation, it is critical to understand how to reduce the amount of surface acting someone is engaging in. Suggestions include knowledge around communication and boundaries (Bakker et al., 2021), promotion of positive emotions through storytelling (Wei et al., 2024) and a transition to deep acting or ENFE as these are associated with the conservation and gain of resources instead of increasing job demands (Zhu et al., 2025). '''Table 3. Treatment phases of clinical burnout (based on Table 2 from van Dam, A. 2021)''' {| class="wikitable" style="margin: auto; |- ! Phase !! Goal !! Interventions |- | '''Phase 1: Crisis''' || Recognition of the patient that the problems are serious, and that serious action is required. These interventions are primarily focused on reducing job demands. || 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. These interventions are primarily focused on increasing job resources. || 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. These interventions are focused on reducing job demands and increasing job resources. || 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. These interventions are focused on reducing job demands and increasing job resources. || Setting priorities regarding quality of life and interpersonal relationships. |} ==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= == References == Alarcon, G. M. (2011). A meta-analysis of burnout with job demands, resources, and attitudes. ''Journal of Vocational Behavior, 79''(2), 549–562. https://doi.org/10.1016/j.jvb.2011.03.007 Bakker, A. B., & de Vries, J. D. (2021). Job demands–resources theory and self-regulation: New explanations and remedies for job burnout. ''Anxiety, Stress, & Coping, 34''(1), 1–21. https://doi.org/10.1080/10615806.2020.1797695 Bakker, A. B., Demerouti, E., & Sanz-Vergel, A. I. (2014). Burnout and work engagement: The JD–R approach. ''Annual Review of Organizational Psychology and Organizational Behavior, 1'', 389–411. https://doi.org/10.1146/annurev-orgpsych-031413-091235 Chan, F. M. (2026). ''The state of workplace burnout 2026: The workplace divide''. Infinite Potential. https://www.ahri.com.au/wp-content/uploads/The-State-of-Workplace-Burnout-2026-Final.pdf Cigrand, C., Quaglia, J. T., Lawlor, S., & Bailey, C. (2025). Healing the healers: The role of contemplative practice in mitigating counselor burnout and empathy fatigue. ''Journal of Mental Health Counseling, 47''(2), 241–261. https://doi.org/10.17744/mehc.47.3.04 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, 32''(23), 5330–5342. https://doi.org/10.1093/cercor/bhac017 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 Ferguson, A. M., Cameron, C. D., & Inzlicht, M. (2021). When does empathy feel good? ''Current Opinion in Behavioral Sciences, 39'', 125–129. https://doi.org/10.1016/j.cobeha.2021.03.011 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 Gynning, B. E., Karlsson, E., Teoh, K., Gustavsson, P., Christiansen, F., & Brulin, E. (2025). Do job resources buffer the harmful effects of job demands on burnout complaints? A 1-year cohort study of Swedish healthcare professionals. ''International Journal of Nursing Studies Advances, 9'', Article 100397. https://doi.org/10.1016/j.ijnsa.2025.100397 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, 2''(3), 96–99. https://doi.org/10.1111/1467-8721.ep10770953 Hobfoll, S. E. (1989). Conservation of resources: A new attempt at conceptualizing stress. ''American Psychologist, 44''(3), 513–524. https://doi.org/10.1037/0003-066X.44.3.513 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 Hunt, P., Denieffe, S., & Gooney, M. (2019). Running on empathy: Relationship of empathy to compassion satisfaction and compassion fatigue in cancer healthcare professionals. ''European Journal of Cancer Care, 28''(4), Article e13124. https://doi.org/10.1111/ecc.13124 Ito, J. K., & Brotheridge, C. M. (2003). Resources, coping strategies, and emotional exhaustion: A conservation of resources perspective. ''Journal of Vocational Behavior, 63''(3), 490–509. https://doi.org/10.1016/S0001-8791(02)00033-7 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/10.1111/jasp.12211 Lesener, T., Gusy, B., & Wolter, C. (2019). The job demands–resources model: A meta-analytic review of longitudinal studies. ''Work & Stress, 33''(1), 76–103. https://doi.org/10.1080/02678373.2018.1529065 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/10.1007/s00127-004-0808-7 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 Nangle, M. R., Henry, J. D., von Hippel, C., & Kjelsaas, K. (2019). An empirical study of how emotion dysregulation and social cognition relate to occupational burnout in dentistry. ''British Dental Journal, 227''(4), 285–290. https://doi.org/10.1038/s41415-019-0620-0 Panksepp, J. B., & Lahvis, G. P. (2011). Rodent empathy and affective neuroscience. ''Neuroscience & 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), Article 04023060. https://doi.org/10.1061/JCEMD4.COENG-13339 Ruby, P., & Decety, J. (2004). How would you feel versus how do you think she would feel? A neuroimaging study of perspective-taking with social emotions. ''Journal of Cognitive Neuroscience, 16''(6), 988–999. 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'', 607–628. 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'', Article 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, 34''(1), Article bhad441. https://doi.org/10.1093/cercor/bhad441 Thaler, R. H. (2000). From homo economicus to homo sapiens. ''The Journal of Economic Perspectives, 14''(1), 133–141. https://www.proquest.com/scholarly-journals/homo-economicus-sapiens/docview/212067365 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'', Article 849566. https://doi.org/10.3389/fpsyg.2022.849566 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 Yu, C. L., & Chou, T. L. (2018). A dual route model of empathy: A neurobiological prospective. ''Frontiers in Psychology, 9'', Article 2212. https://doi.org/10.3389/fpsyg.2018.02212 Zhu, X., Tian, G., Liu, R., & Li, Y. (2025). Emotional leadership and teachers’ psychological well-being: The mediating role of emotional labour and moderating roles of cultural values in JD-R theory. ''European Journal of Education, 60''(1), Article e70006. https://doi.org/10.1111/ejed.70006 }} ==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]] qj5wwoetblsv1l20f23f68vmqhr2xvs 2844679 2844678 2026-11-05T21:51:07Z U3143751 3106785 Refining to word count 2844679 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 at a non-profit organisation providing mental health services to young people aged 18–25, many of whom have experienced challenging living conditions. They support approximately 20 clients each week. Initially, Aelin is highly motivated, demonstrating strong active listening skills, empathy and an ability to create a safe, welcoming environment for clients. They remember personal details and adapt their approach to respond sensitively to clients' emotional needs. However, as their caseload grows and clients present with increasingly complex needs and trauma histories, Aelin becomes increasingly invested in their work. They begin taking work home, with work-related thoughts gradually extending into weekends. Over time, they experience persistent exhaustion, even before the working day begins. Several months later, Aelin notices a shift in their interactions with others, including clients. They feel increasingly detached, frustrated and less empathetic, rather than engaged and attentive. Work leaves them emotionally drained, with little energy for friends or family. As their irritability increases, Aelin begins questioning whether they can 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. 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. {{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 == 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 (Lloyd 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; Seidler et al., 2014). 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]] and the [[wikipedia:Job_demands-resources_model|Job Demands-Resources model]]. === Conservation of resources stress model === The Conservation of resources stress model (CORT; Hobfoll, 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 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. 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). 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 & 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 & 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). 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). It's important to note that job resources 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 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. 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 == At its core, [[empathy]] is the ability to imagine or understand another person's feelings and/or emotions (Shao et al., 2024). Research around empathy is diverse, which is especially apparent when looking at theories considering the constructs of empathy. A dual model consisting of affective and cognitive empathy (Yu and Chou, 2018) appears to be the simplest, where affective empathy occurs as an emotional reaction to someone else's mental state (Shao et al., 2024), and cognitive empathy occurs when a person understands someone else's feelings or emotions (Cui et al., 2022). However, a tripartite model will be the primary focus in this chapter. === Cognitive empathy === Cognitive empathy is the "what" and focuses on a person tangibly understanding what another person feels or, simply, "understanding the mental states of others" (Martingano et al., 2026 p. 802). It is associated with activation in the dorsomedial prefrontal cortex and the right inferior parietal cortex (Ruby & Decety, 2004) and, considered as a [[wikipedia:Attention#Orienting|top-down attention]] process, requires a conscious, sustained focus (Shao et al., 2024). In relation to the JD-R and CORT models, engaging in cognitive empathy is negatively associated with emotional exhaustion and may be labelled as a job resource (Ferguson et al., 2021). === Affective empathy, compassion, and emotional contagion === Affective empathy is the "how" and focuses on the feeling aspect of empathy. It is associated with mirror neuron activity inferior frontal cortex, the premotor areas and the insula (Yu and Chou, 2018). Through the tripartite model, affective empathy is further categorised as either other- or self-focused (Mantinago et al. 2026). When the affective response is self-oriented, the boundaries between a person and the target of empathy blur, leading to [[wikipedia:Emotional_contagion|emotional contagion]] - a theory that suggests emotions are contagious and spread from one person to another through an automatic [[wikipedia:Mimicry|mimicry]] process (Panksepp and Lahvis, 2011). Contagion is negatively associated with emotional exhaustion and may be labelled as a job demand (Hunt and Gooney, 2019). However, when the affective response is other-oriented and Observers understand that the emotions they are reacting to belong to another person, rather than themselves, it leads to empathic concern in the form of sympathy or compassion (Mantinago et al. 2026). Much like cognitive empathy, empathic concern, primarily the compassion aspect, is negatively associated with emotional exhaustion and is considered a job resource. Table 1. outlines the primary differences between the different types of empathy. {| class="wikitable" |+ '''Table 1. Differences Between Empathy Types''' ! Type of empathy ! Description ! What this looks like |- | '''Cognitive empathy''' | Empathy that focuses on understanding and recognising another person's thoughts, feelings and perspective without necessarily sharing their emotions. | Understanding why a friend feels upset after receiving bad news. |- | '''Affective empathy''' | The ability to emotionally respond to and share in another person's emotional experience. | Feeling sad when you see a friend crying. |- | '''Compassion''' (affective-other) | Affective empathy which motivates a desire to take action to make change | Feeling concerned about a struggling friend and offering support. |- | '''Sympathy''' (affective-other) | Affective empathy that does '''not''' motivate a desire to take action. | Feeling sorry for a friend who has lost their job. |- | '''Emotional contagion''' (affective-self) | Mirroring another person's emotional state. | Feeling anxious after spending time with someone who is visibly anxious. |} === Empathy and compassion fatigue === [[wikipedia:Compassion_fatigue|Compassion fatigue]], also known as [[wikipedia:Secondary_trauma|secondary traumatic stress]], is a term introduced by Figley (2002) to describe the negative experience affecting healthcare professionals when they are constantly exposed to the trauma of their patients and is associated with burnout. However, looking at more modern literature, it is understood that compassion does not lead to emotional exhaustion. As such, it will be referred to as empathy fatigue - an alternative term to support clarity in its antecedents (Cigrand et al., 2025). {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' *In the terms of the JD-R and CORT models, Compassion and Cognitive empathy function as job resources, thus helping to buffer individuals from burnout, whereas self-focused affective empathy (contagion) functions as a job demand. *Empathy fatigue occurs because of contagion depleting energy faster than resources can restore or mitigate demands. Aelin has engaged in contagion with their clients and is now experiencing empathy fatigue, contributing to their emotional exhaustion. * {{RoundBoxBottom}} == Emotional labour as a mediating factor between empathy fatigue and emotional exhaustion == [[wikipedia:Social_norm|Social norms]] provide guidelines for socially acceptable emotional expression. As social beings, humans rely on these shared expectations to regulate their emotional responses and maintain appropriate social interactions. This is the essence of [[Emotion labour|emotional labour]]: "the regulation of felt and expressed emotions (whether effortful or not) to match these emotional display rules with the objective of fulfilling organisational goals" (Vial & Cowgill, 2022 p. 2). Emotional labour is typically displayed through one of two ways. === Types of acting === Surface acting involves changing how felt emotions are externally expressed, such as adjusting facial expressions and body language to 'fake' the appropriate emotion (Vial & Cowgill, 2022). This creates an internal conflict known as emotional dissonance, where what someone is feeling and what they are expressing is misaligned (Middleton, 1989, as cited in Kenworthy et al., 2014). A meta-analysis from Kenworthy et al. (2014) looked at the relationship between emotional dissonance and burnout. Across 57 articles published between 1998 and 2011, they found a medium effect size (''r'' =.34) across a large sample (n=16165). Demographics were not specifically reported on, however they considered moderating factors and found a significant slope after regressing for gender (''Z'' = 2.24, ''p'' = .03). Under the JD-R, surface acting may be considered as a job demand. Surface and deep acting, the second type of emotional labour, are both [[emotional regulation]] strategies, but differ in what they are regulating. Surface acting regulates the expression of emotion, and deep acting regulates the internal feeling of the emotion through strategies such as [[wikipedia:Cognitive_appraisal|cognitive reappraisal]] (Vial & Cowgill, 2022). It is important to note that some literature discusses a third emotional labour, expression of naturally felt emotions (ENFE), which just describes that a person's genuine feelings are the appropriate to express in the moment and does not require additional emotional regulation (Zhu et al., 2025). === The mediating effect of emotional labour === Pulling together the research, emotional labour is one of the primary bridges between emotional exhaustion and empathy fatigue. When performing surface acting, people become emotionally exhausted. As such, individuals have fewer resources available to manage their emotions leading to emotional dysregulation (Nangle et al., 2019). This creates a job demand spiral, where they are still required to display appropriate emotional responses (Vial & Cowgill, 2022). Under the CORT model, this further depletes already limited emotional resources, intensifying emotional exhaustion (Hobfoll, 1989). Consequently, while this emotional labour is sustained, it reduces an individual's capacity to engage in cognitive and other-oriented affective empathy. Surface acting may exacerbate this effect by increasing emotional demands, while repeated exposure to others' emotions may further contribute to emotional strain through emotional contagion (Hunt & Gooney, 2019). Therefore, emotional labour, particularly surface acting, acts as the mechanism through which empathy fatigue contributes to emotional exhaustion. '''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 {Deep acting is negatively associated with burnout: |type="()"} + True - False </quiz> {{RoundBoxTop|theme=3}} '''What role does emotional labor play in emotional exhaustion and empathy fatigue''' * In relation to the CORT and JD-R theories, surface acting expends energy as a job demand. When energy levels are exhausted, people are more likely to engage in contagion due to emotional dysregulation, leading to a job demands spiral that exacerbates emotional exhaustion. * Aelin has exhausted their energy supplies by surface acting and is stuck in a job demands spiral due to contagion. {{RoundBoxBottom}} == Strategies for combating emotional exhaustion == Fortunately, burnout is not a permanent condition and treatment is available for dealing with emotional exhaustion. Table 2. has been adapated from van Dam (2021) to provide an overview of goals and interventions that are available. Someone who is experiencing burnout would begin at phase 1, whereas someone who is not would begin at phase 3. Importantly, when it comes to psychoeducation, it is critical to understand how to reduce the amount of surface acting someone is engaging in. Suggestions include knowledge around communication and boundaries (Bakker et al., 2021), promotion of positive emotions through storytelling (Wei et al., 2024) and a transition to deep acting or ENFE as these are associated with the conservation and gain of resources instead of increasing job demands (Zhu et al., 2025). '''Table 3. Treatment phases of clinical burnout (based on Table 2 from van Dam, A. 2021)''' {| class="wikitable" style="margin: auto; |- ! Phase !! Goal !! Interventions |- | '''Phase 1: Crisis''' || Recognition of the patient that the problems are serious, and that serious action is required. These interventions are primarily focused on reducing job demands. || 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. These interventions are primarily focused on increasing job resources. || 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. These interventions are focused on reducing job demands and increasing job resources. || 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. These interventions are focused on reducing job demands and increasing job resources. || Setting priorities regarding quality of life and interpersonal relationships. |} ==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= == References == Alarcon, G. M. (2011). A meta-analysis of burnout with job demands, resources, and attitudes. ''Journal of Vocational Behavior, 79''(2), 549–562. https://doi.org/10.1016/j.jvb.2011.03.007 Bakker, A. B., & de Vries, J. D. (2021). Job demands–resources theory and self-regulation: New explanations and remedies for job burnout. ''Anxiety, Stress, & Coping, 34''(1), 1–21. https://doi.org/10.1080/10615806.2020.1797695 Bakker, A. B., Demerouti, E., & Sanz-Vergel, A. I. (2014). Burnout and work engagement: The JD–R approach. ''Annual Review of Organizational Psychology and Organizational Behavior, 1'', 389–411. https://doi.org/10.1146/annurev-orgpsych-031413-091235 Chan, F. M. (2026). ''The state of workplace burnout 2026: The workplace divide''. Infinite Potential. https://www.ahri.com.au/wp-content/uploads/The-State-of-Workplace-Burnout-2026-Final.pdf Cigrand, C., Quaglia, J. T., Lawlor, S., & Bailey, C. (2025). Healing the healers: The role of contemplative practice in mitigating counselor burnout and empathy fatigue. ''Journal of Mental Health Counseling, 47''(2), 241–261. https://doi.org/10.17744/mehc.47.3.04 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, 32''(23), 5330–5342. https://doi.org/10.1093/cercor/bhac017 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 Ferguson, A. M., Cameron, C. D., & Inzlicht, M. (2021). When does empathy feel good? ''Current Opinion in Behavioral Sciences, 39'', 125–129. https://doi.org/10.1016/j.cobeha.2021.03.011 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 Gynning, B. E., Karlsson, E., Teoh, K., Gustavsson, P., Christiansen, F., & Brulin, E. (2025). Do job resources buffer the harmful effects of job demands on burnout complaints? A 1-year cohort study of Swedish healthcare professionals. ''International Journal of Nursing Studies Advances, 9'', Article 100397. https://doi.org/10.1016/j.ijnsa.2025.100397 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, 2''(3), 96–99. https://doi.org/10.1111/1467-8721.ep10770953 Hobfoll, S. E. (1989). Conservation of resources: A new attempt at conceptualizing stress. ''American Psychologist, 44''(3), 513–524. https://doi.org/10.1037/0003-066X.44.3.513 Hunt, P., Denieffe, S., & Gooney, M. (2019). Running on empathy: Relationship of empathy to compassion satisfaction and compassion fatigue in cancer healthcare professionals. ''European Journal of Cancer Care, 28''(4), Article e13124. https://doi.org/10.1111/ecc.13124 Ito, J. K., & Brotheridge, C. M. (2003). Resources, coping strategies, and emotional exhaustion: A conservation of resources perspective. ''Journal of Vocational Behavior, 63''(3), 490–509. https://doi.org/10.1016/S0001-8791(02)00033-7 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/10.1111/jasp.12211 Lesener, T., Gusy, B., & Wolter, C. (2019). The job demands–resources model: A meta-analytic review of longitudinal studies. ''Work & Stress, 33''(1), 76–103. https://doi.org/10.1080/02678373.2018.1529065 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/10.1007/s00127-004-0808-7 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 Nangle, M. R., Henry, J. D., von Hippel, C., & Kjelsaas, K. (2019). An empirical study of how emotion dysregulation and social cognition relate to occupational burnout in dentistry. ''British Dental Journal, 227''(4), 285–290. https://doi.org/10.1038/s41415-019-0620-0 Panksepp, J. B., & Lahvis, G. P. (2011). Rodent empathy and affective neuroscience. ''Neuroscience & 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), Article 04023060. https://doi.org/10.1061/JCEMD4.COENG-13339 Ruby, P., & Decety, J. (2004). How would you feel versus how do you think she would feel? A neuroimaging study of perspective-taking with social emotions. ''Journal of Cognitive Neuroscience, 16''(6), 988–999. 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'', 607–628. 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'', Article 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, 34''(1), Article bhad441. https://doi.org/10.1093/cercor/bhad441 Thaler, R. H. (2000). From homo economicus to homo sapiens. ''The Journal of Economic Perspectives, 14''(1), 133–141. https://www.proquest.com/scholarly-journals/homo-economicus-sapiens/docview/212067365 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'', Article 849566. https://doi.org/10.3389/fpsyg.2022.849566 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 Yu, C. L., & Chou, T. L. (2018). A dual route model of empathy: A neurobiological prospective. ''Frontiers in Psychology, 9'', Article 2212. https://doi.org/10.3389/fpsyg.2018.02212 Zhu, X., Tian, G., Liu, R., & Li, Y. (2025). Emotional leadership and teachers’ psychological well-being: The mediating role of emotional labour and moderating roles of cultural values in JD-R theory. ''European Journal of Education, 60''(1), Article e70006. https://doi.org/10.1111/ejed.70006 }} ==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]] p1fsyyxfs6vt0vhbpn71dzfh1jljg5b 2844680 2844679 2026-11-05T21:55:07Z U3143751 3106785 /* Conclusion */ Used chat gpt to write a conclusion: https://chatgpt.com/share/6aecfb33-5430-83ec-89f0-7d70945a8d04 2844680 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 at a non-profit organisation providing mental health services to young people aged 18–25, many of whom have experienced challenging living conditions. They support approximately 20 clients each week. Initially, Aelin is highly motivated, demonstrating strong active listening skills, empathy and an ability to create a safe, welcoming environment for clients. They remember personal details and adapt their approach to respond sensitively to clients' emotional needs. However, as their caseload grows and clients present with increasingly complex needs and trauma histories, Aelin becomes increasingly invested in their work. They begin taking work home, with work-related thoughts gradually extending into weekends. Over time, they experience persistent exhaustion, even before the working day begins. Several months later, Aelin notices a shift in their interactions with others, including clients. They feel increasingly detached, frustrated and less empathetic, rather than engaged and attentive. Work leaves them emotionally drained, with little energy for friends or family. As their irritability increases, Aelin begins questioning whether they can 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. 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. {{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 == 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 (Lloyd 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; Seidler et al., 2014). 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]] and the [[wikipedia:Job_demands-resources_model|Job Demands-Resources model]]. === Conservation of resources stress model === The Conservation of resources stress model (CORT; Hobfoll, 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 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. 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). 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 & 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 & 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). 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). It's important to note that job resources 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 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. 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 == At its core, [[empathy]] is the ability to imagine or understand another person's feelings and/or emotions (Shao et al., 2024). Research around empathy is diverse, which is especially apparent when looking at theories considering the constructs of empathy. A dual model consisting of affective and cognitive empathy (Yu and Chou, 2018) appears to be the simplest, where affective empathy occurs as an emotional reaction to someone else's mental state (Shao et al., 2024), and cognitive empathy occurs when a person understands someone else's feelings or emotions (Cui et al., 2022). However, a tripartite model will be the primary focus in this chapter. === Cognitive empathy === Cognitive empathy is the "what" and focuses on a person tangibly understanding what another person feels or, simply, "understanding the mental states of others" (Martingano et al., 2026 p. 802). It is associated with activation in the dorsomedial prefrontal cortex and the right inferior parietal cortex (Ruby & Decety, 2004) and, considered as a [[wikipedia:Attention#Orienting|top-down attention]] process, requires a conscious, sustained focus (Shao et al., 2024). In relation to the JD-R and CORT models, engaging in cognitive empathy is negatively associated with emotional exhaustion and may be labelled as a job resource (Ferguson et al., 2021). === Affective empathy, compassion, and emotional contagion === Affective empathy is the "how" and focuses on the feeling aspect of empathy. It is associated with mirror neuron activity inferior frontal cortex, the premotor areas and the insula (Yu and Chou, 2018). Through the tripartite model, affective empathy is further categorised as either other- or self-focused (Mantinago et al. 2026). When the affective response is self-oriented, the boundaries between a person and the target of empathy blur, leading to [[wikipedia:Emotional_contagion|emotional contagion]] - a theory that suggests emotions are contagious and spread from one person to another through an automatic [[wikipedia:Mimicry|mimicry]] process (Panksepp and Lahvis, 2011). Contagion is negatively associated with emotional exhaustion and may be labelled as a job demand (Hunt and Gooney, 2019). However, when the affective response is other-oriented and Observers understand that the emotions they are reacting to belong to another person, rather than themselves, it leads to empathic concern in the form of sympathy or compassion (Mantinago et al. 2026). Much like cognitive empathy, empathic concern, primarily the compassion aspect, is negatively associated with emotional exhaustion and is considered a job resource. Table 1. outlines the primary differences between the different types of empathy. {| class="wikitable" |+ '''Table 1. Differences Between Empathy Types''' ! Type of empathy ! Description ! What this looks like |- | '''Cognitive empathy''' | Empathy that focuses on understanding and recognising another person's thoughts, feelings and perspective without necessarily sharing their emotions. | Understanding why a friend feels upset after receiving bad news. |- | '''Affective empathy''' | The ability to emotionally respond to and share in another person's emotional experience. | Feeling sad when you see a friend crying. |- | '''Compassion''' (affective-other) | Affective empathy which motivates a desire to take action to make change | Feeling concerned about a struggling friend and offering support. |- | '''Sympathy''' (affective-other) | Affective empathy that does '''not''' motivate a desire to take action. | Feeling sorry for a friend who has lost their job. |- | '''Emotional contagion''' (affective-self) | Mirroring another person's emotional state. | Feeling anxious after spending time with someone who is visibly anxious. |} === Empathy and compassion fatigue === [[wikipedia:Compassion_fatigue|Compassion fatigue]], also known as [[wikipedia:Secondary_trauma|secondary traumatic stress]], is a term introduced by Figley (2002) to describe the negative experience affecting healthcare professionals when they are constantly exposed to the trauma of their patients and is associated with burnout. However, looking at more modern literature, it is understood that compassion does not lead to emotional exhaustion. As such, it will be referred to as empathy fatigue - an alternative term to support clarity in its antecedents (Cigrand et al., 2025). {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' *In the terms of the JD-R and CORT models, Compassion and Cognitive empathy function as job resources, thus helping to buffer individuals from burnout, whereas self-focused affective empathy (contagion) functions as a job demand. *Empathy fatigue occurs because of contagion depleting energy faster than resources can restore or mitigate demands. Aelin has engaged in contagion with their clients and is now experiencing empathy fatigue, contributing to their emotional exhaustion. * {{RoundBoxBottom}} == Emotional labour as a mediating factor between empathy fatigue and emotional exhaustion == [[wikipedia:Social_norm|Social norms]] provide guidelines for socially acceptable emotional expression. As social beings, humans rely on these shared expectations to regulate their emotional responses and maintain appropriate social interactions. This is the essence of [[Emotion labour|emotional labour]]: "the regulation of felt and expressed emotions (whether effortful or not) to match these emotional display rules with the objective of fulfilling organisational goals" (Vial & Cowgill, 2022 p. 2). Emotional labour is typically displayed through one of two ways. === Types of acting === Surface acting involves changing how felt emotions are externally expressed, such as adjusting facial expressions and body language to 'fake' the appropriate emotion (Vial & Cowgill, 2022). This creates an internal conflict known as emotional dissonance, where what someone is feeling and what they are expressing is misaligned (Middleton, 1989, as cited in Kenworthy et al., 2014). A meta-analysis from Kenworthy et al. (2014) looked at the relationship between emotional dissonance and burnout. Across 57 articles published between 1998 and 2011, they found a medium effect size (''r'' =.34) across a large sample (n=16165). Demographics were not specifically reported on, however they considered moderating factors and found a significant slope after regressing for gender (''Z'' = 2.24, ''p'' = .03). Under the JD-R, surface acting may be considered as a job demand. Surface and deep acting, the second type of emotional labour, are both [[emotional regulation]] strategies, but differ in what they are regulating. Surface acting regulates the expression of emotion, and deep acting regulates the internal feeling of the emotion through strategies such as [[wikipedia:Cognitive_appraisal|cognitive reappraisal]] (Vial & Cowgill, 2022). It is important to note that some literature discusses a third emotional labour, expression of naturally felt emotions (ENFE), which just describes that a person's genuine feelings are the appropriate to express in the moment and does not require additional emotional regulation (Zhu et al., 2025). === The mediating effect of emotional labour === Pulling together the research, emotional labour is one of the primary bridges between emotional exhaustion and empathy fatigue. When performing surface acting, people become emotionally exhausted. As such, individuals have fewer resources available to manage their emotions leading to emotional dysregulation (Nangle et al., 2019). This creates a job demand spiral, where they are still required to display appropriate emotional responses (Vial & Cowgill, 2022). Under the CORT model, this further depletes already limited emotional resources, intensifying emotional exhaustion (Hobfoll, 1989). Consequently, while this emotional labour is sustained, it reduces an individual's capacity to engage in cognitive and other-oriented affective empathy. Surface acting may exacerbate this effect by increasing emotional demands, while repeated exposure to others' emotions may further contribute to emotional strain through emotional contagion (Hunt & Gooney, 2019). Therefore, emotional labour, particularly surface acting, acts as the mechanism through which empathy fatigue contributes to emotional exhaustion. '''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 {Deep acting is negatively associated with burnout: |type="()"} + True - False </quiz> {{RoundBoxTop|theme=3}} '''What role does emotional labor play in emotional exhaustion and empathy fatigue''' * In relation to the CORT and JD-R theories, surface acting expends energy as a job demand. When energy levels are exhausted, people are more likely to engage in contagion due to emotional dysregulation, leading to a job demands spiral that exacerbates emotional exhaustion. * Aelin has exhausted their energy supplies by surface acting and is stuck in a job demands spiral due to contagion. {{RoundBoxBottom}} == Strategies for combating emotional exhaustion == Fortunately, burnout is not a permanent condition and treatment is available for dealing with emotional exhaustion. Table 2. has been adapated from van Dam (2021) to provide an overview of goals and interventions that are available. Someone who is experiencing burnout would begin at phase 1, whereas someone who is not would begin at phase 3. Importantly, when it comes to psychoeducation, it is critical to understand how to reduce the amount of surface acting someone is engaging in. Suggestions include knowledge around communication and boundaries (Bakker et al., 2021), promotion of positive emotions through storytelling (Wei et al., 2024) and a transition to deep acting or ENFE as these are associated with the conservation and gain of resources instead of increasing job demands (Zhu et al., 2025). '''Table 3. Treatment phases of clinical burnout (based on Table 2 from van Dam, A. 2021)''' {| class="wikitable" style="margin: auto; |- ! Phase !! Goal !! Interventions |- | '''Phase 1: Crisis''' || Recognition of the patient that the problems are serious, and that serious action is required. These interventions are primarily focused on reducing job demands. || 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. These interventions are primarily focused on increasing job resources. || 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. These interventions are focused on reducing job demands and increasing job resources. || 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. These interventions are focused on reducing job demands and increasing job resources. || Setting priorities regarding quality of life and interpersonal relationships. |} ==Conclusion== Sustained empathic engagement can contribute to emotional exhaustion when the emotional demands of supporting others continually exceed an individual's available resources for recovery. While cognitive empathy and compassion can act as protective job resources, self-focused affective empathy, particularly emotional contagion, can increase emotional demands by causing individuals to absorb the emotions of others. This is further intensified by emotional labour, particularly surface acting, which requires individuals to suppress or alter their genuine emotions to meet workplace expectations, creating emotional dissonance and further depleting their resources. The Conservation of Resources and Job Demands-Resources models demonstrate how ongoing resource loss, high job demands and insufficient opportunities for recovery can create a cycle of increasing stress and exhaustion. As demonstrated through Aelin's experience, recognising the distinction between different forms of empathy and understanding the role of emotional labour is essential to identifying the causes of empathy fatigue and emotional exhaustion. Ultimately, reducing excessive job demands, establishing healthy boundaries, prioritising recovery and developing more sustainable emotional regulation strategies can help preserve emotional resources, prevent burnout and support long-term wellbeing. ==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= == References == Alarcon, G. M. (2011). A meta-analysis of burnout with job demands, resources, and attitudes. ''Journal of Vocational Behavior, 79''(2), 549–562. https://doi.org/10.1016/j.jvb.2011.03.007 Bakker, A. B., & de Vries, J. D. (2021). Job demands–resources theory and self-regulation: New explanations and remedies for job burnout. ''Anxiety, Stress, & Coping, 34''(1), 1–21. https://doi.org/10.1080/10615806.2020.1797695 Bakker, A. B., Demerouti, E., & Sanz-Vergel, A. I. (2014). Burnout and work engagement: The JD–R approach. ''Annual Review of Organizational Psychology and Organizational Behavior, 1'', 389–411. https://doi.org/10.1146/annurev-orgpsych-031413-091235 Chan, F. M. (2026). ''The state of workplace burnout 2026: The workplace divide''. Infinite Potential. https://www.ahri.com.au/wp-content/uploads/The-State-of-Workplace-Burnout-2026-Final.pdf Cigrand, C., Quaglia, J. T., Lawlor, S., & Bailey, C. (2025). Healing the healers: The role of contemplative practice in mitigating counselor burnout and empathy fatigue. ''Journal of Mental Health Counseling, 47''(2), 241–261. https://doi.org/10.17744/mehc.47.3.04 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, 32''(23), 5330–5342. https://doi.org/10.1093/cercor/bhac017 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 Ferguson, A. M., Cameron, C. D., & Inzlicht, M. (2021). When does empathy feel good? ''Current Opinion in Behavioral Sciences, 39'', 125–129. https://doi.org/10.1016/j.cobeha.2021.03.011 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 Gynning, B. E., Karlsson, E., Teoh, K., Gustavsson, P., Christiansen, F., & Brulin, E. (2025). Do job resources buffer the harmful effects of job demands on burnout complaints? A 1-year cohort study of Swedish healthcare professionals. ''International Journal of Nursing Studies Advances, 9'', Article 100397. https://doi.org/10.1016/j.ijnsa.2025.100397 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, 2''(3), 96–99. https://doi.org/10.1111/1467-8721.ep10770953 Hobfoll, S. E. (1989). Conservation of resources: A new attempt at conceptualizing stress. ''American Psychologist, 44''(3), 513–524. https://doi.org/10.1037/0003-066X.44.3.513 Hunt, P., Denieffe, S., & Gooney, M. (2019). Running on empathy: Relationship of empathy to compassion satisfaction and compassion fatigue in cancer healthcare professionals. ''European Journal of Cancer Care, 28''(4), Article e13124. https://doi.org/10.1111/ecc.13124 Ito, J. K., & Brotheridge, C. M. (2003). Resources, coping strategies, and emotional exhaustion: A conservation of resources perspective. ''Journal of Vocational Behavior, 63''(3), 490–509. https://doi.org/10.1016/S0001-8791(02)00033-7 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/10.1111/jasp.12211 Lesener, T., Gusy, B., & Wolter, C. (2019). The job demands–resources model: A meta-analytic review of longitudinal studies. ''Work & Stress, 33''(1), 76–103. https://doi.org/10.1080/02678373.2018.1529065 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/10.1007/s00127-004-0808-7 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 Nangle, M. R., Henry, J. D., von Hippel, C., & Kjelsaas, K. (2019). An empirical study of how emotion dysregulation and social cognition relate to occupational burnout in dentistry. ''British Dental Journal, 227''(4), 285–290. https://doi.org/10.1038/s41415-019-0620-0 Panksepp, J. B., & Lahvis, G. P. (2011). Rodent empathy and affective neuroscience. ''Neuroscience & 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), Article 04023060. https://doi.org/10.1061/JCEMD4.COENG-13339 Ruby, P., & Decety, J. (2004). How would you feel versus how do you think she would feel? A neuroimaging study of perspective-taking with social emotions. ''Journal of Cognitive Neuroscience, 16''(6), 988–999. 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'', 607–628. 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'', Article 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, 34''(1), Article bhad441. https://doi.org/10.1093/cercor/bhad441 Thaler, R. H. (2000). From homo economicus to homo sapiens. ''The Journal of Economic Perspectives, 14''(1), 133–141. https://www.proquest.com/scholarly-journals/homo-economicus-sapiens/docview/212067365 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'', Article 849566. https://doi.org/10.3389/fpsyg.2022.849566 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 Yu, C. L., & Chou, T. L. (2018). A dual route model of empathy: A neurobiological prospective. ''Frontiers in Psychology, 9'', Article 2212. https://doi.org/10.3389/fpsyg.2018.02212 Zhu, X., Tian, G., Liu, R., & Li, Y. (2025). Emotional leadership and teachers’ psychological well-being: The mediating role of emotional labour and moderating roles of cultural values in JD-R theory. ''European Journal of Education, 60''(1), Article e70006. https://doi.org/10.1111/ejed.70006 }} ==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]] c7jyy6pl15qpk79ks0j91hauoyu0s05 2844681 2844680 2026-11-05T21:57:34Z U3143751 3106785 /* See also */ 2844681 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 at a non-profit organisation providing mental health services to young people aged 18–25, many of whom have experienced challenging living conditions. They support approximately 20 clients each week. Initially, Aelin is highly motivated, demonstrating strong active listening skills, empathy and an ability to create a safe, welcoming environment for clients. They remember personal details and adapt their approach to respond sensitively to clients' emotional needs. However, as their caseload grows and clients present with increasingly complex needs and trauma histories, Aelin becomes increasingly invested in their work. They begin taking work home, with work-related thoughts gradually extending into weekends. Over time, they experience persistent exhaustion, even before the working day begins. Several months later, Aelin notices a shift in their interactions with others, including clients. They feel increasingly detached, frustrated and less empathetic, rather than engaged and attentive. Work leaves them emotionally drained, with little energy for friends or family. As their irritability increases, Aelin begins questioning whether they can 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. 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. {{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 == 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 (Lloyd 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; Seidler et al., 2014). 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]] and the [[wikipedia:Job_demands-resources_model|Job Demands-Resources model]]. === Conservation of resources stress model === The Conservation of resources stress model (CORT; Hobfoll, 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 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. 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). 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 & 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 & 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). 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). It's important to note that job resources 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 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. 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 == At its core, [[empathy]] is the ability to imagine or understand another person's feelings and/or emotions (Shao et al., 2024). Research around empathy is diverse, which is especially apparent when looking at theories considering the constructs of empathy. A dual model consisting of affective and cognitive empathy (Yu and Chou, 2018) appears to be the simplest, where affective empathy occurs as an emotional reaction to someone else's mental state (Shao et al., 2024), and cognitive empathy occurs when a person understands someone else's feelings or emotions (Cui et al., 2022). However, a tripartite model will be the primary focus in this chapter. === Cognitive empathy === Cognitive empathy is the "what" and focuses on a person tangibly understanding what another person feels or, simply, "understanding the mental states of others" (Martingano et al., 2026 p. 802). It is associated with activation in the dorsomedial prefrontal cortex and the right inferior parietal cortex (Ruby & Decety, 2004) and, considered as a [[wikipedia:Attention#Orienting|top-down attention]] process, requires a conscious, sustained focus (Shao et al., 2024). In relation to the JD-R and CORT models, engaging in cognitive empathy is negatively associated with emotional exhaustion and may be labelled as a job resource (Ferguson et al., 2021). === Affective empathy, compassion, and emotional contagion === Affective empathy is the "how" and focuses on the feeling aspect of empathy. It is associated with mirror neuron activity inferior frontal cortex, the premotor areas and the insula (Yu and Chou, 2018). Through the tripartite model, affective empathy is further categorised as either other- or self-focused (Mantinago et al. 2026). When the affective response is self-oriented, the boundaries between a person and the target of empathy blur, leading to [[wikipedia:Emotional_contagion|emotional contagion]] - a theory that suggests emotions are contagious and spread from one person to another through an automatic [[wikipedia:Mimicry|mimicry]] process (Panksepp and Lahvis, 2011). Contagion is negatively associated with emotional exhaustion and may be labelled as a job demand (Hunt and Gooney, 2019). However, when the affective response is other-oriented and Observers understand that the emotions they are reacting to belong to another person, rather than themselves, it leads to empathic concern in the form of sympathy or compassion (Mantinago et al. 2026). Much like cognitive empathy, empathic concern, primarily the compassion aspect, is negatively associated with emotional exhaustion and is considered a job resource. Table 1. outlines the primary differences between the different types of empathy. {| class="wikitable" |+ '''Table 1. Differences Between Empathy Types''' ! Type of empathy ! Description ! What this looks like |- | '''Cognitive empathy''' | Empathy that focuses on understanding and recognising another person's thoughts, feelings and perspective without necessarily sharing their emotions. | Understanding why a friend feels upset after receiving bad news. |- | '''Affective empathy''' | The ability to emotionally respond to and share in another person's emotional experience. | Feeling sad when you see a friend crying. |- | '''Compassion''' (affective-other) | Affective empathy which motivates a desire to take action to make change | Feeling concerned about a struggling friend and offering support. |- | '''Sympathy''' (affective-other) | Affective empathy that does '''not''' motivate a desire to take action. | Feeling sorry for a friend who has lost their job. |- | '''Emotional contagion''' (affective-self) | Mirroring another person's emotional state. | Feeling anxious after spending time with someone who is visibly anxious. |} === Empathy and compassion fatigue === [[wikipedia:Compassion_fatigue|Compassion fatigue]], also known as [[wikipedia:Secondary_trauma|secondary traumatic stress]], is a term introduced by Figley (2002) to describe the negative experience affecting healthcare professionals when they are constantly exposed to the trauma of their patients and is associated with burnout. However, looking at more modern literature, it is understood that compassion does not lead to emotional exhaustion. As such, it will be referred to as empathy fatigue - an alternative term to support clarity in its antecedents (Cigrand et al., 2025). {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' *In the terms of the JD-R and CORT models, Compassion and Cognitive empathy function as job resources, thus helping to buffer individuals from burnout, whereas self-focused affective empathy (contagion) functions as a job demand. *Empathy fatigue occurs because of contagion depleting energy faster than resources can restore or mitigate demands. Aelin has engaged in contagion with their clients and is now experiencing empathy fatigue, contributing to their emotional exhaustion. * {{RoundBoxBottom}} == Emotional labour as a mediating factor between empathy fatigue and emotional exhaustion == [[wikipedia:Social_norm|Social norms]] provide guidelines for socially acceptable emotional expression. As social beings, humans rely on these shared expectations to regulate their emotional responses and maintain appropriate social interactions. This is the essence of [[Emotion labour|emotional labour]]: "the regulation of felt and expressed emotions (whether effortful or not) to match these emotional display rules with the objective of fulfilling organisational goals" (Vial & Cowgill, 2022 p. 2). Emotional labour is typically displayed through one of two ways. === Types of acting === Surface acting involves changing how felt emotions are externally expressed, such as adjusting facial expressions and body language to 'fake' the appropriate emotion (Vial & Cowgill, 2022). This creates an internal conflict known as emotional dissonance, where what someone is feeling and what they are expressing is misaligned (Middleton, 1989, as cited in Kenworthy et al., 2014). A meta-analysis from Kenworthy et al. (2014) looked at the relationship between emotional dissonance and burnout. Across 57 articles published between 1998 and 2011, they found a medium effect size (''r'' =.34) across a large sample (n=16165). Demographics were not specifically reported on, however they considered moderating factors and found a significant slope after regressing for gender (''Z'' = 2.24, ''p'' = .03). Under the JD-R, surface acting may be considered as a job demand. Surface and deep acting, the second type of emotional labour, are both [[emotional regulation]] strategies, but differ in what they are regulating. Surface acting regulates the expression of emotion, and deep acting regulates the internal feeling of the emotion through strategies such as [[wikipedia:Cognitive_appraisal|cognitive reappraisal]] (Vial & Cowgill, 2022). It is important to note that some literature discusses a third emotional labour, expression of naturally felt emotions (ENFE), which just describes that a person's genuine feelings are the appropriate to express in the moment and does not require additional emotional regulation (Zhu et al., 2025). === The mediating effect of emotional labour === Pulling together the research, emotional labour is one of the primary bridges between emotional exhaustion and empathy fatigue. When performing surface acting, people become emotionally exhausted. As such, individuals have fewer resources available to manage their emotions leading to emotional dysregulation (Nangle et al., 2019). This creates a job demand spiral, where they are still required to display appropriate emotional responses (Vial & Cowgill, 2022). Under the CORT model, this further depletes already limited emotional resources, intensifying emotional exhaustion (Hobfoll, 1989). Consequently, while this emotional labour is sustained, it reduces an individual's capacity to engage in cognitive and other-oriented affective empathy. Surface acting may exacerbate this effect by increasing emotional demands, while repeated exposure to others' emotions may further contribute to emotional strain through emotional contagion (Hunt & Gooney, 2019). Therefore, emotional labour, particularly surface acting, acts as the mechanism through which empathy fatigue contributes to emotional exhaustion. '''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 {Deep acting is negatively associated with burnout: |type="()"} + True - False </quiz> {{RoundBoxTop|theme=3}} '''What role does emotional labor play in emotional exhaustion and empathy fatigue''' * In relation to the CORT and JD-R theories, surface acting expends energy as a job demand. When energy levels are exhausted, people are more likely to engage in contagion due to emotional dysregulation, leading to a job demands spiral that exacerbates emotional exhaustion. * Aelin has exhausted their energy supplies by surface acting and is stuck in a job demands spiral due to contagion. {{RoundBoxBottom}} == Strategies for combating emotional exhaustion == Fortunately, burnout is not a permanent condition and treatment is available for dealing with emotional exhaustion. Table 2. has been adapated from van Dam (2021) to provide an overview of goals and interventions that are available. Someone who is experiencing burnout would begin at phase 1, whereas someone who is not would begin at phase 3. Importantly, when it comes to psychoeducation, it is critical to understand how to reduce the amount of surface acting someone is engaging in. Suggestions include knowledge around communication and boundaries (Bakker et al., 2021), promotion of positive emotions through storytelling (Wei et al., 2024) and a transition to deep acting or ENFE as these are associated with the conservation and gain of resources instead of increasing job demands (Zhu et al., 2025). '''Table 3. Treatment phases of clinical burnout (based on Table 2 from van Dam, A. 2021)''' {| class="wikitable" style="margin: auto; |- ! Phase !! Goal !! Interventions |- | '''Phase 1: Crisis''' || Recognition that the problems are serious, and that serious action is required. These interventions are primarily focused on reducing job demands. || Psychoeducation Sick leave Dropping domestic tasks and social obligations Inform social network |- | '''Phase 2: Recovery''' || Recovery. These interventions are primarily focused on increasing job resources. || 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. These interventions are focused on reducing job demands and increasing job resources. || 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. These interventions are focused on reducing job demands and increasing job resources. || Setting priorities regarding quality of life and interpersonal relationships. |} ==Conclusion== Sustained empathic engagement can contribute to emotional exhaustion when the emotional demands of supporting others continually exceed an individual's available resources for recovery. While cognitive empathy and compassion can act as protective job resources, self-focused affective empathy, particularly emotional contagion, can increase emotional demands by causing individuals to absorb the emotions of others. This is further intensified by emotional labour, particularly surface acting, which requires individuals to suppress or alter their genuine emotions to meet workplace expectations, creating emotional dissonance and further depleting their resources. The Conservation of Resources and Job Demands-Resources models demonstrate how ongoing resource loss, high job demands and insufficient opportunities for recovery can create a cycle of increasing stress and exhaustion. As demonstrated through Aelin's experience, recognising the distinction between different forms of empathy and understanding the role of emotional labour is essential to identifying the causes of empathy fatigue and emotional exhaustion. Ultimately, reducing excessive job demands, establishing healthy boundaries, prioritising recovery and developing more sustainable emotional regulation strategies can help preserve emotional resources, prevent burnout and support long-term wellbeing. ==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) *[[wikipedia:List_of_psychological_research_methods|List of psychological research methods]] (Wikipedia) ==References== {{Hanging indent|1= == References == Alarcon, G. M. (2011). A meta-analysis of burnout with job demands, resources, and attitudes. ''Journal of Vocational Behavior, 79''(2), 549–562. https://doi.org/10.1016/j.jvb.2011.03.007 Bakker, A. B., & de Vries, J. D. (2021). Job demands–resources theory and self-regulation: New explanations and remedies for job burnout. ''Anxiety, Stress, & Coping, 34''(1), 1–21. https://doi.org/10.1080/10615806.2020.1797695 Bakker, A. B., Demerouti, E., & Sanz-Vergel, A. I. (2014). Burnout and work engagement: The JD–R approach. ''Annual Review of Organizational Psychology and Organizational Behavior, 1'', 389–411. https://doi.org/10.1146/annurev-orgpsych-031413-091235 Chan, F. M. (2026). ''The state of workplace burnout 2026: The workplace divide''. Infinite Potential. https://www.ahri.com.au/wp-content/uploads/The-State-of-Workplace-Burnout-2026-Final.pdf Cigrand, C., Quaglia, J. T., Lawlor, S., & Bailey, C. (2025). Healing the healers: The role of contemplative practice in mitigating counselor burnout and empathy fatigue. ''Journal of Mental Health Counseling, 47''(2), 241–261. https://doi.org/10.17744/mehc.47.3.04 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, 32''(23), 5330–5342. https://doi.org/10.1093/cercor/bhac017 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 Ferguson, A. M., Cameron, C. D., & Inzlicht, M. (2021). When does empathy feel good? ''Current Opinion in Behavioral Sciences, 39'', 125–129. https://doi.org/10.1016/j.cobeha.2021.03.011 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 Gynning, B. E., Karlsson, E., Teoh, K., Gustavsson, P., Christiansen, F., & Brulin, E. (2025). Do job resources buffer the harmful effects of job demands on burnout complaints? A 1-year cohort study of Swedish healthcare professionals. ''International Journal of Nursing Studies Advances, 9'', Article 100397. https://doi.org/10.1016/j.ijnsa.2025.100397 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, 2''(3), 96–99. https://doi.org/10.1111/1467-8721.ep10770953 Hobfoll, S. E. (1989). Conservation of resources: A new attempt at conceptualizing stress. ''American Psychologist, 44''(3), 513–524. https://doi.org/10.1037/0003-066X.44.3.513 Hunt, P., Denieffe, S., & Gooney, M. (2019). Running on empathy: Relationship of empathy to compassion satisfaction and compassion fatigue in cancer healthcare professionals. ''European Journal of Cancer Care, 28''(4), Article e13124. https://doi.org/10.1111/ecc.13124 Ito, J. K., & Brotheridge, C. M. (2003). Resources, coping strategies, and emotional exhaustion: A conservation of resources perspective. ''Journal of Vocational Behavior, 63''(3), 490–509. https://doi.org/10.1016/S0001-8791(02)00033-7 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/10.1111/jasp.12211 Lesener, T., Gusy, B., & Wolter, C. (2019). The job demands–resources model: A meta-analytic review of longitudinal studies. ''Work & Stress, 33''(1), 76–103. https://doi.org/10.1080/02678373.2018.1529065 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/10.1007/s00127-004-0808-7 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 Nangle, M. R., Henry, J. D., von Hippel, C., & Kjelsaas, K. (2019). An empirical study of how emotion dysregulation and social cognition relate to occupational burnout in dentistry. ''British Dental Journal, 227''(4), 285–290. https://doi.org/10.1038/s41415-019-0620-0 Panksepp, J. B., & Lahvis, G. P. (2011). Rodent empathy and affective neuroscience. ''Neuroscience & 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), Article 04023060. https://doi.org/10.1061/JCEMD4.COENG-13339 Ruby, P., & Decety, J. (2004). How would you feel versus how do you think she would feel? A neuroimaging study of perspective-taking with social emotions. ''Journal of Cognitive Neuroscience, 16''(6), 988–999. 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'', 607–628. 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'', Article 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, 34''(1), Article bhad441. https://doi.org/10.1093/cercor/bhad441 Thaler, R. H. (2000). From homo economicus to homo sapiens. ''The Journal of Economic Perspectives, 14''(1), 133–141. https://www.proquest.com/scholarly-journals/homo-economicus-sapiens/docview/212067365 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'', Article 849566. https://doi.org/10.3389/fpsyg.2022.849566 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 Yu, C. L., & Chou, T. L. (2018). A dual route model of empathy: A neurobiological prospective. ''Frontiers in Psychology, 9'', Article 2212. https://doi.org/10.3389/fpsyg.2018.02212 Zhu, X., Tian, G., Liu, R., & Li, Y. (2025). Emotional leadership and teachers’ psychological well-being: The mediating role of emotional labour and moderating roles of cultural values in JD-R theory. ''European Journal of Education, 60''(1), Article e70006. https://doi.org/10.1111/ejed.70006 }} ==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]] jmfpxijr6rby986g9bxxz45b16g6g85 2844682 2844681 2026-11-05T21:59:34Z U3143751 3106785 /* Conservation of resources stress model */ 2844682 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 at a non-profit organisation providing mental health services to young people aged 18–25, many of whom have experienced challenging living conditions. They support approximately 20 clients each week. Initially, Aelin is highly motivated, demonstrating strong active listening skills, empathy and an ability to create a safe, welcoming environment for clients. They remember personal details and adapt their approach to respond sensitively to clients' emotional needs. However, as their caseload grows and clients present with increasingly complex needs and trauma histories, Aelin becomes increasingly invested in their work. They begin taking work home, with work-related thoughts gradually extending into weekends. Over time, they experience persistent exhaustion, even before the working day begins. Several months later, Aelin notices a shift in their interactions with others, including clients. They feel increasingly detached, frustrated and less empathetic, rather than engaged and attentive. Work leaves them emotionally drained, with little energy for friends or family. As their irritability increases, Aelin begins questioning whether they can 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. 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. {{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 == 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 (Lloyd 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; Seidler et al., 2014). 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]] and the [[wikipedia:Job_demands-resources_model|Job Demands-Resources model]]. === Conservation of resources stress model === The Conservation of resources stress model (CORT; Hobfoll, 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 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. The CORT model focuses primarily on two principles. The first principle is loss salience: there is more harm in losing resources than there is in gaining them back (Halbesleben et al., 2014). Research suggests that the pain of loss is felt twice as strong as the pleasure of gain (Thaler, 2000). The second principle is that people must invest 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 & 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 & 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). 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). It's important to note that job resources 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 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. 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 == At its core, [[empathy]] is the ability to imagine or understand another person's feelings and/or emotions (Shao et al., 2024). Research around empathy is diverse. A dual model consisting of affective and cognitive empathy (Yu and Chou, 2018) is relatively simple, where affective empathy occurs as an emotional reaction to someone else's mental state (Shao et al., 2024), and cognitive empathy occurs when a person understands someone else's feelings or emotions (Cui et al., 2022). However, a tripartite model will be the primary focus in this chapter. === Cognitive empathy === Cognitive empathy is the "what" and focuses on a person tangibly understanding what another person feels or, simply, "understanding the mental states of others" (Martingano et al., 2026 p. 802). It is associated with activation in the dorsomedial prefrontal cortex and the right inferior parietal cortex (Ruby & Decety, 2004) and, considered as a [[wikipedia:Attention#Orienting|top-down attention]] process, requires a conscious, sustained focus (Shao et al., 2024). In relation to the JD-R and CORT models, engaging in cognitive empathy is negatively associated with emotional exhaustion and may be labelled as a job resource (Ferguson et al., 2021). === Affective empathy, compassion, and emotional contagion === Affective empathy is the "how" and focuses on the feeling aspect of empathy. It is associated with mirror neuron activity inferior frontal cortex, the premotor areas and the insula (Yu and Chou, 2018). Through the tripartite model, affective empathy is further categorised as either other- or self-focused (Mantinago et al. 2026). When the affective response is self-oriented, the boundaries between a person and the target of empathy blur, leading to [[wikipedia:Emotional_contagion|emotional contagion]] - a theory that suggests emotions are contagious and spread from one person to another through an automatic [[wikipedia:Mimicry|mimicry]] process (Panksepp and Lahvis, 2011). Contagion is negatively associated with emotional exhaustion and may be labelled as a job demand (Hunt and Gooney, 2019). However, when the affective response is other-oriented and Observers understand that the emotions they are reacting to belong to another person, rather than themselves, it leads to empathic concern in the form of sympathy or compassion (Mantinago et al. 2026). Much like cognitive empathy, empathic concern, primarily the compassion aspect, is negatively associated with emotional exhaustion and is considered a job resource. Table 1. outlines the primary differences between the different types of empathy. {| class="wikitable" |+ '''Table 1. Differences Between Empathy Types''' ! Type of empathy ! Description ! What this looks like |- | '''Cognitive empathy''' | Empathy that focuses on understanding and recognising another person's thoughts, feelings and perspective without necessarily sharing their emotions. | Understanding why a friend feels upset after receiving bad news. |- | '''Affective empathy''' | The ability to emotionally respond to and share in another person's emotional experience. | Feeling sad when you see a friend crying. |- | '''Compassion''' (affective-other) | Affective empathy which motivates a desire to take action to make change | Feeling concerned about a struggling friend and offering support. |- | '''Sympathy''' (affective-other) | Affective empathy that does '''not''' motivate a desire to take action. | Feeling sorry for a friend who has lost their job. |- | '''Emotional contagion''' (affective-self) | Mirroring another person's emotional state. | Feeling anxious after spending time with someone who is visibly anxious. |} === Empathy and compassion fatigue === [[wikipedia:Compassion_fatigue|Compassion fatigue]], also known as [[wikipedia:Secondary_trauma|secondary traumatic stress]], is a term introduced by Figley (2002) to describe the negative experience affecting healthcare professionals when they are constantly exposed to the trauma of their patients and is associated with burnout. However, looking at more modern literature, it is understood that compassion does not lead to emotional exhaustion. As such, it will be referred to as empathy fatigue - an alternative term to support clarity in its antecedents (Cigrand et al., 2025). {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' *In the terms of the JD-R and CORT models, Compassion and Cognitive empathy function as job resources, thus helping to buffer individuals from burnout, whereas self-focused affective empathy (contagion) functions as a job demand. *Empathy fatigue occurs because of contagion depleting energy faster than resources can restore or mitigate demands. Aelin has engaged in contagion with their clients and is now experiencing empathy fatigue, contributing to their emotional exhaustion. * {{RoundBoxBottom}} == Emotional labour as a mediating factor between empathy fatigue and emotional exhaustion == [[wikipedia:Social_norm|Social norms]] provide guidelines for socially acceptable emotional expression. As social beings, humans rely on these shared expectations to regulate their emotional responses and maintain appropriate social interactions. This is the essence of [[Emotion labour|emotional labour]]: "the regulation of felt and expressed emotions (whether effortful or not) to match these emotional display rules with the objective of fulfilling organisational goals" (Vial & Cowgill, 2022 p. 2). Emotional labour is typically displayed through one of two ways. === Types of acting === Surface acting involves changing how felt emotions are externally expressed, such as adjusting facial expressions and body language to 'fake' the appropriate emotion (Vial & Cowgill, 2022). This creates an internal conflict known as emotional dissonance, where what someone is feeling and what they are expressing is misaligned (Middleton, 1989, as cited in Kenworthy et al., 2014). A meta-analysis from Kenworthy et al. (2014) looked at the relationship between emotional dissonance and burnout. Across 57 articles published between 1998 and 2011, they found a medium effect size (''r'' =.34) across a large sample (n=16165). Demographics were not specifically reported on, however they considered moderating factors and found a significant slope after regressing for gender (''Z'' = 2.24, ''p'' = .03). Under the JD-R, surface acting may be considered as a job demand. Surface and deep acting, the second type of emotional labour, are both [[emotional regulation]] strategies, but differ in what they are regulating. Surface acting regulates the expression of emotion, and deep acting regulates the internal feeling of the emotion through strategies such as [[wikipedia:Cognitive_appraisal|cognitive reappraisal]] (Vial & Cowgill, 2022). It is important to note that some literature discusses a third emotional labour, expression of naturally felt emotions (ENFE), which just describes that a person's genuine feelings are the appropriate to express in the moment and does not require additional emotional regulation (Zhu et al., 2025). === The mediating effect of emotional labour === Pulling together the research, emotional labour is one of the primary bridges between emotional exhaustion and empathy fatigue. When performing surface acting, people become emotionally exhausted. As such, individuals have fewer resources available to manage their emotions leading to emotional dysregulation (Nangle et al., 2019). This creates a job demand spiral, where they are still required to display appropriate emotional responses (Vial & Cowgill, 2022). Under the CORT model, this further depletes already limited emotional resources, intensifying emotional exhaustion (Hobfoll, 1989). Consequently, while this emotional labour is sustained, it reduces an individual's capacity to engage in cognitive and other-oriented affective empathy. Surface acting may exacerbate this effect by increasing emotional demands, while repeated exposure to others' emotions may further contribute to emotional strain through emotional contagion (Hunt & Gooney, 2019). Therefore, emotional labour, particularly surface acting, acts as the mechanism through which empathy fatigue contributes to emotional exhaustion. '''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 {Deep acting is negatively associated with burnout: |type="()"} + True - False </quiz> {{RoundBoxTop|theme=3}} '''What role does emotional labor play in emotional exhaustion and empathy fatigue''' * In relation to the CORT and JD-R theories, surface acting expends energy as a job demand. When energy levels are exhausted, people are more likely to engage in contagion due to emotional dysregulation, leading to a job demands spiral that exacerbates emotional exhaustion. * Aelin has exhausted their energy supplies by surface acting and is stuck in a job demands spiral due to contagion. {{RoundBoxBottom}} == Strategies for combating emotional exhaustion == Fortunately, burnout is not a permanent condition and treatment is available for dealing with emotional exhaustion. Table 2. has been adapated from van Dam (2021) to provide an overview of goals and interventions that are available. Someone who is experiencing burnout would begin at phase 1, whereas someone who is not would begin at phase 3. Importantly, when it comes to psychoeducation, it is critical to understand how to reduce the amount of surface acting someone is engaging in. Suggestions include knowledge around communication and boundaries (Bakker et al., 2021), promotion of positive emotions through storytelling (Wei et al., 2024) and a transition to deep acting or ENFE as these are associated with the conservation and gain of resources instead of increasing job demands (Zhu et al., 2025). '''Table 3. Treatment phases of clinical burnout (based on Table 2 from van Dam, A. 2021)''' {| class="wikitable" style="margin: auto; |- ! Phase !! Goal !! Interventions |- | '''Phase 1: Crisis''' || Recognition that the problems are serious, and that serious action is required. These interventions are primarily focused on reducing job demands. || Psychoeducation Sick leave Dropping domestic tasks and social obligations Inform social network |- | '''Phase 2: Recovery''' || Recovery. These interventions are primarily focused on increasing job resources. || 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. These interventions are focused on reducing job demands and increasing job resources. || 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. These interventions are focused on reducing job demands and increasing job resources. || Setting priorities regarding quality of life and interpersonal relationships. |} ==Conclusion== Sustained empathic engagement can contribute to emotional exhaustion when the emotional demands of supporting others continually exceed an individual's available resources for recovery. While cognitive empathy and compassion can act as protective job resources, self-focused affective empathy, particularly emotional contagion, can increase emotional demands by causing individuals to absorb the emotions of others. This is further intensified by emotional labour, particularly surface acting, which requires individuals to suppress or alter their genuine emotions to meet workplace expectations, creating emotional dissonance and further depleting their resources. The Conservation of Resources and Job Demands-Resources models demonstrate how ongoing resource loss, high job demands and insufficient opportunities for recovery can create a cycle of increasing stress and exhaustion. As demonstrated through Aelin's experience, recognising the distinction between different forms of empathy and understanding the role of emotional labour is essential to identifying the causes of empathy fatigue and emotional exhaustion. Ultimately, reducing excessive job demands, establishing healthy boundaries, prioritising recovery and developing more sustainable emotional regulation strategies can help preserve emotional resources, prevent burnout and support long-term wellbeing. ==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) *[[wikipedia:List_of_psychological_research_methods|List of psychological research methods]] (Wikipedia) ==References== {{Hanging indent|1= == References == Alarcon, G. M. (2011). A meta-analysis of burnout with job demands, resources, and attitudes. ''Journal of Vocational Behavior, 79''(2), 549–562. https://doi.org/10.1016/j.jvb.2011.03.007 Bakker, A. B., & de Vries, J. D. (2021). Job demands–resources theory and self-regulation: New explanations and remedies for job burnout. ''Anxiety, Stress, & Coping, 34''(1), 1–21. https://doi.org/10.1080/10615806.2020.1797695 Bakker, A. B., Demerouti, E., & Sanz-Vergel, A. I. (2014). Burnout and work engagement: The JD–R approach. ''Annual Review of Organizational Psychology and Organizational Behavior, 1'', 389–411. https://doi.org/10.1146/annurev-orgpsych-031413-091235 Chan, F. M. (2026). ''The state of workplace burnout 2026: The workplace divide''. Infinite Potential. https://www.ahri.com.au/wp-content/uploads/The-State-of-Workplace-Burnout-2026-Final.pdf Cigrand, C., Quaglia, J. T., Lawlor, S., & Bailey, C. (2025). Healing the healers: The role of contemplative practice in mitigating counselor burnout and empathy fatigue. ''Journal of Mental Health Counseling, 47''(2), 241–261. https://doi.org/10.17744/mehc.47.3.04 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, 32''(23), 5330–5342. https://doi.org/10.1093/cercor/bhac017 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 Ferguson, A. M., Cameron, C. D., & Inzlicht, M. (2021). When does empathy feel good? ''Current Opinion in Behavioral Sciences, 39'', 125–129. https://doi.org/10.1016/j.cobeha.2021.03.011 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 Gynning, B. E., Karlsson, E., Teoh, K., Gustavsson, P., Christiansen, F., & Brulin, E. (2025). Do job resources buffer the harmful effects of job demands on burnout complaints? A 1-year cohort study of Swedish healthcare professionals. ''International Journal of Nursing Studies Advances, 9'', Article 100397. https://doi.org/10.1016/j.ijnsa.2025.100397 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, 2''(3), 96–99. https://doi.org/10.1111/1467-8721.ep10770953 Hobfoll, S. E. (1989). Conservation of resources: A new attempt at conceptualizing stress. ''American Psychologist, 44''(3), 513–524. https://doi.org/10.1037/0003-066X.44.3.513 Hunt, P., Denieffe, S., & Gooney, M. (2019). Running on empathy: Relationship of empathy to compassion satisfaction and compassion fatigue in cancer healthcare professionals. ''European Journal of Cancer Care, 28''(4), Article e13124. https://doi.org/10.1111/ecc.13124 Ito, J. K., & Brotheridge, C. M. (2003). Resources, coping strategies, and emotional exhaustion: A conservation of resources perspective. ''Journal of Vocational Behavior, 63''(3), 490–509. https://doi.org/10.1016/S0001-8791(02)00033-7 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/10.1111/jasp.12211 Lesener, T., Gusy, B., & Wolter, C. (2019). The job demands–resources model: A meta-analytic review of longitudinal studies. ''Work & Stress, 33''(1), 76–103. https://doi.org/10.1080/02678373.2018.1529065 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/10.1007/s00127-004-0808-7 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 Nangle, M. R., Henry, J. D., von Hippel, C., & Kjelsaas, K. (2019). An empirical study of how emotion dysregulation and social cognition relate to occupational burnout in dentistry. ''British Dental Journal, 227''(4), 285–290. https://doi.org/10.1038/s41415-019-0620-0 Panksepp, J. B., & Lahvis, G. P. (2011). Rodent empathy and affective neuroscience. ''Neuroscience & 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), Article 04023060. https://doi.org/10.1061/JCEMD4.COENG-13339 Ruby, P., & Decety, J. (2004). How would you feel versus how do you think she would feel? A neuroimaging study of perspective-taking with social emotions. ''Journal of Cognitive Neuroscience, 16''(6), 988–999. 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'', 607–628. 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'', Article 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, 34''(1), Article bhad441. https://doi.org/10.1093/cercor/bhad441 Thaler, R. H. (2000). From homo economicus to homo sapiens. ''The Journal of Economic Perspectives, 14''(1), 133–141. https://www.proquest.com/scholarly-journals/homo-economicus-sapiens/docview/212067365 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'', Article 849566. https://doi.org/10.3389/fpsyg.2022.849566 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 Yu, C. L., & Chou, T. L. (2018). A dual route model of empathy: A neurobiological prospective. ''Frontiers in Psychology, 9'', Article 2212. https://doi.org/10.3389/fpsyg.2018.02212 Zhu, X., Tian, G., Liu, R., & Li, Y. (2025). Emotional leadership and teachers’ psychological well-being: The mediating role of emotional labour and moderating roles of cultural values in JD-R theory. ''European Journal of Education, 60''(1), Article e70006. https://doi.org/10.1111/ejed.70006 }} ==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]] kruc4ljhqipjkr39gey1cgny5jnf48t 2844683 2844682 2026-11-05T21:59:55Z U3143751 3106785 /* Affective empathy, compassion, and emotional contagion */ 2844683 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 at a non-profit organisation providing mental health services to young people aged 18–25, many of whom have experienced challenging living conditions. They support approximately 20 clients each week. Initially, Aelin is highly motivated, demonstrating strong active listening skills, empathy and an ability to create a safe, welcoming environment for clients. They remember personal details and adapt their approach to respond sensitively to clients' emotional needs. However, as their caseload grows and clients present with increasingly complex needs and trauma histories, Aelin becomes increasingly invested in their work. They begin taking work home, with work-related thoughts gradually extending into weekends. Over time, they experience persistent exhaustion, even before the working day begins. Several months later, Aelin notices a shift in their interactions with others, including clients. They feel increasingly detached, frustrated and less empathetic, rather than engaged and attentive. Work leaves them emotionally drained, with little energy for friends or family. As their irritability increases, Aelin begins questioning whether they can 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. 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. {{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 == 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 (Lloyd 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; Seidler et al., 2014). 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]] and the [[wikipedia:Job_demands-resources_model|Job Demands-Resources model]]. === Conservation of resources stress model === The Conservation of resources stress model (CORT; Hobfoll, 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 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. The CORT model focuses primarily on two principles. The first principle is loss salience: there is more harm in losing resources than there is in gaining them back (Halbesleben et al., 2014). Research suggests that the pain of loss is felt twice as strong as the pleasure of gain (Thaler, 2000). The second principle is that people must invest 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 & 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 & 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). 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). It's important to note that job resources 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 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. 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 == At its core, [[empathy]] is the ability to imagine or understand another person's feelings and/or emotions (Shao et al., 2024). Research around empathy is diverse. A dual model consisting of affective and cognitive empathy (Yu and Chou, 2018) is relatively simple, where affective empathy occurs as an emotional reaction to someone else's mental state (Shao et al., 2024), and cognitive empathy occurs when a person understands someone else's feelings or emotions (Cui et al., 2022). However, a tripartite model will be the primary focus in this chapter. === Cognitive empathy === Cognitive empathy is the "what" and focuses on a person tangibly understanding what another person feels or, simply, "understanding the mental states of others" (Martingano et al., 2026 p. 802). It is associated with activation in the dorsomedial prefrontal cortex and the right inferior parietal cortex (Ruby & Decety, 2004) and, considered as a [[wikipedia:Attention#Orienting|top-down attention]] process, requires a conscious, sustained focus (Shao et al., 2024). In relation to the JD-R and CORT models, engaging in cognitive empathy is negatively associated with emotional exhaustion and may be labelled as a job resource (Ferguson et al., 2021). === Affective empathy, compassion, and emotional contagion === Affective empathy is the "how" and focuses on the feeling aspect of empathy. It is associated with mirror neuron activity inferior frontal cortex, the premotor areas and the insula (Yu and Chou, 2018). Through the tripartite model, affective empathy is further categorised as either other- or self-focused (Mantinago et al. 2026). When the affective response is self-oriented, the boundaries between a person and the target of empathy blur, leading to [[wikipedia:Emotional_contagion|emotional contagion]] - a theory that suggests emotions are contagious and spread from one person to another through an automatic [[wikipedia:Mimicry|mimicry]] process (Panksepp and Lahvis, 2011). Contagion is negatively associated with emotional exhaustion and may be labelled as a job demand (Hunt and Gooney, 2019). However, when the affective response is other-oriented and observers understand that the emotions they are reacting to belong to another person, rather than themselves, it leads to empathic concern in the form of sympathy or compassion (Mantinago et al. 2026). Much like cognitive empathy, empathic concern, primarily the compassion aspect, is negatively associated with emotional exhaustion and is considered a job resource. Table 1. outlines the primary differences between the different types of empathy. {| class="wikitable" |+ '''Table 1. Differences Between Empathy Types''' ! Type of empathy ! Description ! What this looks like |- | '''Cognitive empathy''' | Empathy that focuses on understanding and recognising another person's thoughts, feelings and perspective without necessarily sharing their emotions. | Understanding why a friend feels upset after receiving bad news. |- | '''Affective empathy''' | The ability to emotionally respond to and share in another person's emotional experience. | Feeling sad when you see a friend crying. |- | '''Compassion''' (affective-other) | Affective empathy which motivates a desire to take action to make change | Feeling concerned about a struggling friend and offering support. |- | '''Sympathy''' (affective-other) | Affective empathy that does '''not''' motivate a desire to take action. | Feeling sorry for a friend who has lost their job. |- | '''Emotional contagion''' (affective-self) | Mirroring another person's emotional state. | Feeling anxious after spending time with someone who is visibly anxious. |} === Empathy and compassion fatigue === [[wikipedia:Compassion_fatigue|Compassion fatigue]], also known as [[wikipedia:Secondary_trauma|secondary traumatic stress]], is a term introduced by Figley (2002) to describe the negative experience affecting healthcare professionals when they are constantly exposed to the trauma of their patients and is associated with burnout. However, looking at more modern literature, it is understood that compassion does not lead to emotional exhaustion. As such, it will be referred to as empathy fatigue - an alternative term to support clarity in its antecedents (Cigrand et al., 2025). {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' *In the terms of the JD-R and CORT models, Compassion and Cognitive empathy function as job resources, thus helping to buffer individuals from burnout, whereas self-focused affective empathy (contagion) functions as a job demand. *Empathy fatigue occurs because of contagion depleting energy faster than resources can restore or mitigate demands. Aelin has engaged in contagion with their clients and is now experiencing empathy fatigue, contributing to their emotional exhaustion. * {{RoundBoxBottom}} == Emotional labour as a mediating factor between empathy fatigue and emotional exhaustion == [[wikipedia:Social_norm|Social norms]] provide guidelines for socially acceptable emotional expression. As social beings, humans rely on these shared expectations to regulate their emotional responses and maintain appropriate social interactions. This is the essence of [[Emotion labour|emotional labour]]: "the regulation of felt and expressed emotions (whether effortful or not) to match these emotional display rules with the objective of fulfilling organisational goals" (Vial & Cowgill, 2022 p. 2). Emotional labour is typically displayed through one of two ways. === Types of acting === Surface acting involves changing how felt emotions are externally expressed, such as adjusting facial expressions and body language to 'fake' the appropriate emotion (Vial & Cowgill, 2022). This creates an internal conflict known as emotional dissonance, where what someone is feeling and what they are expressing is misaligned (Middleton, 1989, as cited in Kenworthy et al., 2014). A meta-analysis from Kenworthy et al. (2014) looked at the relationship between emotional dissonance and burnout. Across 57 articles published between 1998 and 2011, they found a medium effect size (''r'' =.34) across a large sample (n=16165). Demographics were not specifically reported on, however they considered moderating factors and found a significant slope after regressing for gender (''Z'' = 2.24, ''p'' = .03). Under the JD-R, surface acting may be considered as a job demand. Surface and deep acting, the second type of emotional labour, are both [[emotional regulation]] strategies, but differ in what they are regulating. Surface acting regulates the expression of emotion, and deep acting regulates the internal feeling of the emotion through strategies such as [[wikipedia:Cognitive_appraisal|cognitive reappraisal]] (Vial & Cowgill, 2022). It is important to note that some literature discusses a third emotional labour, expression of naturally felt emotions (ENFE), which just describes that a person's genuine feelings are the appropriate to express in the moment and does not require additional emotional regulation (Zhu et al., 2025). === The mediating effect of emotional labour === Pulling together the research, emotional labour is one of the primary bridges between emotional exhaustion and empathy fatigue. When performing surface acting, people become emotionally exhausted. As such, individuals have fewer resources available to manage their emotions leading to emotional dysregulation (Nangle et al., 2019). This creates a job demand spiral, where they are still required to display appropriate emotional responses (Vial & Cowgill, 2022). Under the CORT model, this further depletes already limited emotional resources, intensifying emotional exhaustion (Hobfoll, 1989). Consequently, while this emotional labour is sustained, it reduces an individual's capacity to engage in cognitive and other-oriented affective empathy. Surface acting may exacerbate this effect by increasing emotional demands, while repeated exposure to others' emotions may further contribute to emotional strain through emotional contagion (Hunt & Gooney, 2019). Therefore, emotional labour, particularly surface acting, acts as the mechanism through which empathy fatigue contributes to emotional exhaustion. '''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 {Deep acting is negatively associated with burnout: |type="()"} + True - False </quiz> {{RoundBoxTop|theme=3}} '''What role does emotional labor play in emotional exhaustion and empathy fatigue''' * In relation to the CORT and JD-R theories, surface acting expends energy as a job demand. When energy levels are exhausted, people are more likely to engage in contagion due to emotional dysregulation, leading to a job demands spiral that exacerbates emotional exhaustion. * Aelin has exhausted their energy supplies by surface acting and is stuck in a job demands spiral due to contagion. {{RoundBoxBottom}} == Strategies for combating emotional exhaustion == Fortunately, burnout is not a permanent condition and treatment is available for dealing with emotional exhaustion. Table 2. has been adapated from van Dam (2021) to provide an overview of goals and interventions that are available. Someone who is experiencing burnout would begin at phase 1, whereas someone who is not would begin at phase 3. Importantly, when it comes to psychoeducation, it is critical to understand how to reduce the amount of surface acting someone is engaging in. Suggestions include knowledge around communication and boundaries (Bakker et al., 2021), promotion of positive emotions through storytelling (Wei et al., 2024) and a transition to deep acting or ENFE as these are associated with the conservation and gain of resources instead of increasing job demands (Zhu et al., 2025). '''Table 3. Treatment phases of clinical burnout (based on Table 2 from van Dam, A. 2021)''' {| class="wikitable" style="margin: auto; |- ! Phase !! Goal !! Interventions |- | '''Phase 1: Crisis''' || Recognition that the problems are serious, and that serious action is required. These interventions are primarily focused on reducing job demands. || Psychoeducation Sick leave Dropping domestic tasks and social obligations Inform social network |- | '''Phase 2: Recovery''' || Recovery. These interventions are primarily focused on increasing job resources. || 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. These interventions are focused on reducing job demands and increasing job resources. || 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. These interventions are focused on reducing job demands and increasing job resources. || Setting priorities regarding quality of life and interpersonal relationships. |} ==Conclusion== Sustained empathic engagement can contribute to emotional exhaustion when the emotional demands of supporting others continually exceed an individual's available resources for recovery. While cognitive empathy and compassion can act as protective job resources, self-focused affective empathy, particularly emotional contagion, can increase emotional demands by causing individuals to absorb the emotions of others. This is further intensified by emotional labour, particularly surface acting, which requires individuals to suppress or alter their genuine emotions to meet workplace expectations, creating emotional dissonance and further depleting their resources. The Conservation of Resources and Job Demands-Resources models demonstrate how ongoing resource loss, high job demands and insufficient opportunities for recovery can create a cycle of increasing stress and exhaustion. As demonstrated through Aelin's experience, recognising the distinction between different forms of empathy and understanding the role of emotional labour is essential to identifying the causes of empathy fatigue and emotional exhaustion. Ultimately, reducing excessive job demands, establishing healthy boundaries, prioritising recovery and developing more sustainable emotional regulation strategies can help preserve emotional resources, prevent burnout and support long-term wellbeing. ==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) *[[wikipedia:List_of_psychological_research_methods|List of psychological research methods]] (Wikipedia) ==References== {{Hanging indent|1= == References == Alarcon, G. M. (2011). A meta-analysis of burnout with job demands, resources, and attitudes. ''Journal of Vocational Behavior, 79''(2), 549–562. https://doi.org/10.1016/j.jvb.2011.03.007 Bakker, A. B., & de Vries, J. D. (2021). Job demands–resources theory and self-regulation: New explanations and remedies for job burnout. ''Anxiety, Stress, & Coping, 34''(1), 1–21. https://doi.org/10.1080/10615806.2020.1797695 Bakker, A. B., Demerouti, E., & Sanz-Vergel, A. I. (2014). Burnout and work engagement: The JD–R approach. ''Annual Review of Organizational Psychology and Organizational Behavior, 1'', 389–411. https://doi.org/10.1146/annurev-orgpsych-031413-091235 Chan, F. M. (2026). ''The state of workplace burnout 2026: The workplace divide''. Infinite Potential. https://www.ahri.com.au/wp-content/uploads/The-State-of-Workplace-Burnout-2026-Final.pdf Cigrand, C., Quaglia, J. T., Lawlor, S., & Bailey, C. (2025). Healing the healers: The role of contemplative practice in mitigating counselor burnout and empathy fatigue. ''Journal of Mental Health Counseling, 47''(2), 241–261. https://doi.org/10.17744/mehc.47.3.04 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, 32''(23), 5330–5342. https://doi.org/10.1093/cercor/bhac017 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 Ferguson, A. M., Cameron, C. D., & Inzlicht, M. (2021). When does empathy feel good? ''Current Opinion in Behavioral Sciences, 39'', 125–129. https://doi.org/10.1016/j.cobeha.2021.03.011 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 Gynning, B. E., Karlsson, E., Teoh, K., Gustavsson, P., Christiansen, F., & Brulin, E. (2025). Do job resources buffer the harmful effects of job demands on burnout complaints? A 1-year cohort study of Swedish healthcare professionals. ''International Journal of Nursing Studies Advances, 9'', Article 100397. https://doi.org/10.1016/j.ijnsa.2025.100397 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, 2''(3), 96–99. https://doi.org/10.1111/1467-8721.ep10770953 Hobfoll, S. E. (1989). Conservation of resources: A new attempt at conceptualizing stress. ''American Psychologist, 44''(3), 513–524. https://doi.org/10.1037/0003-066X.44.3.513 Hunt, P., Denieffe, S., & Gooney, M. (2019). Running on empathy: Relationship of empathy to compassion satisfaction and compassion fatigue in cancer healthcare professionals. ''European Journal of Cancer Care, 28''(4), Article e13124. https://doi.org/10.1111/ecc.13124 Ito, J. K., & Brotheridge, C. M. (2003). Resources, coping strategies, and emotional exhaustion: A conservation of resources perspective. ''Journal of Vocational Behavior, 63''(3), 490–509. https://doi.org/10.1016/S0001-8791(02)00033-7 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/10.1111/jasp.12211 Lesener, T., Gusy, B., & Wolter, C. (2019). The job demands–resources model: A meta-analytic review of longitudinal studies. ''Work & Stress, 33''(1), 76–103. https://doi.org/10.1080/02678373.2018.1529065 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/10.1007/s00127-004-0808-7 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 Nangle, M. R., Henry, J. D., von Hippel, C., & Kjelsaas, K. (2019). An empirical study of how emotion dysregulation and social cognition relate to occupational burnout in dentistry. ''British Dental Journal, 227''(4), 285–290. https://doi.org/10.1038/s41415-019-0620-0 Panksepp, J. B., & Lahvis, G. P. (2011). Rodent empathy and affective neuroscience. ''Neuroscience & 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), Article 04023060. https://doi.org/10.1061/JCEMD4.COENG-13339 Ruby, P., & Decety, J. (2004). How would you feel versus how do you think she would feel? A neuroimaging study of perspective-taking with social emotions. ''Journal of Cognitive Neuroscience, 16''(6), 988–999. 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'', 607–628. 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'', Article 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, 34''(1), Article bhad441. https://doi.org/10.1093/cercor/bhad441 Thaler, R. H. (2000). From homo economicus to homo sapiens. ''The Journal of Economic Perspectives, 14''(1), 133–141. https://www.proquest.com/scholarly-journals/homo-economicus-sapiens/docview/212067365 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'', Article 849566. https://doi.org/10.3389/fpsyg.2022.849566 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 Yu, C. L., & Chou, T. L. (2018). A dual route model of empathy: A neurobiological prospective. ''Frontiers in Psychology, 9'', Article 2212. https://doi.org/10.3389/fpsyg.2018.02212 Zhu, X., Tian, G., Liu, R., & Li, Y. (2025). Emotional leadership and teachers’ psychological well-being: The mediating role of emotional labour and moderating roles of cultural values in JD-R theory. ''European Journal of Education, 60''(1), Article e70006. https://doi.org/10.1111/ejed.70006 }} ==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]] 8g5babcoyns3x9fvl407gl2m5wnjw23 2844685 2844683 2026-11-05T22:00:48Z U3143751 3106785 /* Strategies for combating emotional exhaustion */ 2844685 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 at a non-profit organisation providing mental health services to young people aged 18–25, many of whom have experienced challenging living conditions. They support approximately 20 clients each week. Initially, Aelin is highly motivated, demonstrating strong active listening skills, empathy and an ability to create a safe, welcoming environment for clients. They remember personal details and adapt their approach to respond sensitively to clients' emotional needs. However, as their caseload grows and clients present with increasingly complex needs and trauma histories, Aelin becomes increasingly invested in their work. They begin taking work home, with work-related thoughts gradually extending into weekends. Over time, they experience persistent exhaustion, even before the working day begins. Several months later, Aelin notices a shift in their interactions with others, including clients. They feel increasingly detached, frustrated and less empathetic, rather than engaged and attentive. Work leaves them emotionally drained, with little energy for friends or family. As their irritability increases, Aelin begins questioning whether they can 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. 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. {{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 == 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 (Lloyd 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; Seidler et al., 2014). 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]] and the [[wikipedia:Job_demands-resources_model|Job Demands-Resources model]]. === Conservation of resources stress model === The Conservation of resources stress model (CORT; Hobfoll, 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 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. The CORT model focuses primarily on two principles. The first principle is loss salience: there is more harm in losing resources than there is in gaining them back (Halbesleben et al., 2014). Research suggests that the pain of loss is felt twice as strong as the pleasure of gain (Thaler, 2000). The second principle is that people must invest 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 & 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 & 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). 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). It's important to note that job resources 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 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. 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 == At its core, [[empathy]] is the ability to imagine or understand another person's feelings and/or emotions (Shao et al., 2024). Research around empathy is diverse. A dual model consisting of affective and cognitive empathy (Yu and Chou, 2018) is relatively simple, where affective empathy occurs as an emotional reaction to someone else's mental state (Shao et al., 2024), and cognitive empathy occurs when a person understands someone else's feelings or emotions (Cui et al., 2022). However, a tripartite model will be the primary focus in this chapter. === Cognitive empathy === Cognitive empathy is the "what" and focuses on a person tangibly understanding what another person feels or, simply, "understanding the mental states of others" (Martingano et al., 2026 p. 802). It is associated with activation in the dorsomedial prefrontal cortex and the right inferior parietal cortex (Ruby & Decety, 2004) and, considered as a [[wikipedia:Attention#Orienting|top-down attention]] process, requires a conscious, sustained focus (Shao et al., 2024). In relation to the JD-R and CORT models, engaging in cognitive empathy is negatively associated with emotional exhaustion and may be labelled as a job resource (Ferguson et al., 2021). === Affective empathy, compassion, and emotional contagion === Affective empathy is the "how" and focuses on the feeling aspect of empathy. It is associated with mirror neuron activity inferior frontal cortex, the premotor areas and the insula (Yu and Chou, 2018). Through the tripartite model, affective empathy is further categorised as either other- or self-focused (Mantinago et al. 2026). When the affective response is self-oriented, the boundaries between a person and the target of empathy blur, leading to [[wikipedia:Emotional_contagion|emotional contagion]] - a theory that suggests emotions are contagious and spread from one person to another through an automatic [[wikipedia:Mimicry|mimicry]] process (Panksepp and Lahvis, 2011). Contagion is negatively associated with emotional exhaustion and may be labelled as a job demand (Hunt and Gooney, 2019). However, when the affective response is other-oriented and observers understand that the emotions they are reacting to belong to another person, rather than themselves, it leads to empathic concern in the form of sympathy or compassion (Mantinago et al. 2026). Much like cognitive empathy, empathic concern, primarily the compassion aspect, is negatively associated with emotional exhaustion and is considered a job resource. Table 1. outlines the primary differences between the different types of empathy. {| class="wikitable" |+ '''Table 1. Differences Between Empathy Types''' ! Type of empathy ! Description ! What this looks like |- | '''Cognitive empathy''' | Empathy that focuses on understanding and recognising another person's thoughts, feelings and perspective without necessarily sharing their emotions. | Understanding why a friend feels upset after receiving bad news. |- | '''Affective empathy''' | The ability to emotionally respond to and share in another person's emotional experience. | Feeling sad when you see a friend crying. |- | '''Compassion''' (affective-other) | Affective empathy which motivates a desire to take action to make change | Feeling concerned about a struggling friend and offering support. |- | '''Sympathy''' (affective-other) | Affective empathy that does '''not''' motivate a desire to take action. | Feeling sorry for a friend who has lost their job. |- | '''Emotional contagion''' (affective-self) | Mirroring another person's emotional state. | Feeling anxious after spending time with someone who is visibly anxious. |} === Empathy and compassion fatigue === [[wikipedia:Compassion_fatigue|Compassion fatigue]], also known as [[wikipedia:Secondary_trauma|secondary traumatic stress]], is a term introduced by Figley (2002) to describe the negative experience affecting healthcare professionals when they are constantly exposed to the trauma of their patients and is associated with burnout. However, looking at more modern literature, it is understood that compassion does not lead to emotional exhaustion. As such, it will be referred to as empathy fatigue - an alternative term to support clarity in its antecedents (Cigrand et al., 2025). {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' *In the terms of the JD-R and CORT models, Compassion and Cognitive empathy function as job resources, thus helping to buffer individuals from burnout, whereas self-focused affective empathy (contagion) functions as a job demand. *Empathy fatigue occurs because of contagion depleting energy faster than resources can restore or mitigate demands. Aelin has engaged in contagion with their clients and is now experiencing empathy fatigue, contributing to their emotional exhaustion. * {{RoundBoxBottom}} == Emotional labour as a mediating factor between empathy fatigue and emotional exhaustion == [[wikipedia:Social_norm|Social norms]] provide guidelines for socially acceptable emotional expression. As social beings, humans rely on these shared expectations to regulate their emotional responses and maintain appropriate social interactions. This is the essence of [[Emotion labour|emotional labour]]: "the regulation of felt and expressed emotions (whether effortful or not) to match these emotional display rules with the objective of fulfilling organisational goals" (Vial & Cowgill, 2022 p. 2). Emotional labour is typically displayed through one of two ways. === Types of acting === Surface acting involves changing how felt emotions are externally expressed, such as adjusting facial expressions and body language to 'fake' the appropriate emotion (Vial & Cowgill, 2022). This creates an internal conflict known as emotional dissonance, where what someone is feeling and what they are expressing is misaligned (Middleton, 1989, as cited in Kenworthy et al., 2014). A meta-analysis from Kenworthy et al. (2014) looked at the relationship between emotional dissonance and burnout. Across 57 articles published between 1998 and 2011, they found a medium effect size (''r'' =.34) across a large sample (n=16165). Demographics were not specifically reported on, however they considered moderating factors and found a significant slope after regressing for gender (''Z'' = 2.24, ''p'' = .03). Under the JD-R, surface acting may be considered as a job demand. Surface and deep acting, the second type of emotional labour, are both [[emotional regulation]] strategies, but differ in what they are regulating. Surface acting regulates the expression of emotion, and deep acting regulates the internal feeling of the emotion through strategies such as [[wikipedia:Cognitive_appraisal|cognitive reappraisal]] (Vial & Cowgill, 2022). It is important to note that some literature discusses a third emotional labour, expression of naturally felt emotions (ENFE), which just describes that a person's genuine feelings are the appropriate to express in the moment and does not require additional emotional regulation (Zhu et al., 2025). === The mediating effect of emotional labour === Pulling together the research, emotional labour is one of the primary bridges between emotional exhaustion and empathy fatigue. When performing surface acting, people become emotionally exhausted. As such, individuals have fewer resources available to manage their emotions leading to emotional dysregulation (Nangle et al., 2019). This creates a job demand spiral, where they are still required to display appropriate emotional responses (Vial & Cowgill, 2022). Under the CORT model, this further depletes already limited emotional resources, intensifying emotional exhaustion (Hobfoll, 1989). Consequently, while this emotional labour is sustained, it reduces an individual's capacity to engage in cognitive and other-oriented affective empathy. Surface acting may exacerbate this effect by increasing emotional demands, while repeated exposure to others' emotions may further contribute to emotional strain through emotional contagion (Hunt & Gooney, 2019). Therefore, emotional labour, particularly surface acting, acts as the mechanism through which empathy fatigue contributes to emotional exhaustion. '''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 {Deep acting is negatively associated with burnout: |type="()"} + True - False </quiz> {{RoundBoxTop|theme=3}} '''What role does emotional labor play in emotional exhaustion and empathy fatigue''' * In relation to the CORT and JD-R theories, surface acting expends energy as a job demand. When energy levels are exhausted, people are more likely to engage in contagion due to emotional dysregulation, leading to a job demands spiral that exacerbates emotional exhaustion. * Aelin has exhausted their energy supplies by surface acting and is stuck in a job demands spiral due to contagion. {{RoundBoxBottom}} == Strategies for combating emotional exhaustion == Burnout treatment is available. Table 2. has been adapated from van Dam (2021) to provide an overview of goals and interventions. Someone experiencing burnout would begin at phase 1. Importantly, when it comes to psychoeducation, it is critical to understand how to reduce the amount of surface acting someone is engaging in. Suggestions include knowledge around communication and boundaries (Bakker et al., 2021), promotion of positive emotions through storytelling (Wei et al., 2024) and a transition to deep acting or ENFE as these are associated with the conservation and gain of resources instead of increasing job demands (Zhu et al., 2025). '''Table 3. Treatment phases of clinical burnout (based on Table 2 from van Dam, A. 2021)''' {| class="wikitable" style="margin: auto; |- ! Phase !! Goal !! Interventions |- | '''Phase 1: Crisis''' || Recognition that the problems are serious, and that serious action is required. These interventions are primarily focused on reducing job demands. || Psychoeducation Sick leave Dropping domestic tasks and social obligations Inform social network |- | '''Phase 2: Recovery''' || Recovery. These interventions are primarily focused on increasing job resources. || 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. These interventions are focused on reducing job demands and increasing job resources. || 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. These interventions are focused on reducing job demands and increasing job resources. || Setting priorities regarding quality of life and interpersonal relationships. |} ==Conclusion== Sustained empathic engagement can contribute to emotional exhaustion when the emotional demands of supporting others continually exceed an individual's available resources for recovery. While cognitive empathy and compassion can act as protective job resources, self-focused affective empathy, particularly emotional contagion, can increase emotional demands by causing individuals to absorb the emotions of others. This is further intensified by emotional labour, particularly surface acting, which requires individuals to suppress or alter their genuine emotions to meet workplace expectations, creating emotional dissonance and further depleting their resources. The Conservation of Resources and Job Demands-Resources models demonstrate how ongoing resource loss, high job demands and insufficient opportunities for recovery can create a cycle of increasing stress and exhaustion. As demonstrated through Aelin's experience, recognising the distinction between different forms of empathy and understanding the role of emotional labour is essential to identifying the causes of empathy fatigue and emotional exhaustion. Ultimately, reducing excessive job demands, establishing healthy boundaries, prioritising recovery and developing more sustainable emotional regulation strategies can help preserve emotional resources, prevent burnout and support long-term wellbeing. ==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) *[[wikipedia:List_of_psychological_research_methods|List of psychological research methods]] (Wikipedia) ==References== {{Hanging indent|1= == References == Alarcon, G. M. (2011). A meta-analysis of burnout with job demands, resources, and attitudes. ''Journal of Vocational Behavior, 79''(2), 549–562. https://doi.org/10.1016/j.jvb.2011.03.007 Bakker, A. B., & de Vries, J. D. (2021). Job demands–resources theory and self-regulation: New explanations and remedies for job burnout. ''Anxiety, Stress, & Coping, 34''(1), 1–21. https://doi.org/10.1080/10615806.2020.1797695 Bakker, A. B., Demerouti, E., & Sanz-Vergel, A. I. (2014). Burnout and work engagement: The JD–R approach. ''Annual Review of Organizational Psychology and Organizational Behavior, 1'', 389–411. https://doi.org/10.1146/annurev-orgpsych-031413-091235 Chan, F. M. (2026). ''The state of workplace burnout 2026: The workplace divide''. Infinite Potential. https://www.ahri.com.au/wp-content/uploads/The-State-of-Workplace-Burnout-2026-Final.pdf Cigrand, C., Quaglia, J. T., Lawlor, S., & Bailey, C. (2025). Healing the healers: The role of contemplative practice in mitigating counselor burnout and empathy fatigue. ''Journal of Mental Health Counseling, 47''(2), 241–261. https://doi.org/10.17744/mehc.47.3.04 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, 32''(23), 5330–5342. https://doi.org/10.1093/cercor/bhac017 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 Ferguson, A. M., Cameron, C. D., & Inzlicht, M. (2021). When does empathy feel good? ''Current Opinion in Behavioral Sciences, 39'', 125–129. https://doi.org/10.1016/j.cobeha.2021.03.011 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 Gynning, B. E., Karlsson, E., Teoh, K., Gustavsson, P., Christiansen, F., & Brulin, E. (2025). Do job resources buffer the harmful effects of job demands on burnout complaints? A 1-year cohort study of Swedish healthcare professionals. ''International Journal of Nursing Studies Advances, 9'', Article 100397. https://doi.org/10.1016/j.ijnsa.2025.100397 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, 2''(3), 96–99. https://doi.org/10.1111/1467-8721.ep10770953 Hobfoll, S. E. (1989). Conservation of resources: A new attempt at conceptualizing stress. ''American Psychologist, 44''(3), 513–524. https://doi.org/10.1037/0003-066X.44.3.513 Hunt, P., Denieffe, S., & Gooney, M. (2019). Running on empathy: Relationship of empathy to compassion satisfaction and compassion fatigue in cancer healthcare professionals. ''European Journal of Cancer Care, 28''(4), Article e13124. https://doi.org/10.1111/ecc.13124 Ito, J. K., & Brotheridge, C. M. (2003). Resources, coping strategies, and emotional exhaustion: A conservation of resources perspective. ''Journal of Vocational Behavior, 63''(3), 490–509. https://doi.org/10.1016/S0001-8791(02)00033-7 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/10.1111/jasp.12211 Lesener, T., Gusy, B., & Wolter, C. (2019). The job demands–resources model: A meta-analytic review of longitudinal studies. ''Work & Stress, 33''(1), 76–103. https://doi.org/10.1080/02678373.2018.1529065 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/10.1007/s00127-004-0808-7 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 Nangle, M. R., Henry, J. D., von Hippel, C., & Kjelsaas, K. (2019). An empirical study of how emotion dysregulation and social cognition relate to occupational burnout in dentistry. ''British Dental Journal, 227''(4), 285–290. https://doi.org/10.1038/s41415-019-0620-0 Panksepp, J. B., & Lahvis, G. P. (2011). Rodent empathy and affective neuroscience. ''Neuroscience & 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), Article 04023060. https://doi.org/10.1061/JCEMD4.COENG-13339 Ruby, P., & Decety, J. (2004). How would you feel versus how do you think she would feel? A neuroimaging study of perspective-taking with social emotions. ''Journal of Cognitive Neuroscience, 16''(6), 988–999. 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'', 607–628. 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'', Article 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, 34''(1), Article bhad441. https://doi.org/10.1093/cercor/bhad441 Thaler, R. H. (2000). From homo economicus to homo sapiens. ''The Journal of Economic Perspectives, 14''(1), 133–141. https://www.proquest.com/scholarly-journals/homo-economicus-sapiens/docview/212067365 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'', Article 849566. https://doi.org/10.3389/fpsyg.2022.849566 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 Yu, C. L., & Chou, T. L. (2018). A dual route model of empathy: A neurobiological prospective. ''Frontiers in Psychology, 9'', Article 2212. https://doi.org/10.3389/fpsyg.2018.02212 Zhu, X., Tian, G., Liu, R., & Li, Y. (2025). Emotional leadership and teachers’ psychological well-being: The mediating role of emotional labour and moderating roles of cultural values in JD-R theory. ''European Journal of Education, 60''(1), Article e70006. https://doi.org/10.1111/ejed.70006 }} ==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]] bvbq3x19h6pw3skkc9yh3wl5zff6767 2844697 2844685 2026-11-06T00:41:30Z U3143751 3106785 /* External links */ 2844697 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 at a non-profit organisation providing mental health services to young people aged 18–25, many of whom have experienced challenging living conditions. They support approximately 20 clients each week. Initially, Aelin is highly motivated, demonstrating strong active listening skills, empathy and an ability to create a safe, welcoming environment for clients. They remember personal details and adapt their approach to respond sensitively to clients' emotional needs. However, as their caseload grows and clients present with increasingly complex needs and trauma histories, Aelin becomes increasingly invested in their work. They begin taking work home, with work-related thoughts gradually extending into weekends. Over time, they experience persistent exhaustion, even before the working day begins. Several months later, Aelin notices a shift in their interactions with others, including clients. They feel increasingly detached, frustrated and less empathetic, rather than engaged and attentive. Work leaves them emotionally drained, with little energy for friends or family. As their irritability increases, Aelin begins questioning whether they can 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. 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. {{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 == 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 (Lloyd 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; Seidler et al., 2014). 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]] and the [[wikipedia:Job_demands-resources_model|Job Demands-Resources model]]. === Conservation of resources stress model === The Conservation of resources stress model (CORT; Hobfoll, 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 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. The CORT model focuses primarily on two principles. The first principle is loss salience: there is more harm in losing resources than there is in gaining them back (Halbesleben et al., 2014). Research suggests that the pain of loss is felt twice as strong as the pleasure of gain (Thaler, 2000). The second principle is that people must invest 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 & 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 & 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). 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). It's important to note that job resources 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 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. 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 == At its core, [[empathy]] is the ability to imagine or understand another person's feelings and/or emotions (Shao et al., 2024). Research around empathy is diverse. A dual model consisting of affective and cognitive empathy (Yu and Chou, 2018) is relatively simple, where affective empathy occurs as an emotional reaction to someone else's mental state (Shao et al., 2024), and cognitive empathy occurs when a person understands someone else's feelings or emotions (Cui et al., 2022). However, a tripartite model will be the primary focus in this chapter. === Cognitive empathy === Cognitive empathy is the "what" and focuses on a person tangibly understanding what another person feels or, simply, "understanding the mental states of others" (Martingano et al., 2026 p. 802). It is associated with activation in the dorsomedial prefrontal cortex and the right inferior parietal cortex (Ruby & Decety, 2004) and, considered as a [[wikipedia:Attention#Orienting|top-down attention]] process, requires a conscious, sustained focus (Shao et al., 2024). In relation to the JD-R and CORT models, engaging in cognitive empathy is negatively associated with emotional exhaustion and may be labelled as a job resource (Ferguson et al., 2021). === Affective empathy, compassion, and emotional contagion === Affective empathy is the "how" and focuses on the feeling aspect of empathy. It is associated with mirror neuron activity inferior frontal cortex, the premotor areas and the insula (Yu and Chou, 2018). Through the tripartite model, affective empathy is further categorised as either other- or self-focused (Mantinago et al. 2026). When the affective response is self-oriented, the boundaries between a person and the target of empathy blur, leading to [[wikipedia:Emotional_contagion|emotional contagion]] - a theory that suggests emotions are contagious and spread from one person to another through an automatic [[wikipedia:Mimicry|mimicry]] process (Panksepp and Lahvis, 2011). Contagion is negatively associated with emotional exhaustion and may be labelled as a job demand (Hunt and Gooney, 2019). However, when the affective response is other-oriented and observers understand that the emotions they are reacting to belong to another person, rather than themselves, it leads to empathic concern in the form of sympathy or compassion (Mantinago et al. 2026). Much like cognitive empathy, empathic concern, primarily the compassion aspect, is negatively associated with emotional exhaustion and is considered a job resource. Table 1. outlines the primary differences between the different types of empathy. {| class="wikitable" |+ '''Table 1. Differences Between Empathy Types''' ! Type of empathy ! Description ! What this looks like |- | '''Cognitive empathy''' | Empathy that focuses on understanding and recognising another person's thoughts, feelings and perspective without necessarily sharing their emotions. | Understanding why a friend feels upset after receiving bad news. |- | '''Affective empathy''' | The ability to emotionally respond to and share in another person's emotional experience. | Feeling sad when you see a friend crying. |- | '''Compassion''' (affective-other) | Affective empathy which motivates a desire to take action to make change | Feeling concerned about a struggling friend and offering support. |- | '''Sympathy''' (affective-other) | Affective empathy that does '''not''' motivate a desire to take action. | Feeling sorry for a friend who has lost their job. |- | '''Emotional contagion''' (affective-self) | Mirroring another person's emotional state. | Feeling anxious after spending time with someone who is visibly anxious. |} === Empathy and compassion fatigue === [[wikipedia:Compassion_fatigue|Compassion fatigue]], also known as [[wikipedia:Secondary_trauma|secondary traumatic stress]], is a term introduced by Figley (2002) to describe the negative experience affecting healthcare professionals when they are constantly exposed to the trauma of their patients and is associated with burnout. However, looking at more modern literature, it is understood that compassion does not lead to emotional exhaustion. As such, it will be referred to as empathy fatigue - an alternative term to support clarity in its antecedents (Cigrand et al., 2025). {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' *In the terms of the JD-R and CORT models, Compassion and Cognitive empathy function as job resources, thus helping to buffer individuals from burnout, whereas self-focused affective empathy (contagion) functions as a job demand. *Empathy fatigue occurs because of contagion depleting energy faster than resources can restore or mitigate demands. Aelin has engaged in contagion with their clients and is now experiencing empathy fatigue, contributing to their emotional exhaustion. * {{RoundBoxBottom}} == Emotional labour as a mediating factor between empathy fatigue and emotional exhaustion == [[wikipedia:Social_norm|Social norms]] provide guidelines for socially acceptable emotional expression. As social beings, humans rely on these shared expectations to regulate their emotional responses and maintain appropriate social interactions. This is the essence of [[Emotion labour|emotional labour]]: "the regulation of felt and expressed emotions (whether effortful or not) to match these emotional display rules with the objective of fulfilling organisational goals" (Vial & Cowgill, 2022 p. 2). Emotional labour is typically displayed through one of two ways. === Types of acting === Surface acting involves changing how felt emotions are externally expressed, such as adjusting facial expressions and body language to 'fake' the appropriate emotion (Vial & Cowgill, 2022). This creates an internal conflict known as emotional dissonance, where what someone is feeling and what they are expressing is misaligned (Middleton, 1989, as cited in Kenworthy et al., 2014). A meta-analysis from Kenworthy et al. (2014) looked at the relationship between emotional dissonance and burnout. Across 57 articles published between 1998 and 2011, they found a medium effect size (''r'' =.34) across a large sample (n=16165). Demographics were not specifically reported on, however they considered moderating factors and found a significant slope after regressing for gender (''Z'' = 2.24, ''p'' = .03). Under the JD-R, surface acting may be considered as a job demand. Surface and deep acting, the second type of emotional labour, are both [[emotional regulation]] strategies, but differ in what they are regulating. Surface acting regulates the expression of emotion, and deep acting regulates the internal feeling of the emotion through strategies such as [[wikipedia:Cognitive_appraisal|cognitive reappraisal]] (Vial & Cowgill, 2022). It is important to note that some literature discusses a third emotional labour, expression of naturally felt emotions (ENFE), which just describes that a person's genuine feelings are the appropriate to express in the moment and does not require additional emotional regulation (Zhu et al., 2025). === The mediating effect of emotional labour === Pulling together the research, emotional labour is one of the primary bridges between emotional exhaustion and empathy fatigue. When performing surface acting, people become emotionally exhausted. As such, individuals have fewer resources available to manage their emotions leading to emotional dysregulation (Nangle et al., 2019). This creates a job demand spiral, where they are still required to display appropriate emotional responses (Vial & Cowgill, 2022). Under the CORT model, this further depletes already limited emotional resources, intensifying emotional exhaustion (Hobfoll, 1989). Consequently, while this emotional labour is sustained, it reduces an individual's capacity to engage in cognitive and other-oriented affective empathy. Surface acting may exacerbate this effect by increasing emotional demands, while repeated exposure to others' emotions may further contribute to emotional strain through emotional contagion (Hunt & Gooney, 2019). Therefore, emotional labour, particularly surface acting, acts as the mechanism through which empathy fatigue contributes to emotional exhaustion. '''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 {Deep acting is negatively associated with burnout: |type="()"} + True - False </quiz> {{RoundBoxTop|theme=3}} '''What role does emotional labor play in emotional exhaustion and empathy fatigue''' * In relation to the CORT and JD-R theories, surface acting expends energy as a job demand. When energy levels are exhausted, people are more likely to engage in contagion due to emotional dysregulation, leading to a job demands spiral that exacerbates emotional exhaustion. * Aelin has exhausted their energy supplies by surface acting and is stuck in a job demands spiral due to contagion. {{RoundBoxBottom}} == Strategies for combating emotional exhaustion == Burnout treatment is available. Table 2. has been adapated from van Dam (2021) to provide an overview of goals and interventions. Someone experiencing burnout would begin at phase 1. Importantly, when it comes to psychoeducation, it is critical to understand how to reduce the amount of surface acting someone is engaging in. Suggestions include knowledge around communication and boundaries (Bakker et al., 2021), promotion of positive emotions through storytelling (Wei et al., 2024) and a transition to deep acting or ENFE as these are associated with the conservation and gain of resources instead of increasing job demands (Zhu et al., 2025). '''Table 3. Treatment phases of clinical burnout (based on Table 2 from van Dam, A. 2021)''' {| class="wikitable" style="margin: auto; |- ! Phase !! Goal !! Interventions |- | '''Phase 1: Crisis''' || Recognition that the problems are serious, and that serious action is required. These interventions are primarily focused on reducing job demands. || Psychoeducation Sick leave Dropping domestic tasks and social obligations Inform social network |- | '''Phase 2: Recovery''' || Recovery. These interventions are primarily focused on increasing job resources. || 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. These interventions are focused on reducing job demands and increasing job resources. || 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. These interventions are focused on reducing job demands and increasing job resources. || Setting priorities regarding quality of life and interpersonal relationships. |} ==Conclusion== Sustained empathic engagement can contribute to emotional exhaustion when the emotional demands of supporting others continually exceed an individual's available resources for recovery. While cognitive empathy and compassion can act as protective job resources, self-focused affective empathy, particularly emotional contagion, can increase emotional demands by causing individuals to absorb the emotions of others. This is further intensified by emotional labour, particularly surface acting, which requires individuals to suppress or alter their genuine emotions to meet workplace expectations, creating emotional dissonance and further depleting their resources. The Conservation of Resources and Job Demands-Resources models demonstrate how ongoing resource loss, high job demands and insufficient opportunities for recovery can create a cycle of increasing stress and exhaustion. As demonstrated through Aelin's experience, recognising the distinction between different forms of empathy and understanding the role of emotional labour is essential to identifying the causes of empathy fatigue and emotional exhaustion. Ultimately, reducing excessive job demands, establishing healthy boundaries, prioritising recovery and developing more sustainable emotional regulation strategies can help preserve emotional resources, prevent burnout and support long-term wellbeing. ==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) *[[wikipedia:List_of_psychological_research_methods|List of psychological research methods]] (Wikipedia) ==References== {{Hanging indent|1= == References == Alarcon, G. M. (2011). A meta-analysis of burnout with job demands, resources, and attitudes. ''Journal of Vocational Behavior, 79''(2), 549–562. https://doi.org/10.1016/j.jvb.2011.03.007 Bakker, A. B., & de Vries, J. D. (2021). Job demands–resources theory and self-regulation: New explanations and remedies for job burnout. ''Anxiety, Stress, & Coping, 34''(1), 1–21. https://doi.org/10.1080/10615806.2020.1797695 Bakker, A. B., Demerouti, E., & Sanz-Vergel, A. I. (2014). Burnout and work engagement: The JD–R approach. ''Annual Review of Organizational Psychology and Organizational Behavior, 1'', 389–411. https://doi.org/10.1146/annurev-orgpsych-031413-091235 Chan, F. M. (2026). ''The state of workplace burnout 2026: The workplace divide''. Infinite Potential. https://www.ahri.com.au/wp-content/uploads/The-State-of-Workplace-Burnout-2026-Final.pdf Cigrand, C., Quaglia, J. T., Lawlor, S., & Bailey, C. (2025). Healing the healers: The role of contemplative practice in mitigating counselor burnout and empathy fatigue. ''Journal of Mental Health Counseling, 47''(2), 241–261. https://doi.org/10.17744/mehc.47.3.04 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, 32''(23), 5330–5342. https://doi.org/10.1093/cercor/bhac017 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 Ferguson, A. M., Cameron, C. D., & Inzlicht, M. (2021). When does empathy feel good? ''Current Opinion in Behavioral Sciences, 39'', 125–129. https://doi.org/10.1016/j.cobeha.2021.03.011 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 Gynning, B. E., Karlsson, E., Teoh, K., Gustavsson, P., Christiansen, F., & Brulin, E. (2025). Do job resources buffer the harmful effects of job demands on burnout complaints? A 1-year cohort study of Swedish healthcare professionals. ''International Journal of Nursing Studies Advances, 9'', Article 100397. https://doi.org/10.1016/j.ijnsa.2025.100397 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, 2''(3), 96–99. https://doi.org/10.1111/1467-8721.ep10770953 Hobfoll, S. E. (1989). Conservation of resources: A new attempt at conceptualizing stress. ''American Psychologist, 44''(3), 513–524. https://doi.org/10.1037/0003-066X.44.3.513 Hunt, P., Denieffe, S., & Gooney, M. (2019). Running on empathy: Relationship of empathy to compassion satisfaction and compassion fatigue in cancer healthcare professionals. ''European Journal of Cancer Care, 28''(4), Article e13124. https://doi.org/10.1111/ecc.13124 Ito, J. K., & Brotheridge, C. M. (2003). Resources, coping strategies, and emotional exhaustion: A conservation of resources perspective. ''Journal of Vocational Behavior, 63''(3), 490–509. https://doi.org/10.1016/S0001-8791(02)00033-7 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/10.1111/jasp.12211 Lesener, T., Gusy, B., & Wolter, C. (2019). The job demands–resources model: A meta-analytic review of longitudinal studies. ''Work & Stress, 33''(1), 76–103. https://doi.org/10.1080/02678373.2018.1529065 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/10.1007/s00127-004-0808-7 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 Nangle, M. R., Henry, J. D., von Hippel, C., & Kjelsaas, K. (2019). An empirical study of how emotion dysregulation and social cognition relate to occupational burnout in dentistry. ''British Dental Journal, 227''(4), 285–290. https://doi.org/10.1038/s41415-019-0620-0 Panksepp, J. B., & Lahvis, G. P. (2011). Rodent empathy and affective neuroscience. ''Neuroscience & 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), Article 04023060. https://doi.org/10.1061/JCEMD4.COENG-13339 Ruby, P., & Decety, J. (2004). How would you feel versus how do you think she would feel? A neuroimaging study of perspective-taking with social emotions. ''Journal of Cognitive Neuroscience, 16''(6), 988–999. 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'', 607–628. 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'', Article 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, 34''(1), Article bhad441. https://doi.org/10.1093/cercor/bhad441 Thaler, R. H. (2000). From homo economicus to homo sapiens. ''The Journal of Economic Perspectives, 14''(1), 133–141. https://www.proquest.com/scholarly-journals/homo-economicus-sapiens/docview/212067365 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'', Article 849566. https://doi.org/10.3389/fpsyg.2022.849566 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 Yu, C. L., & Chou, T. L. (2018). A dual route model of empathy: A neurobiological prospective. ''Frontiers in Psychology, 9'', Article 2212. https://doi.org/10.3389/fpsyg.2018.02212 Zhu, X., Tian, G., Liu, R., & Li, Y. (2025). Emotional leadership and teachers’ psychological well-being: The mediating role of emotional labour and moderating roles of cultural values in JD-R theory. ''European Journal of Education, 60''(1), Article e70006. https://doi.org/10.1111/ejed.70006 }} ==External links== * [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]] n7di6ghj7lr9m5fltyi6181am58cywp 2844700 2844697 2026-11-06T01:03:38Z U3143751 3106785 /* References */ 2844700 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 at a non-profit organisation providing mental health services to young people aged 18–25, many of whom have experienced challenging living conditions. They support approximately 20 clients each week. Initially, Aelin is highly motivated, demonstrating strong active listening skills, empathy and an ability to create a safe, welcoming environment for clients. They remember personal details and adapt their approach to respond sensitively to clients' emotional needs. However, as their caseload grows and clients present with increasingly complex needs and trauma histories, Aelin becomes increasingly invested in their work. They begin taking work home, with work-related thoughts gradually extending into weekends. Over time, they experience persistent exhaustion, even before the working day begins. Several months later, Aelin notices a shift in their interactions with others, including clients. They feel increasingly detached, frustrated and less empathetic, rather than engaged and attentive. Work leaves them emotionally drained, with little energy for friends or family. As their irritability increases, Aelin begins questioning whether they can 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. 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. {{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 == 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 (Lloyd 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; Seidler et al., 2014). 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]] and the [[wikipedia:Job_demands-resources_model|Job Demands-Resources model]]. === Conservation of resources stress model === The Conservation of resources stress model (CORT; Hobfoll, 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 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. The CORT model focuses primarily on two principles. The first principle is loss salience: there is more harm in losing resources than there is in gaining them back (Halbesleben et al., 2014). Research suggests that the pain of loss is felt twice as strong as the pleasure of gain (Thaler, 2000). The second principle is that people must invest 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 & 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 & 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). 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). It's important to note that job resources 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 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. 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 == At its core, [[empathy]] is the ability to imagine or understand another person's feelings and/or emotions (Shao et al., 2024). Research around empathy is diverse. A dual model consisting of affective and cognitive empathy (Yu and Chou, 2018) is relatively simple, where affective empathy occurs as an emotional reaction to someone else's mental state (Shao et al., 2024), and cognitive empathy occurs when a person understands someone else's feelings or emotions (Cui et al., 2022). However, a tripartite model will be the primary focus in this chapter. === Cognitive empathy === Cognitive empathy is the "what" and focuses on a person tangibly understanding what another person feels or, simply, "understanding the mental states of others" (Martingano et al., 2026 p. 802). It is associated with activation in the dorsomedial prefrontal cortex and the right inferior parietal cortex (Ruby & Decety, 2004) and, considered as a [[wikipedia:Attention#Orienting|top-down attention]] process, requires a conscious, sustained focus (Shao et al., 2024). In relation to the JD-R and CORT models, engaging in cognitive empathy is negatively associated with emotional exhaustion and may be labelled as a job resource (Ferguson et al., 2021). === Affective empathy, compassion, and emotional contagion === Affective empathy is the "how" and focuses on the feeling aspect of empathy. It is associated with mirror neuron activity inferior frontal cortex, the premotor areas and the insula (Yu and Chou, 2018). Through the tripartite model, affective empathy is further categorised as either other- or self-focused (Mantinago et al. 2026). When the affective response is self-oriented, the boundaries between a person and the target of empathy blur, leading to [[wikipedia:Emotional_contagion|emotional contagion]] - a theory that suggests emotions are contagious and spread from one person to another through an automatic [[wikipedia:Mimicry|mimicry]] process (Panksepp and Lahvis, 2011). Contagion is negatively associated with emotional exhaustion and may be labelled as a job demand (Hunt and Gooney, 2019). However, when the affective response is other-oriented and observers understand that the emotions they are reacting to belong to another person, rather than themselves, it leads to empathic concern in the form of sympathy or compassion (Mantinago et al. 2026). Much like cognitive empathy, empathic concern, primarily the compassion aspect, is negatively associated with emotional exhaustion and is considered a job resource. Table 1. outlines the primary differences between the different types of empathy. {| class="wikitable" |+ '''Table 1. Differences Between Empathy Types''' ! Type of empathy ! Description ! What this looks like |- | '''Cognitive empathy''' | Empathy that focuses on understanding and recognising another person's thoughts, feelings and perspective without necessarily sharing their emotions. | Understanding why a friend feels upset after receiving bad news. |- | '''Affective empathy''' | The ability to emotionally respond to and share in another person's emotional experience. | Feeling sad when you see a friend crying. |- | '''Compassion''' (affective-other) | Affective empathy which motivates a desire to take action to make change | Feeling concerned about a struggling friend and offering support. |- | '''Sympathy''' (affective-other) | Affective empathy that does '''not''' motivate a desire to take action. | Feeling sorry for a friend who has lost their job. |- | '''Emotional contagion''' (affective-self) | Mirroring another person's emotional state. | Feeling anxious after spending time with someone who is visibly anxious. |} === Empathy and compassion fatigue === [[wikipedia:Compassion_fatigue|Compassion fatigue]], also known as [[wikipedia:Secondary_trauma|secondary traumatic stress]], is a term introduced by Figley (2002) to describe the negative experience affecting healthcare professionals when they are constantly exposed to the trauma of their patients and is associated with burnout. However, looking at more modern literature, it is understood that compassion does not lead to emotional exhaustion. As such, it will be referred to as empathy fatigue - an alternative term to support clarity in its antecedents (Cigrand et al., 2025). {{RoundBoxTop|theme=3}} '''What is the relationship between empathy fatigue and emotional exhaustion''' *In the terms of the JD-R and CORT models, Compassion and Cognitive empathy function as job resources, thus helping to buffer individuals from burnout, whereas self-focused affective empathy (contagion) functions as a job demand. *Empathy fatigue occurs because of contagion depleting energy faster than resources can restore or mitigate demands. Aelin has engaged in contagion with their clients and is now experiencing empathy fatigue, contributing to their emotional exhaustion. * {{RoundBoxBottom}} == Emotional labour as a mediating factor between empathy fatigue and emotional exhaustion == [[wikipedia:Social_norm|Social norms]] provide guidelines for socially acceptable emotional expression. As social beings, humans rely on these shared expectations to regulate their emotional responses and maintain appropriate social interactions. This is the essence of [[Emotion labour|emotional labour]]: "the regulation of felt and expressed emotions (whether effortful or not) to match these emotional display rules with the objective of fulfilling organisational goals" (Vial & Cowgill, 2022 p. 2). Emotional labour is typically displayed through one of two ways. === Types of acting === Surface acting involves changing how felt emotions are externally expressed, such as adjusting facial expressions and body language to 'fake' the appropriate emotion (Vial & Cowgill, 2022). This creates an internal conflict known as emotional dissonance, where what someone is feeling and what they are expressing is misaligned (Middleton, 1989, as cited in Kenworthy et al., 2014). A meta-analysis from Kenworthy et al. (2014) looked at the relationship between emotional dissonance and burnout. Across 57 articles published between 1998 and 2011, they found a medium effect size (''r'' =.34) across a large sample (n=16165). Demographics were not specifically reported on, however they considered moderating factors and found a significant slope after regressing for gender (''Z'' = 2.24, ''p'' = .03). Under the JD-R, surface acting may be considered as a job demand. Surface and deep acting, the second type of emotional labour, are both [[emotional regulation]] strategies, but differ in what they are regulating. Surface acting regulates the expression of emotion, and deep acting regulates the internal feeling of the emotion through strategies such as [[wikipedia:Cognitive_appraisal|cognitive reappraisal]] (Vial & Cowgill, 2022). It is important to note that some literature discusses a third emotional labour, expression of naturally felt emotions (ENFE), which just describes that a person's genuine feelings are the appropriate to express in the moment and does not require additional emotional regulation (Zhu et al., 2025). === The mediating effect of emotional labour === Pulling together the research, emotional labour is one of the primary bridges between emotional exhaustion and empathy fatigue. When performing surface acting, people become emotionally exhausted. As such, individuals have fewer resources available to manage their emotions leading to emotional dysregulation (Nangle et al., 2019). This creates a job demand spiral, where they are still required to display appropriate emotional responses (Vial & Cowgill, 2022). Under the CORT model, this further depletes already limited emotional resources, intensifying emotional exhaustion (Hobfoll, 1989). Consequently, while this emotional labour is sustained, it reduces an individual's capacity to engage in cognitive and other-oriented affective empathy. Surface acting may exacerbate this effect by increasing emotional demands, while repeated exposure to others' emotions may further contribute to emotional strain through emotional contagion (Hunt & Gooney, 2019). Therefore, emotional labour, particularly surface acting, acts as the mechanism through which empathy fatigue contributes to emotional exhaustion. '''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 {Deep acting is negatively associated with burnout: |type="()"} + True - False </quiz> {{RoundBoxTop|theme=3}} '''What role does emotional labor play in emotional exhaustion and empathy fatigue''' * In relation to the CORT and JD-R theories, surface acting expends energy as a job demand. When energy levels are exhausted, people are more likely to engage in contagion due to emotional dysregulation, leading to a job demands spiral that exacerbates emotional exhaustion. * Aelin has exhausted their energy supplies by surface acting and is stuck in a job demands spiral due to contagion. {{RoundBoxBottom}} == Strategies for combating emotional exhaustion == Burnout treatment is available. Table 2. has been adapated from van Dam (2021) to provide an overview of goals and interventions. Someone experiencing burnout would begin at phase 1. Importantly, when it comes to psychoeducation, it is critical to understand how to reduce the amount of surface acting someone is engaging in. Suggestions include knowledge around communication and boundaries (Bakker et al., 2021), promotion of positive emotions through storytelling (Wei et al., 2024) and a transition to deep acting or ENFE as these are associated with the conservation and gain of resources instead of increasing job demands (Zhu et al., 2025). '''Table 3. Treatment phases of clinical burnout (based on Table 2 from van Dam, A. 2021)''' {| class="wikitable" style="margin: auto; |- ! Phase !! Goal !! Interventions |- | '''Phase 1: Crisis''' || Recognition that the problems are serious, and that serious action is required. These interventions are primarily focused on reducing job demands. || Psychoeducation Sick leave Dropping domestic tasks and social obligations Inform social network |- | '''Phase 2: Recovery''' || Recovery. These interventions are primarily focused on increasing job resources. || 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. These interventions are focused on reducing job demands and increasing job resources. || 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. These interventions are focused on reducing job demands and increasing job resources. || Setting priorities regarding quality of life and interpersonal relationships. |} ==Conclusion== Sustained empathic engagement can contribute to emotional exhaustion when the emotional demands of supporting others continually exceed an individual's available resources for recovery. While cognitive empathy and compassion can act as protective job resources, self-focused affective empathy, particularly emotional contagion, can increase emotional demands by causing individuals to absorb the emotions of others. This is further intensified by emotional labour, particularly surface acting, which requires individuals to suppress or alter their genuine emotions to meet workplace expectations, creating emotional dissonance and further depleting their resources. The Conservation of Resources and Job Demands-Resources models demonstrate how ongoing resource loss, high job demands and insufficient opportunities for recovery can create a cycle of increasing stress and exhaustion. As demonstrated through Aelin's experience, recognising the distinction between different forms of empathy and understanding the role of emotional labour is essential to identifying the causes of empathy fatigue and emotional exhaustion. Ultimately, reducing excessive job demands, establishing healthy boundaries, prioritising recovery and developing more sustainable emotional regulation strategies can help preserve emotional resources, prevent burnout and support long-term wellbeing. ==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) *[[wikipedia:List_of_psychological_research_methods|List of psychological research methods]] (Wikipedia) ==References== {{Hanging indent|1= Alarcon, G. M. (2011). A meta-analysis of burnout with job demands, resources, and attitudes. ''Journal of Vocational Behavior, 79''(2), 549–562. https://doi.org/10.1016/j.jvb.2011.03.007 Bakker, A. B., & de Vries, J. D. (2021). Job demands–resources theory and self-regulation: New explanations and remedies for job burnout. ''Anxiety, Stress, & Coping, 34''(1), 1–21. https://doi.org/10.1080/10615806.2020.1797695 Bakker, A. B., Demerouti, E., & Sanz-Vergel, A. I. (2014). Burnout and work engagement: The JD–R approach. ''Annual Review of Organizational Psychology and Organizational Behavior, 1'', 389–411. https://doi.org/10.1146/annurev-orgpsych-031413-091235 Chan, F. M. (2026). ''The state of workplace burnout 2026: The workplace divide''. Infinite Potential. https://www.ahri.com.au/wp-content/uploads/The-State-of-Workplace-Burnout-2026-Final.pdf Cigrand, C., Quaglia, J. T., Lawlor, S., & Bailey, C. (2025). Healing the healers: The role of contemplative practice in mitigating counselor burnout and empathy fatigue. ''Journal of Mental Health Counseling, 47''(2), 241–261. https://doi.org/10.17744/mehc.47.3.04 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, 32''(23), 5330–5342. https://doi.org/10.1093/cercor/bhac017 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 Ferguson, A. M., Cameron, C. D., & Inzlicht, M. (2021). When does empathy feel good? ''Current Opinion in Behavioral Sciences, 39'', 125–129. https://doi.org/10.1016/j.cobeha.2021.03.011 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 Gynning, B. E., Karlsson, E., Teoh, K., Gustavsson, P., Christiansen, F., & Brulin, E. (2025). Do job resources buffer the harmful effects of job demands on burnout complaints? A 1-year cohort study of Swedish healthcare professionals. ''International Journal of Nursing Studies Advances, 9'', Article 100397. https://doi.org/10.1016/j.ijnsa.2025.100397 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, 2''(3), 96–99. https://doi.org/10.1111/1467-8721.ep10770953 Hobfoll, S. E. (1989). Conservation of resources: A new attempt at conceptualizing stress. ''American Psychologist, 44''(3), 513–524. https://doi.org/10.1037/0003-066X.44.3.513 Hunt, P., Denieffe, S., & Gooney, M. (2019). Running on empathy: Relationship of empathy to compassion satisfaction and compassion fatigue in cancer healthcare professionals. ''European Journal of Cancer Care, 28''(4), Article e13124. https://doi.org/10.1111/ecc.13124 Ito, J. K., & Brotheridge, C. M. (2003). Resources, coping strategies, and emotional exhaustion: A conservation of resources perspective. ''Journal of Vocational Behavior, 63''(3), 490–509. https://doi.org/10.1016/S0001-8791(02)00033-7 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/10.1111/jasp.12211 Lesener, T., Gusy, B., & Wolter, C. (2019). The job demands–resources model: A meta-analytic review of longitudinal studies. ''Work & Stress, 33''(1), 76–103. https://doi.org/10.1080/02678373.2018.1529065 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/10.1007/s00127-004-0808-7 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 Nangle, M. R., Henry, J. D., von Hippel, C., & Kjelsaas, K. (2019). An empirical study of how emotion dysregulation and social cognition relate to occupational burnout in dentistry. ''British Dental Journal, 227''(4), 285–290. https://doi.org/10.1038/s41415-019-0620-0 Panksepp, J. B., & Lahvis, G. P. (2011). Rodent empathy and affective neuroscience. ''Neuroscience & 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), Article 04023060. https://doi.org/10.1061/JCEMD4.COENG-13339 Ruby, P., & Decety, J. (2004). How would you feel versus how do you think she would feel? A neuroimaging study of perspective-taking with social emotions. ''Journal of Cognitive Neuroscience, 16''(6), 988–999. 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'', 607–628. 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'', Article 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, 34''(1), Article bhad441. https://doi.org/10.1093/cercor/bhad441 Thaler, R. H. (2000). From homo economicus to homo sapiens. ''The Journal of Economic Perspectives, 14''(1), 133–141. https://www.proquest.com/scholarly-journals/homo-economicus-sapiens/docview/212067365 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'', Article 849566. https://doi.org/10.3389/fpsyg.2022.849566 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 Yu, C. L., & Chou, T. L. (2018). A dual route model of empathy: A neurobiological prospective. ''Frontiers in Psychology, 9'', Article 2212. https://doi.org/10.3389/fpsyg.2018.02212 Zhu, X., Tian, G., Liu, R., & Li, Y. (2025). Emotional leadership and teachers’ psychological well-being: The mediating role of emotional labour and moderating roles of cultural values in JD-R theory. ''European Journal of Education, 60''(1), Article e70006. https://doi.org/10.1111/ejed.70006 }} ==External links== * [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]] 1m60jexhxfyw9k3ef31weodqpi7og0m 2 331327 2844688 2828565 2026-11-05T22:04:59Z U3143751 3106785 /* Social Contributions */ 2844688 wikitext text/x-wiki == About me == I am '''Mitchel Swann'''. I am a third year [[Motivation and emotion|Motivation and Emotion]] student at the [https://www.canberra.edu.au/ University of Canberra]. === Hobbies === [[File:Man bouldering at The Hive, in Winnipeg, Manitoba.jpg|thumb|300x300px|'''Figure 1'''. Bouldering is good exercise.]] Some of my hobbies are: * [[w:Bouldering|Climbing/bouldering]] * [[w:Reading|Reading]] * [[w:Video_game|Video games]] * [[w:Board_game|Boardgames]] ** [[w:Dungeons_&_Dragons|Dungeons and Dragons]] == Book chapter I'm working on == [[Motivation and emotion/Book/2026/Empathy fatigue and emotional exhaustion|Empathy fatigue and emotional exhaustion]] == Social Contributions == # [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2026%2FAdaptive_versus_maladaptive_self-reflection&diff=2823494&oldid=2823481 Adjusted title casing to match Motivation and Emotion book 2026] # [[Talk:Motivation and emotion/Book/2026/Adaptive versus maladaptive self-reflection#Suggested resources|Left a comment suggesting a resource to use]] # [https://uclearn.canberra.edu.au/courses/20143/discussion_topics/457906 Discussion forum post on fixing RoundBottomBox template issue] #* [https://en.wikiversity.org/w/index.php?title=Template%3ARoundBoxBottom&diff=2828561&oldid=2828398 Fixed the RoundBottomBox template issue] # [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2026/Self-determination_theory_and_dementia_care&oldid=2839971 Provided advice to fix RoundBox issue on self-determination theory and dementia care page] 4uyot87zbe5lphur62t2l9nyua90h6w 0 331771 2844630 2844602 2026-11-05T15:55:39Z Dick Bos 24466 /* The Works and Correspondence of David Ricardo */ corr ref 2844630 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/n4/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]] 9d4m47li3pt7wl52t1yu00io9w52vx2 0 331787 2844636 2844480 2026-11-05T16:35:28Z Dick Bos 24466 /* Sources */ Benetti lang + Hollander 1975 2844636 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 | language = 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)]])}} * {{cite journal | last = Hollander | first = Samuel | date = 1975 | title = Ricardo and the Corn Profit Model : Reply to Eatwell | journal = Economica | volume = 42 | issue = 166 | pages = 188-202 | doi = 10.2307/2553594 | url = https://www.jstor.org/stable/2553594 | 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 | 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)]])}} gbfhuga2r8sybemhodtfqudolu507hk 2844666 2844636 2026-11-05T20:05:59Z Dick Bos 24466 small corr. 2844666 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 (in fact: "wheat"). 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 studied Ricardo, the term “corn” will be mainly 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 "neat produce" (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 | language = 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)]])}} * {{cite journal | last = Hollander | first = Samuel | date = 1975 | title = Ricardo and the Corn Profit Model : Reply to Eatwell | journal = Economica | volume = 42 | issue = 166 | pages = 188-202 | doi = 10.2307/2553594 | url = https://www.jstor.org/stable/2553594 | 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 | 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)]])}} 4hv66py2eq4hc42xw8c1bvm907bj5pe 2 332158 2844702 2841778 2026-11-06T01:48:18Z The Citer 3110681 Added phylogeny! 2844702 wikitext text/x-wiki My template, text, and phrase storage! ==Templates== [[File:|thumb|300x300px|[Insert_text_here]]], [[File:|60px]] ==Text== – ==Phrases== <nowiki>''Incertae sedis''</nowiki> ==Purpose== It's a storage space so I don't have to copy text in source editor; I just can copy it in reading mode. 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150 155 160 165 170 175 180 185 190 195 200 205 210 215 220 225 230 235 240 245 250 255 260 265 270 275 280 285 290 295 300 305 310 315 320 325 330 335 340 345 350 355 360 365 370 375 380 385 390 395 400 405 410 415 420 425 430 435 440 445 450 455 460 465 470 475 480 485 490 495 500 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101 106 111 116 121 126 131 136 141 146 151 156 161 166 171 176 181 186 191 196 201 206 211 216 221 226 231 236 241 246 251 256 261 266 271 276 281 286 291 296 301 306 311 316 321 326 331 336 341 346 351 356 361 366 371 376 381 386 391 396 401 406 411 416 421 426 431 436 441 446 451 456 461 466 471 476 481 486 491 496 2 7 12 17 22 27 32 37 42 47 52 57 62 67 72 77 82 87 92 User:[[The Citer]] [[User:The Citer]]dendron =[Unused NAME]= [[Wikipedia:Superrosids|Superrosidia]], [[Wikipedia:Rosids|Rosida]], [[Wikipedia:Rosids|Eurosida]], Fabidae, Malvidae<br> [[Wikipedia:Superasterids|Superasteridia]], [[Wikipedia:Asterids|Asterida]], [[Wikipedia:Asterids|Euasterida]], campanulidae, lamiidae ==phylogeny?== Hmmmmm... I have a speculative evolution project [https://sites.google.com/view/worldofaxellia/home here]. {{clade |label1=FUCA |1={{clade |label1=Twokaidae |1={{clade |1=''Twoka |2=''Fortka}} |label2=Fifida |2={{clade |1=''Quinnafifca |label2=Dakaidae |2={{clade |1=''Trififca |2=''Vigidaka}} }} |label3=Hepticommida |3={{clade |1=''Trihepticommina |label2=Bihepticommidae |2={{clade |1=''Quadrihepticommina}} }} |label4=Theekazoa |4={{clade |label1=Ennida |1={{clade |label1=Trivigidae |1={{clade |1=''Trigentidaka}} }} |label2=Undekcommida |2={{clade |1=''Hepterosexla |2=''Bivigidaka |label3=Hepterodidae |3={{clade |1=''Enndaka}} }} |3=Sexla}} }} }} 10ln8olj8tiqc09xevy0msgbwiggthh 0 332412 2844690 2844533 2026-11-05T22:20:24Z Scogdill 1331941 2844690 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 I followed him on deck. A Babel of voices and a clutch at my bag and rug thoroughly aroused me, and jumping into a trap the Pelican and I were in about three minutes set down at Gardner's Royal Hotel. A venerable-looking elder, whom Fleming voted a "[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 mucch needed rest. Up and doing soon after tea, a delicate task was before us. Fleming had a friend on the land, a gallant captain, who knew of a spot where we could fight for a month. Guernsey, dear to the "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 Alderney [?], and 25 from Jersey. I mention these distances, for we explored the whole group. Encompassed with rocks, almost hidden in high water, navigation is exceedingly dangerous at all times near the island, the knowledge of which quickly knocked one of our plans on the head. Standing at the door of Gardner's Royal Hotel on Thursday morning Castle Coronet came in view, while in the distance, almost hidden by a heavy sea mist, could just be discerned the islands of Herm [?], Jethon, and Sark. Guernsey, it should be mentioned, though under the protection of England, boasts a sort of Home Rule government, rather irksome to the stranger within its gates, especially if that stranger is on mischief bent and gets caught. Making, in a cautious manner, inquiries at the breakfast table, we found a friend in "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 made "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 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 heard 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 a 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 M., 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. On returning to town we received satisfactory reports of the work done by each of the contestants. Slavin, looked after by Josh Cosnett, the Birmingham heavy-weight, and his friend Mr Michael Taylor, of Australia, though rather light, was in good health. At one time he scaled only 12st 8lb, but a judicious spell of light work brought him back to 13st odd. Considering the importance of the business in hand, and its bearing on future events, Slavin had some  cause for anxiety. In addition to this he was fighting for 300''l.'' of his own money, and consequently wished to do himself justice in the international battle. The next question was the selection of someone capable and willing to fill the all-important post of referee. According to the articles it was left to the stakeholder to appoint the official named providing the representatives failed to agree, and in the end this course had to be adopted, solely owing to the objections lodged by Smith. The first gentleman suggested by Slavin's side was Mr B. J. Angle, whom Smith promptly objected to. Next a well-known owner of racehorses (a gentleman who adjudicated upon the knuckle fight between George Camp and "Click" Soles) was proposed with no better result. Mr Bat Murphy, strange to say, would not suit, and then in succession "No" was said to the names of  Messrs G. H. Vise, A. F. Bettinson, and [B.?] Bayley. All these gentlemen, it must be stated, were proposed by Slavin and objected to by Smith, who in the meantime had named a member of the Pelican Club, a gentleman of limited experience in these matters, and the landlord of the house from which he was training, Mr W. Riley, Newmarket. This brought about a deadlock, and the state of affairs getting wind, ugly rumours were soon afloat, and even money, and in many cases good odds, were betted that there would be "no fight." All this while Slavin was chafing and fuming at the delay, whilst at the same time his  representative and Smith's were at work almost night and day endeavouring to bring about a settlement of the difficulty. For a long time it looked as though the matter would end in smoke, as after the services of the stakeholder had been called upon great difficulty was experienced in getting anyone to accept the office of referee. So late as Friday at noon nothing had been done, and it was not until the evening of that day that Mr Joe Vesey, consenting to stand, was nominated by the stakeholder. A Scare. Though the battle-ground was only known to two persons, the country where the fight was to be decided being, in fact, or supposed to be, a perfect secret, on Friday morning a paragraph said to have been sent out by a news agency appeared in several papers, and created intense astonishment and consternation. This, set out in large type by those who we should have thought would have suppressed the account, stated that Smith on the previous evening had arrived in London from his training quarters, and even went so far as to name the street he slept in. That Smith's visit to town was extremely injudicious there is not the slightest doubt, but this was not the worst. The time of departure for the scene of action was given, with a description of the place selected. This publicity had the effect of keeping away many who bad expressed a wish to be present, and on Slavin's side no less than eight gentlemen, each holding a high position in the sporting world, declined to have anything further to do with the affair. Who divulged the secret is at present a mystery, but there is not the slightest doubt that the information so freely put forward not only opened the eyes of the authorities, but gave the "mob," to a certain extent, a key to the [Col. 5c–6a] situation. Slavin received the news with no small amount of astonishment, and a couple of telegrams informing him that a batch of the "boys" had gone across to make things warm for him did not give peace to his troubled mind. Smith made his way out of England on Friday by the short sea route, Dover to Calais, which he reached in safety. Slavin, not so fortunate, was arrested on Satuday, as already reported, but, after being bound over to keep the peace, he got away the same evening. FERFORIAICII OF THE MEN.  JEN =ITN was born in Whitecross-street, in the parish of Cripple-gate, 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>?</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>1</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 mates he was taken in hand by Jack Knifton ("81-tonner"), and later by Mr F. Grimm, of the Central Club. Smith has never been defeated, though in his first engagement he did not shape like a coming Champion. His first success was in a competition at the Griffin, Shoreditch. '''Later he''' beat 11= / .  Welieway. al is matched to bw  with glee* kr • 1111. pea Arabi a class  sad a half tigar then dab. ia peed  eandark bean asap lbs. restensly sepia in  dais Jack Essay brae hapeedurit agegement. The  sear was lead epee se • geed Meg lee Bar, at  It teed him nay as hear to gash oft the  Sea nweeselletire, though he had slightly  the bed et the eneaster der dart to air.  the *eau Gale a .an on  =il la tiell  to near him to Massy, Maim but  Shia wee tlelellud. Smith's neat agegene* was  with Wed, Dresia Is Ins to a tea with glens far  lfti. seta what ore edit a Emhart she. In this  easel rah Wend Mt right sea asarthele*  rad seesead et re ea *tan rear Smith else  wan Ter Ilyneteets Alkessaa Oselfebe=  1111, at XVI Weinedeselkdernag  Weans nhareseller), mad Ten loregel . randere Ith  woo pouselsil Maslow oil yellotiwg el birrolf. easswaell  by • preassa *dee of the spark Sark% as* telles  was wllh at Dna le UAL aide ind re Ohara  skip. as the tat and *me of as the bads at deem  ▪ Si., w Wedeesday. Dana If. Is all de  reads wee lark esewering Tria Mem Davie brag  lashed a natewith  Ws ersesegesay met Alt Oreettlehl. "l.b 3Am il or  111. to likkeartreetret sew Peek. ea Twasy.  16. 1118 Ildeleen Nab wen height, may la  era ith's tsar awl the nee fell, tearer* • ary teal  was nail dessedare pasty, end • and ei  lake lete the rim& etseels , the SOL the ear=  Neer at the prinelpel ashen d the  grad tedeelde n C l ablis  • aw. earderk  ertelied be light leek law, the "lidear." as he in  anima. Ibis revel • deses.• two days' day  lb. Claised hag Mesa by • saes le lbeYap en  eastry. diefter's peak aimed that Smith bed era  than the media mamba. d IbEemos. but Yale Is Mile  doubt that Oft " ads s Mated Itabt. and Mall was set  Me ••11114mearl kr. likatese after meeting me bad.  MINIM ellholeg vn ture  ens  t t r asywbsoa the after dde NW. and  = , oorb i l anael=wore siedete i r e legthe  potls  rvlsizd of ti mid emetaideg Mellen= party war  rowis maw  thee Me arampommis woos  I . 27r " be Mir al si lMillsoes irlas  is, with the meek OM pdasidois. sesseds. sed •  dam Mime were modimed .ally sae waswieg Is  • funikare ti by Ms pollee berm they bd  gees lily yards as art sway. All were wept  as as The.. Parawn, and ertsweemmer  bad ever be ha the to take nabs  The asemosoalebeam NM" eat Mimic.  MMIT id  ad= re Si, esene eal t relnites  ad be the allolo. whirl ales sear is •  re ma had read met ether far WS seals.  as Drama IS, 111111. AGM asst=re me •  Iluassual lime Osalsoll with Seek the Maw  ,plisa haw ildd Ira eras(  ' des Meat wel l ie l le m ans all:. Ohib or  Tare Jaabms, Me Wound ellamoka. la • glove easiest,  kr 16111.  FRANK P. SLAVIN.  the Clrompisa of Asmtulk, was Ina as Tannery D. NO,  at Nabas4 NAIL He steeds eft lin In his stockings,  amt lerers sae ileum er 13st 6111. Wavle served  his Ma or egreediseship to a verseireda ad blurear  and NM arm eat as a pedeelea, H. always bad  • tir es glerea, and qua • rest deal van  did net am la ea ma all the Isar  pee 7B 72 Is. ler mew. Mesgh be wee immel  sempolitime esl2 o a . : mirder. Oak's raga  nes alit alehhe: Pow:  IL ke Mt 64461,mA - ha  was/lee mem bodes thietemadmiss. He Wm  rwiera/mAre Peihri Milt is gimes ler ANL holds ad the  el Masan em throe remits.  ev akwesterOM mil ia she lallo%yom the  Ausloallso lama sal Wile ear.  Mew= mew Sum ho M oma deed  up kr bur eemaik. At this pm' illevia bed ea mop ilese  sf eine sad ma rear ad sere two. bat nem et  i• s !resat the ea elf a be went to  reey, la Mew l aim will  ale as air plain them =ere aria m l erlaylor.  pupil d destmo. ke pass with small sham sod  Mega M He wee Oise Seel with a  Melie d r rune/ IWO a M. adds end beet an  Ala • net el ewe ears Man were er roc .  arl ms . hails. heir bad  elibme mesh Mm Si • rile Manny  bin Mla seam la '  In Yg  ••ea mark Ire ealally Oran an A 111 4 . 614.•  good made his mpressare. sad the yak  were ter WA are. They had • Idg battle  la Bram haw la hews al a belt, and a draw was  a Neer amnia make sr Se SIN COL, sad bet.  sole we bait Si Yelled this. Afterwards  Darla haler malsbod idea to IA raids to step Car.  tie M sit suede, but ibe Isar is • my elerer and  lake essa the the lad Oubsegeently Pak  sot Whig Elbe there. challenged  ert  malt fil Oh me • sea for 1004  are with am eras. welsh Ned, mu to lady  Weak Sala al fhb we has my moat of backeed  lead • sherwe be light Pea ashen ter Ur  s-dde sad the Chereaship. !bola's per* wad a  rat with ewes as ran Tay twat aA am te  lona thee. Ira% laws eared Jails Mt lb as  maims In VIM Asoloolia um sew a IN bare, vele  roma la year w laimarin ate • trip to New Am  hal Nee gat ma pat was IS, are as heal*  sad der remaking he bed Si aegis at the ad el  the is. Is Daft lbw* beet • big fellow le leer  Maas, and jawed • lag be te light Isar  the Charm. Tams all net be weed wee until  they bed ° MOW dads sheet In early weds. The  C ` were then seedited ler Vir aide. sad lathe with  Ilse gnaws at Weeiplal. Wag bliss le Ness op el the  sag at the NW mewl. Alfas 0. we went but to  114MMNIVOI4 ea Vz le AM ea a swish with Jai..  eat um Is is he MI peas le Amorist.  Si ma Ifrafes. but Mhz EWE bed  P hid lb s w il a h MA latheowalest woodoeirarleibew. Nibs  vas wow in !AM. MI bowel saltallimps te  This be midi kr tO will small&es kse whiii  Sd le linhe i a md impolarl oat ha Imia  way bed, and all hy eras to a aunty  half en lam masers be Wasener, al Das  eEwi pet mond liallmoos la Me piper. em Tidier  le V a le as numb; INC sae drla dy  aria il= 4iis Ldemelset Oberla seme arem s s  Er Lista ihrialser air 7 lire ;was lisesilumasio.  lashes Me bused • half. awairially We lee Nkomosim  Mao wee mime. drawee all bobs se, sal Os Ws  =with lea gimes. sod mosed tb. sea.  min wiN mama to the mimic to the ill sena  Pram to this ally_Tame, the tesqemeret  Hula, fee NM., with the result that the Mee  addedleast to hie ale. Tana yam  tat mower*  the w =tisee la the third read. TIP, to  ealdwal to be aet Met perternann. Atom  this tins Wien lleadoE. fie IlliNsobapel. woo le NM  beam, mid fashadyholy of bashms. they pet op • hat  or lOU he a Leaser le ben ler lid. sades  After wafters blue pa* eV Portet however. herder.  Mauls were Pp a dear with the watch. sal Sin  the "army." a ease *say  ever ts, Rill hr . we star followed, end then Star we  Ora IT a hie Yet appear* la &whoa at  Adry's To where he beat " Oberwaid " Goode to  la rear le • Era of twelve reads. for 400/. •  THE PELICAN CLUB AND THE FIGHT.  TO THE EDITOR OP THE SPORTSMAN.  SIR,—As an impreasion prevails that the Smith and  Slann fight wan arranged under the patronage of this  club, 1 should esteem it a favour if you would allow me  to state emphatically that the Pelican Club gave no  countenance to the affair in any way whatever, and  had nothing to do with it.—Yours truly,  CECIL RALEIGH, Hoe. Sec.  Pelican Club, 21, Denman-street, W., Dec. 23.  We hear on good authority that it is the intention  el the mambas of the Pelican Club to present Slavin,  the Australian Champion, with a belt, in recognition  of the manly exhibition he gave yesterday at Bruges.  OF TEE  FRANK SLAVIN TO JOHN L.  SULLIVAN.  FRANI StAvitt, the Champion of Australia, is prepared  to meet John L. Sullivan, under London Prize  Ring Rules, for 500/. or 10004 a-side and the  Championship of the World. Slavin bay lodged  with no a sum of 1001., and will cosiest  the title with Sullivan in either America or  Australia in six mouths from signing articles. We  have '''cabled''' this challenge to Mr R. K. Foe, of New  York, and the money lodged with him or the Editor  of Tsui Smarms's well lead to a match. === Dundee ''Evening Telegraph'' === This Wednesday, 25 December 1889 article describes Slavin's reception in London after the fight. Viscount Mandeville was there, suggesting that he was a Slavin supporter:<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. 8¹/₂in. 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. 8¹/₂in. high, and weighs in condition 12st. 10lb. Round the chest he measures 43¹/₂in.; waist, 36¹/₂in.; hips, 41in.; thigh, 24¹/₂in.; calf, 16³/₄in.; biceps, 15¹/₂in. 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}} 4oktquor04bt6f532j9oyuflmblbooz 2 332474 2844616 2840120 2026-11-05T12:44:38Z MathXplore 2888076 Template:Blocked user 2844616 wikitext text/x-wiki {{Blocked user}} 02r18o8f88pflm5ejs8wavv0k0yq05x 3 332480 2844615 2843641 2026-11-05T12:44:01Z MathXplore 2888076 /* 2026-10-15 */ reply: {{subst:Block}} (-) ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]]) 2844615 wikitext text/x-wiki == [[:Category:Page flag templates]] == [[:Category:Page flag templates]] is a page for the category definition. It is not a page for biblical theology. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 12:18, 13 October 2026 (UTC) ==Welcome== {{Robelbox|theme=9|title='''[[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]], Theologiministry89!'''|width=100%}} <div style="{{Robelbox/pad}}"> You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or get in touch with [[User talk:MathXplore|me personally]] if you would like some [[Help:Contents|help]]. Remember to [[Wikiversity:Signature#How to add your signature|sign]] your comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. Using the signature icon [[File:OOjs UI icon signature-ltr.svg]] makes it simple. We invite you to [[Wikiversity:Be bold|be bold]] and [[Wikiversity|assume good faith]]. Please abide by our [[Wikiversity:Civility|civility]], [[Wikiversity:Privacy policy|privacy]], and [[Foundation:Terms of Use|terms of use]] policies. To find your way around, check out: <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Wikiversity:Introduction|Introduction to Wikiversity]] * [[Help:Guides|Take a guided tour]] and learn [[Help:Editing|how to edit]] * [[Wikiversity:Browse|Browse]] or visit an educational level portal:<br>[[Portal:Pre-school Education|pre-school]] | [[Portal:Primary Education|primary]] | [[Portal:Secondary Education|secondary]] | [[Portal:Tertiary Education|tertiary]] | [[Portal:Non-formal Education|non-formal]] * [[Wikiversity:Introduction explore|Explore]] links in left-hand navigation menu </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] * Learn [[Help:How to write an educational resource|how to write an educational resource]] * Find out about [[Wikiversity:Research|research]] activities * Give [[Wikiversity:Feedback|feedback]] about your observations * Discuss issues or ask questions at the [[Wikiversity:Colloquium|colloquium]] </div> <br clear="both"/> To get started, experiment in the [[wikiversity:sandbox|sandbox]] or on [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 12:18, 13 October 2026 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} : [[File:Information.svg|25px|alt=Information icon]] Hello, Theologiministry89. This is your user talk page; the purpose of this page is notification and communication with other Wikiversity editors. It is not a workspace for articles in progress or for self-promotion. Please use your [[Special:MyPage/sandbox|user sandbox]] or the [[Wikiversity:Drafts|draft article space]] to practice editing or to create new articles. Thank you.<!-- Template:Uw-usertalk --> [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 12:40, 14 October 2026 (UTC) : [[File:Nuvola apps important.svg|25px|alt=Warning icon]] Please stop. If you continue to use talk pages for [[Wikiversity:Talkpage policy|inappropriate discussion]], you may be [[Wikiversity:Blocking policy|blocked from editing]]. <!-- Template:uw-chat3 --> [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 12:44, 15 October 2026 (UTC) == 2026-10-15 == [[Image:Nuvola apps important.svg|25px]] Please stop your '''disruptive editing'''. If you continue to '''vandalize''' Wikiversity{{#if:|, as you did to [[:{{{1}}}]]}}, you may be '''blocked''' from editing. {{#if:|{{{2}}}|}}<!-- Template:uw-vandalism3 --><!-- Glow-vandalism3 @ 1792068314867.4s --><nowiki></nowiki> [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 12:45, 15 October 2026 (UTC) : [[File:Stop hand nuvola.svg|30px|alt=Stop icon]] You may be '''[[Wikiversity:Blocking policy|blocked from editing]] without further warning''' the next time you use talk pages for [[Wikiversity:Talkpage policy|inappropriate discussions]]. <!-- Template:uw-chat4 --> [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 06:36, 1 November 2026 (UTC) : {{tmbox |type = delete |image = [[File:Stop x nuvola.svg|40px]] |text = You have been blocked for a period of indefinite from editing Wikiversity for abuse of editing privileges. If you believe this block is unjustified, you may contest this block by adding <code><nowiki>{{unblock|your reason here}}</nowiki></code> to the top of your talk page to request unblocking, replacing ''"your reason here"'' with the reason you should be unblocked. Thank you. }} [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 12:44, 5 November 2026 (UTC) 9x3z46vwsyl3lmrbx32cka4awy28ut5 2 332486 2844617 2840131 2026-11-05T12:45:34Z MathXplore 2888076 Template:Proposed deletion 2844617 wikitext text/x-wiki <!-- TO CONTEST THIS PROPOSED DELETION, remove the following template, including this comment, up to the CLOSING COMMENT --> {{Ombox | type = delete | image = [[File:Orologio rosso.svg|45x45px|center|No license|link=]] | text = Please [{{fullurl:{{FULLPAGENAME}}|action=edit}} help improve] the [[Wikiversity:What is Wikiversity?|educational quality]] of this resource to increase [[Wikiversity:Learning|engagement]] by [[Wikiversity:Who are Wikiversity participants?|participants]]. Any concrete improvements made by '''{{#time:F j, Y|{{REVISIONTIMESTAMP}} +90 days}}''' may allow it to be [[WV:PROD|kept]]. <br> You may remove {{tl|proposed deletion}} from this resource's source text to contest this proposal, with or without [[{{TALKPAGENAME}}|discussion]]. {{#if:|---- <div style="text-align:center; margin-bottom:0em;"> The nominator's rationale: <p>{{{1}}}</p> </div> }}}}{{center|[[Special:PrefixIndex/{{FULLPAGENAME}}/|''Link to any subpages this page might have'']]}} {{event trigger|date=November 5, 2026|when=90 days|[[Category:Pending deletions|{{PAGENAME}}]]}} {{event trigger|date=November 5, 2026|when=60 days|[[Category:60-day proposed deletions|{{PAGENAME}}]]}} [[Category:Proposed deletions|{{PAGENAME}}]] <!-- CLOSING COMMENT, remove up to and including this comment --> Theologia STEM School TSS is the project of theologia generally and specially the project of theologia research center TRC for STEM science correlation with the theologia as the combination of theological research and analysis by STEM science school for exegetical theology, practical theology, historical theology, systematic theology with the open knowledge, open access, open science, open education, open culture, open software, open school, open library, open research, open journal and the Lab experiments, publications, journal, articles, books, ebooks, images, videos, music, songs, and apps for easy access for the world 🌍 kmng9mrk9akoe2aaoxdj3tk3no4dmuv 0 332549 2844698 2843863 2026-11-06T00:52:30Z DavidMCEddy 218607 add cite to Trainer 2025 2844698 wikitext text/x-wiki {{Research project}} :''This article invites all to contribute references and interpretations to the growth in average annual income, Gross Domestic Product (GDP) per capita at purchasing power parity, as applied to leading countries in the [[w:Maddison Project|Maddison Historical Statistics Project]].'' :''This essay is on Wikiversity to encourage a wide discussion of the issues it raises moderated by the Wikimedia rules that invite contributors to [[w:Wikipedia:Be bold|“be bold but not reckless,”]] contributing revisions written from a [[Wikiversity:Disclosures|neutral point of view]], [[Wikiversity:Cite sources|citing credible sources]] -- while raising other questions and concerns on the associated [[Wikiversity:FAQ|'''{{'}}“Discuss”'''{{'}} page]] -- and treating others with respect.'' ::''This article uses [[w:ISO 8601|ISO 8601]] dates except for References, which are controlled by standard Wikidata formatting, and direct quotes. In the initial author's experience, [[ISO 8601 and computing differences between dates|ISO 8601 dates seem to make it easier to remember dates and to compute differences between them.]]'' == Abstract == This article first identifies and plots GDP per capita for leading countries in the Maddison Project data then discusses comments by Nobel prize winning economists and others to interpret what appears in said plot. No one really knows how to make economies grow. However, growth at the technology frontier requires creative destruction, and immigrants tend to be more entrepreneurial than native born. Also, creative destruction is facilitated and inequality is limited by diversity of ownership and control of the major media. == Maddison Historical Statistics == [[File:MaddisonDataLeaders1349-2022.svg|thumb|''Figure 1. World leaders in GDP per capita 1349-2022 (NLD = initially [[w:Holland|Holland]] to 1807 and the [[w:Netherlands|Netherlands]] after; GBR = initially England to 1700, Great Britain to 1870, then the UK; USA starting in 1650 with the 13 colonies that became the US; SGP = Singapore starting in 1870).<ref>See Bolt and van Zanden (2024) for the Maddison Data generally, Broadberry et al. (2015) for the data on England 1252–1700 and on Great Britain until 1870, Sugimoto (2011) for Singapore to 2007, van Zanden and van Leeuwen (2012) for the data on Holland 1349–1807, and Smits et al. (2000) for the data on the Netherlands 1808-1913.</ref> The dramatic drop in the orange line between 1807 and 1808 is due to the change in reference from Holland to the Netherlands; at that time Holland was the most prosperous region of the Netherlands.<ref name=vanZ_B>Private email exchange with van Zanden and Bolt, 2026-01-08.</ref>'']] The [[w:Maddison Project|Maddison Historical Statistics Project]] (aka Maddison Project) is widely recognized as the premier data source on historical macroeconomic data. It is named after [[w:Angus Maddison|Angus Maddison]] (1926-2010), who was a pioneer in [[w:National accounts|national accounts]], especially estimating [[w:Gros Domestic Product|gross domestic product]] (GDP) per capita (GDPpc) and population (pop) at various times dating back to the year 1 of the [[w:Common Era|Common Era]] (CE / AD). After Maddison's death, the Maddison Project has continued to update the database to include the best sources available on any country for any historical period since year 1. When accessed 2026-10-10 the version of the data available had been released 2024-09-18 containing data to 2022.<ref>Bolt and van Zanden (2024).</ref> This dataset includes 23,280 observations on at least one of GDPpc and pop for an average of 137 years for 169 countries. Only 18 countries had the highest GDPpc for at least one year. Annual data starts in 1252, and that's only for one country, GBR = England / Great Britain / the United Kingdom. NLD (Holland / the Netherlands) had the highest GDPpc for 97 percent of the years between 1349 and 1807. The drop in the orange line in Figure 1 between 1807 and 1808 is due to the change of reference from Holland to the Netherlands.<ref name=vanZ_B/> Visual comparison of the orange and red lines in Figure 1 suggests that the GDPpc of Great Britain overtook that of the Netherlands a few decades earlier and overtook Holland a few decades later. This analysis focuses on data since 1349, in part because data were available for at most 4 countries for most years prior to that date. (The 3-letter country codes used here are [[w:ISO 3166-1 alpha-3|ISO 3166-1 alpha-3]], used in the Maddison data.) We are interested in technology leaders, so we eliminate the [[w:petrostates|petrostates]] of Qatar (QAT), Kuwait (KWT), and United Arab Emirates (ARE), plus Norway (NOR); the latter owes a substantial portion of their wealth to democratic management of [[w:North Sea oil|North Sea oil]].<ref>Dośpiał-Borysiak (2017).</ref> With these restrictions 12 countries had the highest GDPpc for at least one year. However, FRA (France), ITA (Italy), SWE (Sweden), BEL (Belgium), NZL (New Zealand) and CHE (Switzerland) led for at most 7 of the 674 years between 1349 and 2022; we eliminate those, because including them would seem to add more clutter than clarity. We also eliminate AUS (Australia), which led for 17 of the 39 years between 1853 and 1891, when it was competing for the leadership with GBR and USA, for the same reason. And we eliminate LUX (Luxembourg), which has a population under a million. This leaves NLD, GBR, USA, and SGP (Singapore), plotted in Figure 1; Singapore has led since 2010.<ref>More details of this analysis are available in the chapter on "[[Media Literacy and You/The impact of the media on political economy since the time of the Pharaohs|The impact of the media on political economy since the time of the Pharaohs]]"" in the book-in-progress on ''[[Media Literacy and You]]'' on Wikiversity.</ref> This plot has some interesting features, which we discuss in the remainder of this research report. == Nobel economists and others == [[w:Abhijit Banerjee|Banerjee]] and [[w:Esther Duflo|Duflo]] (2019, p. 207) said, "despite the best efforts of generations of economists, the deep mechanisms of persistent economic growth remain elusive." That appeared shortly before they shared the 2019 [[w:List of Nobel Memorial Prize laureates in Economic Sciences|Nobel Memorial Prize in Economics]] with [[w:Michael Kremer|Michael Kremer]] for documenting how [[w:Randomized controlled trial|randomized controlled trial]]s could be used to alleviating poverty in [[w:Development economics|development economics]].<ref>For the present discussion, we assume that enough [[w:Externality|externalities]] have been sufficiently internalized with appropriate [[w:Pigouvian tax|Pigouvian tax]]es to make growth desirable. This converts discussions of [[w:Degrowth|degrowth]], e.g., by Alexander (2026), into evaluations of whether externalities have actually been sufficiently internalized and what to do if they are not. Internalizing externalities appropriately will not be easy as long as those who benefit from not paying for damages they cause to customers, employees and neighbors control substantial portions of the funds for the media. However, for the present discussion, we assume this is not an issue.</ref> While Banerjee and Duflo insisted that no one knows how to make an economy grow, other leading economists have recommended partial solutions. For example, [[w:Joel Mokyr|Mokyr]] (2017, p. 53) said, "The willingness of the Netherlands and later Britain and the United States to tolerate Jews and dissenting Christians contributed a great deal to their economies, especially in high-skilled manufacturing and financial services." In other words, immigrants made an outsized contribution to the growth of NLD, GBR, and US, visible in Figure 1. Mokyr's comment appeared over a year before Banerjee and Duflo (2019) and roughy eight years before Mokyr shared the 2025 Nobel Memorial Prize in Economics with [[w:Philippe Aghion|Aghion]] and [[w:Peter Howitt (economist)|Howitt]]. Mokyr (2017, p. 255) also said that, "intellectual innovation will eventually end up being increasingly resisted by an incumbent technology; if that resistance is not overcome, technological progress may be extinguished." That resistance, he suggested, drives {{quote| the empirical regularity that no society remains at the cutting edge of technological creativity for very long<ref>Mokyr (2017, p. 260).</ref>.}} Mokyr (2017, p. 260; 1994) calls this "Cardwell's Law" after Cardwell (1972), which describes {{quote| the interaction between technological creativity and political competition ... based on the observation that technological progress encounters resistance from various groups that believe they stand to lose from innovation. These pressure groups will try to manipulate the political system to suppress successful innovation.}} [[w:Robert Michels|Michels]] (1911) called this "the [[w:Iron law of oligarchy|iron law of oligarchy]]". Further documentation of this appears in Acemoglu and Robinson (2012) and Acemoglu and Johnson (2023). Trainer (2025) insisted that the oligarchs manipulating the political system have been primarily the elites in colonial and neocolonial powers, who benefit from collaborating with the local elites discussed by Acemoglu, Robinson, and Johnson. Figure 1 begins with growth in England and Holland between 1349 and 1400, likely due to the fact that the [[w:Black Death|Black Death]] (1346-1353) killed between 30 and 60 percent of the population: Since land was not destroyed, survivors were likely able to produce more agricultural products for the same labor. Also, under the feudal system in place before the Black Death, commoners were not allowed to move without the permission of their employer, the Lord of the land where they worked. Efforts by elites to prevent commoners from moving and obtaining higher wages failed in England.<ref>Acemoglu and Robinson (2012, esp. pp. 99-101) note that surviving commoners in England were able to bargain for higher wages. The authorities passed emergency legislation in 1349 and 1351 that made it a crime to offer or accept wages higher than before the plague or to refuse to work or break an existing contract. In 1361 the penalties were increased to include imprisonment and even branding. This led to [[w:Peasants' Revolt|a major insurrection in 1381]] led by [[w:Wat Tyler|Wat Tyler]], which was put down with over 1,300 deaths. After that insurrection, authorities in England allowed commoners to demand and accept higher wages. Commoners were also allowed to move without the permission of their Lord, which had ''not'' been accepted before. By contrast, the treatment of commoners in Russia was reportedly ''worse'' than before the plague. Changes in the treatment of commoners between England and Russia seemed to vary depending on proximity to these extremes, with Austria more like Russia and France more like England.</ref> The growth in Holland from 1492 has been has been attributed to [[w:Expulsion of Jews from Spain|Jews expelled from Spain settling]] in Holland and starting new businesses. The iron law of oligarchy provides an explanation for why growth stopped in England around 1400, Holland around 1600, and why growth has been slowing in recent decades in in all the countries plotted in in Figure 1. It can also explain [[w:History of the Roman Empire|the rise and fall of the Roman Empire]], of the [[w:Maya civilization|Maya civilization]], and other societies that became very rich from allowing commoners to become entrepreneurs and then declined after previous winners changed the rules of the political economy to block further [[w:creative destruction|creative destruction]]. This includes how [[w:History of the Republic of Venice|Venice]] became "possibly the richest place in the world, with the most advanced set of inclusive economic institutions", dominating trade in the [[w:Mediterranean Sea|Mediterranean]] based on a system that encouraged entrepreneurial commoners to undertake risky trading ventures starting after 810 CE and ending when leaders made wealthy by such ventures changed the rules to end that system starting in 1297, destroying their dominance of Mediterranean trade "and converting Venice into a museum", in the words of Acemoglu and Robinson (2012, pp. 152-155). Robinson (2025) provided a related explanation for the poverty of [[w:Zimbabwe|Zimbabwe]], which has a 2026 population of roughly 17 million souls with a 2025 GDP per capital of roughly $5,400 at PPP. '[D]uring fieldwork in Zimbabwe in 2017 I had breakfast with the CEO of a subsidiary of a large multinational company. I was trying to understand the political incentives and mechanisms that had led to extractive economic institutions in the country, leading it to be substantially poorer today than it was at the time of independence in 1980. ... At some point in the conversation the CEO looked up at me and said: “I really don’t know what your problem is, it really isn’t complicated, what is it that you can’t understand? ... [T]here’s plenty of money in this country for 300 people.”' Mokyr (2017, p. 60) also mentioned "entrepreneurs of hate", who convince others to dislike some group in a way that may benefit those "entrepreneurs" and major media organizations, who promote such "entrepreneurs" to please those who control much of the money for the media. For-profit social media have been particularly effective in herding their audiences into echo chambers, reinforcing their preconceptions in ways that increase political polarization and violence and threaten democracy.<ref>See, e.g., DiResta (2026), DiResta and Keinfeld (2025), and the Wikiversity article on "[[Managing the good and the evils of social media]]", accessed 2026-10-14.</ref> When successful this protects current elites from competition that would otherwise drive [[w:creative destruction|creative destruction]], benefitting the public at large while threatening existing elites. For example, most research on immigrants suggest that on average immigrants are more entrepreneurial and no more likely to commit crime than native born,<ref>e.g., Martínez et al. (2017); Muchow and Stanishevska (2026); Ajzenman(2023).</ref> and some studies report that [[w:Sanctuary city|sanctuary jurisdiction]]s have on average higher median incomes and less crime.<ref>Martínez et al. (2017); Wong (2017). See also the Wikipedia article on "[[w:Sanctuary city|sanctuary city]]", accessed 2026-10-12.</ref> Where immigrant crime is higher, it typically result from anti-immigrant labor market restrictions or from young less-educated males with irregular status.<ref>Marie and Paolo (2024). See also the Wikipedia article on "[[w:Immigration and crime|Immigration and crime]]", accessed 2026-10-12.</ref> But major media outlets make money from featuring charismatic figures, who attract an audience without offending people who control much of the money for the media. In 2017 President [[w:Immigration policy of the first Trump administration|Trump]]'s Attorney General [[w:Jeff Sessions|Jeff Sessions]] quoted a research study that supposedly showed that sanctuary cities had more crime than cities without sanctuary policies. The authors of that study insisted that Sessions had ''misrepresented'' their work, which concluded "that sanctuary policies have no effect on crime rates, despite narratives to the contrary."<ref>Roll (2017); Gonzalez O’Brien, et al. (2017).</ref> Similarly, a 2017-09-19 report described how, "The Trump administration rejects its own study finding refugees help the economy". <ref>Swanson (2017).</ref> Just under three months later, the [[w:Union of Concerned Scientists|Union of Concerned Scientists]] (UCS) described how, "The Trump Administration Manipulates Study on Economic Benefits of Refugees in US": The [[w:Refugee Act of 1980|Refugee Act of 1980]] requires the president to set by October 1 the minimum number of refugees to be allowed into the US the following year. To support this, the [[w:Department of Health and Human Services|Department of Health and Human Services]] (HHS) produced a [[w:Cost–benefit analysis|cost-benefit analysis]] on the economic impact of refugees. A draft of this 2017 analysis concluded that “the net financial impact of refugees was positive over the 10-year period, at $63 billion.” Then [[w:Stephen Miller|Stephen Miller]] intervened to delete the benefits -- tax revenues -- generated by refugees. The resulting final report concluded that the "per-capita refugee costs for major HHS programs [averaged over 10 years] totaled $3,300.” Miller’s [[w:United States Domestic Policy Council|Domestic Policy Council]] (DPC) reportedly "maneuvered around the [[w:United States Department of State|State Department]], [[w:United States Department of Defense|Defense Department]], [[w:Joint Chiefs of Staff|Joint Chiefs of Staff]], [[w:Office of the Vice President of the United States|Office of the Vice-]]President, and [[w:Office of Management and Budget|Office of Management and Budget]] to push through his draconian cuts to the refugee cap, dismissing and rewriting the evidence as needed." A State Department official said, "The process has never been this corrupt.” UCS reported that, "Manipulating scientific evidence to fit a political narrative is a clear path toward implementing policies that are detrimental both to the US and to our allies."<ref>Union of Concerned Scientists (2018).</ref> To the extent that these comments provide a balanced summary of the research in this area, it suggests that leaders like Trump, Sessions, and Miller, to name only three, can be fairly describes as "entrepreneurs of hate". We next discuss competition in media markets before discussing immigrants further. === Diversity of ownership of the major media === Acemoglu and Johnson (2023, esp. ch. 4) describe how religious leaders in hierarchical societies prior to [[w:James VI and I|King James]] of the [[w:King James Version|King James Bible]] convinced commoners to live in poverty while giving increasing shares of what they produced, so religious and secular elites could live in opulence. Prior to King James the major media were religious organizations. In such hierarchical societies some religious leaders were entrepreneurs of hate, exhorting the faithful to fear and hate designated evil others and give increasing shares of what they produce so secular elites would "protect" them in wars against designated evil others. Acemoglu and Robinson (2012, p. 461) said, "Empowerment of society at large is difficult to coordinate and maintain without widespread information about whether there are economic and political abuses by those in power. ... Pamphlets and books informing and galvanizing people played an important role during the [[w:Glorious Revolution|Glorious Revolution]] [1688] in England, the French Revolution [1789-1799]" and in the US. ([[w:Daron Acemoglu|Acemoglu]], [[w:Simon Johnson (economist)|Johnson]] and [[w:James A. Robinson|Robinson]] shared the 2024 [[w:List of Nobel Memorial Prize laureates in Economic Sciences|Nobel Memorial Prize in Economics]].)<ref>Acemoglu and Robinson (2012) suggest three primary distinctions between the poorest and wealthiest nations. The poorest tend to be [[w:failed state|failed states]] like [[w:Somalia|Somalia]], which are poor, because of a complete lack of rule of law and continued violent conflict between warring groups. Acemoglu and Robinson divide other nation states between "extractive" and "inclusive" regimes. "Extractive" regimes tend to be poor, because elites write the rules to protect their interests against competition and the creative destruction required for economic growth at the technology frontier. "Inclusive" regimes limit monopolies and encourage enough commoners to become entrepreneurs that over time the economy tends to produce more with less, increasing the wealth of the nation.</ref> But pamphlets started challenging established power structures much earlier. The publication of [[w:Martin Luther|Martin Luther]]'s ''[[w:Ninety-five Theses|Ninety-five Theses]]'' in Latin 1517-10-31 was quickly condemned by [[w:Pope Leo X|Pope Leo X]] and his hierarchy. [[w:Johann Tetzel|Johann Tetzel]], who had been selling [[w:Indulgence|indulgence]]s, published a pamphlet calling for Luther to be burnt for [[w:heresy|heresy]]. Eight hundred copies were reportedly sent to [[w:Wittenberg|Wittenberg]] to be sold, but students seized and burned them.<ref>Hendrix (2015, p. 64), quoted from the Wikipedia article on the ''[[w:Ninety-five Theses|Ninety-five Theses]]'', accessed 2026-10-14.</ref> This led to a pamphlet war. Luther's brief German-language reply was short and easy for commoners to understand<ref>Leppin and Wengert (2015, p. 389), cited from the Wikipedia article on the ''[[w:Ninety-five Theses|Ninety-five Theses]]'', accessed 2026-10-14.</ref> and was much more popular than Tetzel's.<ref>Pettegree (2015, p. 145), cited from the Wikipedia article on the ''[[w:Ninety-five Theses|Ninety-five Theses]]'', accessed 2026-10-14.</ref> Less than twenty years later, in 1534, King [[w:Henry VIII|Henry VIII]] declared himself head of the [[w:Church of England|Church of England]], denying the authority of the Pope. To secure the support of his subjects, he drove a propaganda war that included print literature<ref>Raymond (2006), cited from the Wikipedia article on "[[w:Pamphlet war|pamphlet war]]", accessed 2026-10-15.</ref> and theatre and minstrel troupes funded by the king. The latter traveled around the land to promote the new religious practices. Henry was portrayed as the glorious king of England, bravely and heroically defending the true faith against foreign devils, especially the Pope and Catholic priests, monks, etc.<ref>Hibbert et al. (2010, p. 928), cited from the Wikipedia article on "[[w:Henry VIII|Henry VIII]]", accessed 2026-10-15.</ref> Henry died in 1547. Less than a dozen years later, the crown passed to [[w:Elizabeth I|Elizabeth I]]. Her limited personal wealth and English tradition required her to negotiate with Parliament for tax money. Acemoglu and Robinson (2012, pp. 105-111) noted that she was forced to limit the granting of monopolies and allow a class of merchants to compete in international trade. Many English merchants became wealthy, helped by the defeat of the [[w:Spanish Armada|Spanish Armada]] in 1588. When Elizabeth died in 1603, she was replaced by her cousin, King [[w:James VI and I|James VI of Scotland]], who then became also James I of England. He claimed [[w:Divine right of kings|divine right of kings]], threatening this class of merchants. To raise money without asking Parliament, he granted patents and monopolies to those who would pay. Parliament responded by passing the [[w:Statute of Monopolies|Statute of Monopolies]] on 1624-05-29. This statute repealed some past and future patents and monopolies except for novel inventions. But that Statute of Monopolies did not end the conflict. It ultimately went public, described as, "A Pamphlet War in England, 1641-1643".<ref>Washington (2018).</ref> Some pamphlets supported [[w:Charles I of England|Charles I]], who became King after James died in 1625. Others supported Parliamentarians and [[w:Levellers|Levellers]]. The latter supported radical ideas like [[w:popular sovereignty|popular sovereignty]], extended suffrage, equality before the law and religious tolerance. In 1642 war broke out between Parliament and Charles I, though many in Parliament sided with Charles -- especially those, who had received monopolies from him.<ref>Acemoglu and Robinson (2012, esp. pp. 189, 210).</ref> The war was won by the [[w:New Model Army|New Model Army]], which represented many in Parliament plus Levellers. The latter were excluded from the government that followed the [[w:Execution of Charles I of England|beheading of King Charles I]] in 1649.<ref>Foxley (2013, p. 207), cited from the Wikipedia article on [[w:Levellers|Levellers]], accessed 2026-10-15.</ref> However, that new government stopped granting as many monopolies and allowed so many commoners to become entrepreneurs that the GDP per capita began to grow at rates that are key features of Figure 1 for England and the 13 English colonies that became the US over a century later. One explanation for that economic growth was the diversity of independent publishers of pamphlets, books and newspapers. [[w:Robert W. McChesney|McChesney]] and [[w:John Nichols (journalist)|Nichols]] (2010, 2016) suggested that the US [[w:Postal Service Act|Postal Service Act]] of 1792<ref>Wikipedia "[[w:Postal Service Act|Postal Service Act]]", accessed 2023-07-11.</ref> made a major contribution to making the US what it is today. Under that act, newspapers were delivered up to 100 miles for a penny, when postage for letters was between 6 and 25 cents depending on distance.<ref>Historian, USPS (2010).</ref> McChesney and Nichols estimated that between 1840 and 1844, the US postal subsidy was a little over 0.2% of GDP.<ref name=McC-N2010>McChesney and Nichols (2010, pp. 310-311, note 88).</ref> During the first half of the nineteenth century the US had more independent newspaper publishers (more independent media organizations) per million population than at any other time or place in human history.<ref>John (1995). See also the interview with him at "[[Media concentration per Columbia History Professor Richard John]]" on Wikiversity, accessed 2026-10-12.</ref> In 1932 with official unemployment over 20 percent, the conservative argument that blames the poor for their poverty did not sell newspapers, and the US still had sufficient competition in media markets that presidential candidate Franklin Roosevelt (FDR) got a mandate to spend the money to put humans back to work while taxing the ultra-wealthy like they had never been taxed before or since and during World War II to institute wage and price controls under the [[w:Office of Price Administration|Office of Price Administration]]. The latter prevented price gouging by government contractors that had generated inflation and stifled economic growth in previous major wars (World War I, the Civil War, and the War of 1812). Instead, the US averaged 7 percent per year growth for FDR's 12 years.<ref>The "6 percent per year" growth during FDR's 12 years was computed as the average of log(GDPpc) in the Maddison Data. To be precise, that computation produced 0.0597; converting that from a log scale to percent increase gives 0.0615. Using data from [[w:MeasuringWorth|MeasuringWorth]] instead of Maddison, we get 0.783 and 0.815, respectively. More documentation of this is available in "[[Media Literacy and You/Fox, the Great Depression, the Great Recession, and our future]]" on Wikiversity, accessed 2026-10-13.</ref> One of the most interesting features of Figure 1 is the behavior of the US economy in the administrations of [[w:Herbert Hoover|Herbert Hoover]] and FDR: Under Hoover, the economy declined at 9 percent per year for 4 years, a financial disaster that created the public demand for change that gave FDR the mandate to do what he did. Economic growth under two other presidents are numerically comparable to FDR's numbers but for much shorter periods: (1) [[w: Warren G. Harding|Warren G. Harding]] averaged 7 percent annual growth for two years before he died in 1923 during the [[w:Roaring Twenties|Roaring Twenties]]. (2) [[w:Rutherford B Hayes|Rutherford B Hayes]], who took office in 1877 at the end of the [[w:Panic of 1873|Panic of 1873]], oversaw 5.5 percent growth per year for his 4 years as president. Those impressive sounding average growth figures are so short that they are not noticeably different from other minor features visible in Figure 1. FDR also dramatically reduced inequality. For over a century between about 1820 and 1932 the top 1 percent of the US population owned roughly 45 percent of net personal wealth in the US, while the bottom 90 percent owned roughly 22 percent. Shortly after FDR died, the wealth of the top 1 percent and bottom 90 percent were even at roughly 30 percent. FDR also oversaw dramatic growth in the power of organized labor. Their successes continued to reduce inequality to roughly 22 percent for the top 1 percent vs. 37 percent for the bottom 90 percent when Ronald Reagan became president. Since then, [[w:Reaganomics|Reaganomics]] have been rebuilding the [[w:Robber baron (industrialist)|Robber Baron]]s.<ref>This history of inequality is documented in [[:File:Share of US wealth 90p99.svg|Figure 7. Shares of US wealth - bottom 90 and top 1 percent, 1820-2023]] in the Wikiversity article on "[[w:Media Literacy and You/Fox, the Great Depression, the Great Recession, and our future|Media Literacy and You/Fox, the Great Depression, the Great Recession, and our future]]" in Wikiversity, accessed 2026-10-13.</ref> FDR made enemies among the wealthiest and most powerful. In a [[q:Franklin D. Roosevelt#Address at Madison Square Garden (1936)|speech in Maddison Square Garden in 1936]], FDR mentioned, "the old enemies of peace: business and financial monopoly, speculation, reckless banking, class antagonism, sectionalism, war profiteering. ... They are unanimous in their hate for me -- and I welcome their hatred." They doubtless could see that they could no longer afford as many personal servants -- maids, butlers, cooks, chauffeurs, etc. -- as they had before FDR. Broadcasting was relatively new when FDR took office. The economics of broadcasting naturally favored a very few networks, whose power concerned FDR. In 1941 his [[w:Federal Communications Commission| Federal Communications Commission]] (FCC) adopted the [[w:Mayflower doctrine|Mayflower doctrine]], which prohibited broadcasters from advocacy. When FDR died, people with power were able to get the Mayflower doctrine replaced by the more lenient [[w:Fairness Doctrine|Fairness Doctrine]], which opened the floodgates for entrepreneurs of hate, who drove the [[w:McCarthyism|McCarthy Red Scare]] in the US and many other problems since. Over time this became worse, facilitated by increasing concentration of ownership of the media. That contributed to the end of the Fairness Doctrine starting in 1985. That revocation and especially the more recent rise of for-profit social media have amplified the ability of entrepreneurs of hate to make money both for commercial social media and for themselves by disseminating lies about others, e.g., immigrants.<ref>On how for-profit social media companies promote entrepreneurs of hate, see, e.g., DiResta (2026) and the Wikiversity article on, "[[Managing the good and the evils of social media]]", accessed 2026-10-12. DiResta (2026, p. 1) described the promotion of "influencers" by for-profit social media as essentially unleashed "a war for power, influence, and reality itself—a conflict in which the ends justify any means, and one that has ripped America apart, threatened our very democracy, and consumed my own life for the past decade, leading to death threats, congressional subpoenas, lawsuits, and legal bills running into seven figures." In the interview on Wikiversity, DiResta said her legal bills to that point had exceeded $7 million.</ref> And the rate of economic growth in the US (and the Netherlands and the UK) have been slowing in recent decades, as is visible in Figure 1. Inequality expert [[w:Thomas Piketty|Piketty]] (2022, p. 139) noted, {{quote| In the United States, the national income per inhabitant rose at a rate ... of 2.2% between 1950 to 1990 when the top tax rate reached on average 72%. The top rate was then cut in half, with the announced objective of boosting growth. But in fact, growth fell by half, to 1.1% per annum between 1990 and 2020 ... . [A]ll the data at our disposal today suggest that virtually confiscatory tax rates have been an immense historical success. ...[T]he battle for equality and education ... has made economic development and human progress possible, and not the veneration of property, stability, and inequality.}} Table 1 compares Piketty's growth numbers with computations based on the data from both Maddison and [[w:MeasuringWorth|MeasuringWorth]]: The numbers are close but not exact. {| class="wikitable sortable" style="text-align: center;" |+ Table 1. US growth rate 1950 to 1990 to 2020 |- ! ! colspan=3 | US GDP per capita !! colspan=2 | Growth (% / yr) |- ! source || 1950 !! 1990 !! 2020 !! 1950 - 1990 !! 1990 - 2020 |- | Piketty || - || - || - || 2.2% || 1.1% |- | Maddison || $15,240 || $36,982 || $54,379 || 2.24% || 1.29% |- | MeasuringWorth || $16,208 || $40,191 || $61,127 || 2.30% || 1.41% |} Acemoglu and Robinson (2012, pp. 105-106) provide a historical analysis supporting Piketty's concern about inequality slowing growth. They note that in 1588, England, Spain and France {{quote| were ruled by relatively absolutist monarchs: [[w:Elizabeth I|Elizabeth I]] in England, [[w:Philip II of Spain|Philip II]] in Spain, and [[w:Henry III of France|Henry III]] in France. All were battling with assemblies of citizens -- such as the [[w:Parliament of England#Tudor era (1485–1603)|Parliament]] in England, the [[w:Cortes Generales#Habsburg rule (16th–17th centuries)|Cortes]] in Spain, and the [[w:Estates General (France)#Revival in 1560–1614|Estates-General]] in France -- that were demanding more rights and control over the monarchy. ... [T]he English Parliament and the Spanish Cortes had power over taxation, while the Estates-General did not. In Spain this mattered little, because after 1492 the Spanish Crown had a vast American empire and benefited massively from the gold and silver found there. In England ... Elizabeth I was far less financially independent, so she had to beg Parliament for more taxes. In exchange, Parliament demanded concessions, in particular restrictions of the right of Elizabeth to create monopolies. ... In Spain the Cortes lost a similar conflict.}} Consistent with Acemoglu and Johnson (2023) and the discussion above, inequality tends to increase elite control of the media, which it makes it harder for commoners to get a fair hearing of their concerns. === Media conflicts of interest in IP law === Major media have [[w:Conflict of interest|conflicts of interest]] in honestly reporting on anything involving [[w:Intellectual property|intellectual property]] (IP) law. An example of Cardwell's law, mentioned above, is the [[w:Copyright Term Extension Act|Sonny Bono Copyright Term Extension Act]], also known as the Mickey Mouse Protection Act of 1998. The copyright term on ''[[w:Steamboat Willie|Steamboat Willie]]'', the first [[w:Mickey Mouse|Mickey Mouse]] animated film that Disney distributed, was extended multiple times by the US Congress following extensive lobbying by the film industry led by [[w:The Walt Disney Company|the Walt Disney Company]]. ''Steamboat Willie'', which debuted in 1928, finally entered the [[w:Public domain in the United States|US public domain]] on January 1, 2024. Eric Eldred and others filed suit, claiming that allowing any number of retroactive extensions violated the [[w:Copyright Clause|Copyright Clause]] of the US Constitution, which says, {{quote| [the United States Congress shall have power] To promote the Progress of Science and useful Arts, by securing for limited Times to Authors and Inventors the exclusive Right to their respective Writings and Discoveries.}} In ''[[w:Eldred v. Ashcroft|Eldred v. Ashcroft]]'' (2003), the [[w:Supreme Court of the United States|US Supreme Court]] disagreed with Eldred and allowed the statute to remain in effect. However, that case drew enough public attention that copyright terms have not (yet) been extended further, and ''Steamboat Willie'' is currently in the pubic domain. [[w:Lawrence Lessig|Lessig]] (2004) ''[[w:Free Culture (book)|Free Culture]]'' describes other examples, where US copyright law seems to ''limit'' rather than "promote the Progress of Science and useful Arts". This includes a case where a 2002 freshman, Jesse Jordan, modified a search engine to make it easier for other student to find pictures, research, notes, movie clips, etc., on an [[w:intranet|intranet]] at [[w:Rensselaer Polytechnic Institute|Rensselaer Polytechnic Institute]] (RPI). The [[w:Recording Industry Association of America|Recording Industry Association of America]] (RIAA) sued Jordan and three other students for close to $100 billion for copyright violations -- six times the total profit of the film industry in 2001. Jordan had an uncle, who was an attorney, who explained that the legal bills for fighting that case would be at least $250,000. That would bankrupt his parents; they could not recover that money, even if they won. Jordan had saved $12,000 to cover part of his college expenses. RIAA accepted that and settled the case.<ref>Lessig (2004, p. 42).</ref> This is an example of a [[w:Strategic lawsuit against public participation|strategic lawsuit against public participation]] (SLAPP), which in this case rested on questionable interpretations of copyright law. The major media have conflicts of interest in honestly reporting on anything involving (a) copyright laws, because much of their income is derived from copyright royalties, or (b) anything that might offend their major accounts, including major corporations. And anti-SLAPP laws, which exist in some jurisdictions but not others,<ref>[[w:Robert Corn-Revere|Robert Corn-Revere]], Chief Counsel for the [[w:Foundation for Individual Rights and Expression|Foundation for Individual Rights and Expression]], supports anti-SLAPP laws and the "JAWBONE act", to limit the power of the government to threaten questionable actions, like the Trump administration's threats against the ABC network over a Jimmy Kimmel monolog. See the Wikiversity article on "[[Foundation for Individual Rights and Expression]]", accessed 2026-10-16.</ref> limit the ability of major corporations to abuse their power against potential challengers without such deep pockets. Similarly Graves (2023) asked, "Do copyrights and paywalls on refereed academic journals violate the US constitution?" No one expects to make money from copyright royalties for drafts submitted to refereed academic journals. They do it out of a felt need (a) to contribute to the future of humanity or (b) to build their reputation or (c) because promotions at many universities are based partly on publication records. Graves is not alone in his concern about this: Many European countries do not allow reports of research they fund to be published behind paywalls.<ref>Dimi Zagorski is concerned that copyright exceptions for scientific research are different in different European countries, and that's a problem for the Wikimedia Foundation. See the discussion of this in the Wikiversity article on "[[Wikimedia concerns with European copyright rules including AI and scientific research]]". That's the opposite of US law, which grants patents to universities, research institutions, private companies, and individuals for innovations developed under government contracts, per the Bayh-Dole Act of 1980, described by Baker (2016, p. 80).</ref> Regarding patent law, the major media have conflicts of interest in discussing proposed changes affecting major accounts, but they make little use of patent law themselves. This seems to have contributed to the claim that, "[O]ver the last four decades ... stronger patent protection seems to be associated with slower growth" in the US, according to [[w:Dean Baker|Baker]] (2016, pp. 77-78). A potentially similar phenomenon appears in a comparison of 21 different European countries in an analysis that displays "a positive relationship between anticorruption measures in a country and rate of innovation in that country as measured by patents per capita", discussed by Aghion et al. (2021, Figure 15.1b, pp. 292-293).<ref>Usher and Kim-Leffingwell (2022) documented how each member of the [[w:Institute for Nonprofit News|Institute for Nonprofit News]] (INN), relatively small nonprofits independent of major media organizations and not funded by advertising, in a US federal court district was associated with on average 1.4 additional prosecutions for political corruption per year between 2003 and 2019. They found on average 10 federal prosecutions for political corruption for each federal court district per year. Thus, 1.4 additional prosecutions per year per district represents an increase of 14 percent per member of INN.</ref> === Immigrants === We now return to discussing the initiation of economic growth following the beheading of Charles I in 1649 visible in Figure 1. Charles I was replaced by [[w:English Council of State|Lord Protector Cromwell]], who refused to authorize as many monopolies as his predecessors, as mentioned above. He also encouraged Jews and dissenting Christians to settle in England, as mentioned by Mokyr (2017, p. 53), cited above. But Mokyr is not alone in claiming that immigrants contribute to economic growth. Aghion et al. (2021, p. 266) cite Arkolakis et al. (2020) for explaining how the period 1880–1920 is noteworthy for two parallel evolutions. First, the United States caught up with England and France to become the most technologically advanced country in the world as well as the wealthiest in terms of GDP per capita. Second, the United States welcomed large numbers of immigrants, most notably from Europe. The purpose of the cited study was to demonstrate a causal connection between these two phenomena: in other words, to show that European immigration was responsible for the United States’ taking the lead in technology. ... More precisely, if the United States had completely closed its borders to all foreign immigrants in 1880, its per capita GDP would have been 30 percent lower than the level actually attained in 1920". === Industrial Revolution === There seems to be a broad consensus that the [[w:Industrial Revolution|Industrial Revolution]] began in England. However, there is less agreement on when it began. Mokyr (2017, p. 3) says, "The British Industrial Revolution of the late eighteenth century unleashed a phenomenon never before even remotely experienced by any society." He described "three cultural elements" required for "sustainable economic growth propelled by technological progress": (1, 2) The belief that it was both ''possible'' and ''desirable''. (3) Supportive policy measures had to be implemented. "[B]efore 1750 they did not produce anything like an Industrial Revolution or sustainable economic growth propelled by technological progress.<ref>Mokyr (2017, pp. 19-20).</ref> Acemoglu (2026, p. 53) similarly mentioned the eighteenth century in saying, "The breakthroughs in production technology that started in the middle of the eighteenth century in Britain unleashed the Industrial Revolution." Aghion et al. (2021, esp. pp. 29-30) offer a ''later'' date for the start of the Industrial Revolution: They report that, "European economies prior to 1820 experienced a series of phases of growth and decline. This raises a question: How did the pre-1820 period differ from the takeoff that started with the first Industrial Revolution?" Aghion et al. (2021, p. 20) describe the "first Industrial Revolution" as having been triggered by "the invention of the steam engine by James Watt in the 1770s ... . The invention of electricity set off the second Industrial Revolution, whose golden age was in the first half of the twentieth century. This second revolution originated with the invention of the [[w:Incandescent light bulb|light bulb]] by [[w:Thomas Edison|Thomas Edison]] in 1879 and of the [[w:dynamo|dynamo]] by [[w:Werner von Siemens|Werner von Siemens]] in 1866."<ref>The dynamo was independently invented by [[w:Henry Wilde (engineer)|Henry Wilde]], [[w:Charles Wheatstone|Charles Wheatstone]], [[w:Samuel Alfred Varley|Samuel Alfred Varley]] and Siemens, all about the same time, according to the Wikipedia article on [[w:Dynamo#Practical designs|dynamo]], accessed 2026-10-18.</ref> The image in Figure 1 raises questions about these claims by Mokyr, Acemogu, and Aghion et al.: It shows growth followed by stagnation in Holland but not "phases of growth and decline" with the exception of wars and [[w:Herbert Hoover|Herbert Hoover]]. It also does ''not'' show the slope (i.e., rate of growth) increasing starting around 1750 or the 1770s or 1879 or 1886. Instead it shows sustained growth in GDP per capita for England / Great Britain / the UK starting ''abruptly'' after 1649 with economic disruptions during the reign of [[w:Anne, Queen of Great Britain|Queen Anne]] (1702-1714) and the [[w:War of the Spanish Succession|War of the Spanish Succession]] (1701-1714). This was followed by relatively steady but slower growth as England was ruled by Kings [[w:House of Hanover#Monarchs of Great Britain, Ireland, and Hanover|George I, II, and III]] (1714-1820). Another dramatic increase in the slope started abruptly about 1819. Acemoglu and Johnson (2023, chapter 1) are more consistent with Figure 1. They said that, "the Industrial Revolution ... [was] rooted in institutional changes that started empowering the middling sort of English people from the sixteenth and seventeenth centuries onward." ... However, "the quality of life of a European peasant circa 1700 was not much different from that of an Egyptian peasant two thousand or seven thousand years previously." (chapter 5) They also claim that, "English national production per capita doubled from 1500 to 1700", (chapter 5), and "Growth emerged slowly in the second half of the eighteenth century as the Industrial Revolution, based on major technological breakthroughs and their application in industry, took root." (p. 45, chapter 2) Indeed, Figure 1 shows English GDP per capita doubling between 1500 and 1700, but it appears to have been fairly constant from 1500 to 1649, then roughly doubled in the next half century. [The data on England up to 1700 in Figure 1 came from Broadberry et al. (2015), which was not available when Acemoglu and Robinson were writing ''Why Nations Fail'', which appeared in 2012.] Acemoglu and Robinson (2012, ch. 11) describe the financial instability and slow growth during the reigns of Queen Anne and George I, II, and III. After the Glorious Revolution, France, Spain, the pope, and others refused to recognize [[w:William III of England|William and Mary]] (1689-1702) and then Queen Anne as heads of state of England. Instead they continued to support the claims of James II and then his son, [[w:James Francis Edward Stuart|James Francis Edward Stuart]] (the "Old Pretender", a brother of Queen Anne), after James II died in 1701. During the War of the Spanish Succession, each side tried to exploit the internal divisions of the other. Owen (1938, ch. 8) said France almost succeeded in an attempt in 1708 to [[w:Planned French invasion of Britain (1708)|land between 5 and 6 thousand soldiers in northeast Scotland]]. [[w:Louis XIV|Louis XIV]] had hoped to get the Old Protector named as King James VIII of Scotland. There were still [[w:Jacobitism|Jacobites]] in Scotland, who supported the Old Pretender and likely would have supported the invaders if they had landed. If they had succeeded, it would have created major problems for Queen Anne and her supporters. However, the Royal Navy prevented the invaders from disembarking, and they returned to France. When Queen Anne died in 1702, she was succeeded by George I. His Parliament was dominated by the Whig party, which represented the new mercantile and economic interests and had been the main organization behind the Glorious Revolution. Once in power, they were tempted to use their newly found position to prey on the rights of others but were constrained both by competing groups in Parliament, "and by the very institutions that they had fought to introduce to strengthen Parliament and to prevent the emergence of a new absolutism and the return of the Stuarts",<ref>After James II fled England, the new government worked to block [[w:Jacobitism|Jacobitism]], which attempted to restore the [[w:House of Stuart|House of Stuart]] to the thrones of Scotland and Britain. When James II died, his son became titular head of this movement, who became known as "the [[w:James Francis Edward Stuart|Old Pretender]]". After he died, he was replaced by his son, who became known as "the [[w:Charles Edward Stuart|Young Pretender]]".</ref> according to Acemoglu and Robinson (2012, pp. 302-303). The Whigs gave "[[w:Common land|common land]]" to nobles, confiscating it from commoners. Some commoners, who had traditionally shared that land, joined "Blacks", so called, because they blackened their faces when hunting on former commons or destroying the property of the nobility or extorting money from nobles or their employees. Parliament retaliated by passing the [[w:Black Act 1723|Black Act 1723]], under which someone accused of [[w:Poaching|poaching]] a [[w:Hare|hare]], for example, could under certain circumstances receive a summary judgment of guilty and then be executed without delay if apprehended later.<ref>Newman and Brown, eds. (1997, p. 58). See also the Wikipedia article on "[[w:Black Act 1723|Black Act 1723]]", accessed 2026-10-17.</ref> But not all accused were summarily executed. In 1724 "John Huntridge was accused of aiding deer stealers and abetting known Blacks, both crimes punishable by hanging. The prosecution of Huntridge came right from the top, initiated by the Regency Council of Lords Justices ... . After a trial of eight or nine hours, the jury found Huntridge innocent, partly on procedural grounds, since there were irregularities with the way the evidence had been collected. ... The jury was made up not of Huntridge’s peers, but of major landowners and gentry, who ought to have sympathized" with the noble, whose property had been attacked by Blacks allegedly aided by Huntridge.<ref>Acemoglu and Robinson (2012, pp. 304-305).</ref> These kinds of actions by the Parliaments of George I, II, and III might explain the slower growth during that period compared to that of the last half of the seventeenth century. That slower growth ended abruptly at the time of the [[w:Peterloo massacre|Peterloo Massacre]], 1819-08-16, visible in Figure 1. The growth of factories implied growth of cities, where workers lived closer together and could more easily organize. A campaign arose with multiple demands: They opposed the [[w:Corn Laws|Corn Laws]], which kept prices high for cereal grains and products made from them. They also demanded "universal [adult male] suffrage" and secret ballots in elections. A series of public meetings discussing these grievances led to one 1819-08-16, which attracted roughly 60,000. Acemoglu and Robinson (2012, p. 207) reported that, {{quote| The authorities were very nervous about the meeting, and a force of six hundred cavalry of the [[w:15th The King's Hussars|Fifteenth Hussars]] had been assembled. As the speeches began, a local magistrate decided to issue a warrant for the arrest of the speakers. As police tried to enforce the warrant, they met with the opposition of the crowd, and fighting broke out. At this point the Hussars charged the crowd. Within a few chaotic minutes, eleven people were dead and probably six hundred wounded. The ''[[w:Manchester Observer|Manchester Observer]]'' called it the Peterloo Massacre. But given the changes that had already taken place in economic and political institutions, long-run repression was not a solution in England. The Peterloo Massacre would remain an isolated incident. Following the riot, the political institutions in England gave way to the pressure, and the destabilizing threat of much wider social unrest, particularly after the 1830 revolution in France against [[w:Charles X of France|Charles X]], who had tried to restore the absolutism destroyed by the French Revolution of 1789. In 1832 the government passed the [[w:Reform Act 1832|First Reform Act]]. It enfranchised [[w:Birmingham|Birmingham]], [[w:Leeds|Leeds]], [[w:Manchester|Manchester]], and [[w:Sheffield|Sheffield]], and broadened the base of voting so that manufacturers could be represented in Parliament. ... [I]n 1846 they managed to get the hated Corn Laws repealed ... .}} The ''Manchester Observer'' closed roughly two years later after [[w:Manchester Observer#Closure by repeated prosecution|repeated prosecutions]] for [[w:Seditious libel|seditious libel]]. However, the British government placed enough other limits on its power that enough other commoners were allowed to become successful entrepreneurs that the rate of economic growth in the UK increased immediately, as indicated by the increase in the slope of the UK line in Figure 1 until World Wars I and II. == In sum == The above suggests that the rate of economic growth changes with either (a) a change in administration -- a new head of state -- or (b) some other major event like war, the Peterloo Massacre, or the [[w:Wall Street crash of 1929|Wall Street crash of 1929]]. Policy changes that can increase growth include (1) limits on grants of monopolies, (2) diverse ownership and control of the major media and (3) support for immigrants. However, correlation does not imply causation. [[w:Randomized controlled trial|Randomized controlled trial]]s have made important contributions to human understanding of economic growth, as documented by Banerjee, Duflo, and Kramer, cited above and recommended in the Wikiversity article on "[[Information is a public good: Designing experiments to improve government]]". Failing that, more analyses of data from different political jurisdictions with different media policies could be useful, as described by Neff and Pickard (2024).<ref>Graves (2025) includes an analysis of the data in Neff and Pickard (2024).</ref> == Caveats == We should be more confident of the results of randomized controlled trials, as recommended by Banerjee, Duflo, and Kremer, than of anyone interpreting the historical record, even by Nobel economists. The above attempts to accurately reflect the perspectives of all sources cited. Readers are invited to revise this article, subject to the standard Wikimedia rules of writing from a [[Wikiversity:Disclosures|neutral point of view]], [[Wikiversity:Cite sources|citing credible sources]] -- while raising other questions and concerns on the associated [[Wikiversity:FAQ|'''{{'}}“Discuss”{{'}}''' page]] -- and treating others with respect. == Notes == {{reflist}} == Bibliography == * <!-- Daron Acemoglu (2026-08-11) What Happened to Liberal Democracy?: Remaking a Politics of Shared Prosperity (Function)-->{{cite Q|Q141241291}} * <!--Daron Acemoglu and Simon Johnson (2023) Power and Progress: Our Thousand-Year Struggle Over Technology and Prosperity (PublicAffairs)-->{{cite Q|Q125292212}} * <!--Daron Acemoglu and James A. Robinson (2012) Why Nations Fail (Crown)-->{{cite Q|Q7997840}} * <!--Philippe Aghion, Céline Antonin, and Simon Bunel (2021) The Power of Creative Destruction: Economic Upheaval and the Wealth of Nations (Harvard University Press)-->{{cite Q|Q137641358}} * <!--Ajzenman, Nicolas (29 January 2023). "Migrants don't cause crime rates to increase — but false perceptions endure anyway". The Conversation.-->{{cite Q|Q141758664}} * <!--Samuel Alexander (2026-07) "Game Theory and the Dilemma of Degrowth: Why Embrace Degrowth if Other States or Societies Do Not?"-->{{cite Q|Q141968019}} * <!--Costas Arkolakis, Michael Peters, and Sun Kyoung Lee, (2020) "European Immigrants and the United States’ Rise to the Technological Frontier"-->{{cite Q|Q141678287}} * <!--Dean Baker (2016) Rigged: How globalization and the rules to the modern economy were structured to make the rich richer-->{{cite Q|Q100216001}} * <!--Abhijit V. Banerjee and Esther Duflo (2019) Good Economics for Hard Times (PublicAffairs)-->{{cite Q|Q85764011}} * <!--Jutta Bolt and Jan Luiten van Zanden (2024-09-18) "Maddison style estimates of the evolution of the world economy: A new 2023 update", Journal of Economic Surveys, 1-41-->{{cite Q|Q126723821}} * <!--S. N. Broadberry, B. Campbell, A. Klein, M. Overton and B. van Leeuwen (2015) ''British Economic Growth 1270-1870'' (Cambridge University Press)-->{{cite Q|Q57945943}} * <!--Donald S. L. Cardwell (1972) Turning Points in Western Technology (Neale Watson, Science History Pubs)-->{{cite Q|Q141795083}} * <!--Renée DiResta (2026) Invisible Rulers: The People Who Turn Lies into Reality, updated edition-->{{cite Q|Q141156783}} * <!--Renée DiResta and Rachel Kleinfeld (2025-09-10) For Expertise to Matter, Nonpartisan Institutions Need New Communications Strategies-->{{cite Q|Q141158362}} * <!--Katarzyna Dośpiał-Borysiak (2017) "Model of state management of petroleum sector -- Case of Norway-->{{cite Q|Q141774179}} * <!--Rachel Foxley (2013) The Levellers: Radical Political Thought in the English Revolution (Oxford University Press)-->{{cite Q|Q136227988}} * <!--Benjamin Gonzalez O’Brien, Loren Collingwood, Stephen Omar El-Khatib (2017-05-07) "The Politics of Refuge: Sanctuary Cities, Crime, and Undocumented Immigration-->{{cite Q|Q55932800}} * <!--Graves (2025-01-08) Defend Free Speech For All (Town Hall)-->{{cite Q|Q132855526}} * <!--Graves (2023-03-21) Do copyrights and paywalls on academic journals violate the US Constitution?-->{{cite Q|Q126116187}} * <!--Christopher Hibbert, Ben Weinreb, John Keay, Julia Keay, and Matthew Weinreb (2011-09-09) The London Encyclopaedia, 3rd ed.-->{{cite Q|Q137188993}} * <!-- Historian, USPS (2010) Postage Rates for Periodicals -->{{cite Q|Q112167199}} * <!--Richard R. John (1995) Spreading the News: The American Postal System from Franklin to Morse-->{{cite Q|Q54641943}} * <!--Volker Leppin and Timothy J. Wengert (2015) "Sources for and against the Posting of the Ninety-Five Theses"-->{{cite Q|Q141737347}} * <!--Lawrence Lessig (2004) Free Culture-->{{cite Q|Q830526}} * <!--Olivier Marie and Paolo Pinotti (2024-02-01) "Immigration and Crime: An International Perspective", Journal of Economic Perspectives. 38 (1): 181–200-->{{cite Q|Q128434874}} * <!--Martínez, Daniel E.; Martínez-Schuldt, Ricardo D.; Cantor, Guillermo (2017). " Providing Sanctuary or Fostering Crime? A Review of the Research on 'Sanctuary Cities' and Crime" . Sociology Compass. 12 e12547. doi:10.1111/soc4.12547. ISSN 1751-9020.-->{{cite Q|Q55932802}} * <!-- Robert W. McChesney; John Nichols (2010). The Death and Life of American Journalism (Bold Type Books) -->{{cite Q|Q104888067}}. * <!-- Robert W. McChesney; John Nichols (2016). People get ready-->{{cite Q|Q87619174}} * <!--Robert Michels (1911) Les Partis politiques, essai sur les tendances oligarchiques des démocraties-->{{cite Q|Q7210216}} * <!--Joel Mokyr (2017) A Culture of Growth: The Origins of the Modern Economy (Princeton University Press)-->{{cite Q|Q136118590}} * <!--Joel Mokyr (1994) "Cardwell's Law and the political economy of technological progress", Research Policy 23(5)561-574-->{{cite Q|Q139772607}} * <!--Ashley N. Muchow & Taisiia Stanishevska (2026) "From Headlines to Handcuffs: Local News Coverage of Immigrant Crime and Hispanic Arrest", Justice Quarterly, vol. 43(3) (https://doi.org/10.1080/07418825.2025.2550307)-->{{cite Q|Q141760694}} * <!--Neff and Pickard (2024) "Funding Democracy: Public Media and Democratic Health in 33 Countries"-->{{cite Q|Q131468289}} * <!--Gerald G. Newman and Leslie Ellen Brown, eds. (1997) Britain in the Hanoverian Age, 1714–1837: An Encyclopedia-->{{cite Q|Q141759196|authors = Gerald G. Newman and Leslie Ellen Brown, eds.}} * <!--John Hely Owen (1938) War at Sea Under Queen Anne 1702–1708-->{{cite Q|Q141757101}} * <!--Andrew Pettegree (2015-10-27) Brand Luther: How an Unheralded Monk Turned His Small Town into a Center of Publishing, Made Himself the Most Famous Man in Europe--and Started the Protestant Reformation-->{{cite Q|Q141761223}} * <!--Thomas Piketty (2022) A Brief History of Equality-->{{cite Q|Q139886018}} * <!--Joad Raymond (2006-11-02) Pamphlets and Pamphleteering in Early Modern Britain-->{{cite Q|Q141761849}} * <!--James A. Robinson (2025-04) "Paths to the periphery", NBER Working Paper 33671-->{{cite Q|Q141967690}} * <!--Nick Roll (2017-07-16) "Correcting Jeff Sessions: Attorney general railed against crime in sanctuary cities, citing two professors’ work. But his claims are contrary to their conclusions.-->{{cite Q|Q141735231}} * <!--J.P. Smits, E. Horlings and J.L. van Zanden (2000) Dutch GDP and its Components 1800-1913 (Groningen Growth and Development Centre)-->{{cite Q|Q137669960}} * <!--I. Sugimoto (2011) ''Economic growth of Singapore in the twentieth century: historical GDP estimates and empirical investigations'' ( World Scientific Publishing)-->{{cite Q|Q137669987}} * <!--Kelly Swanson (2017-09-17) "The Trump administration rejects its own study finding refugees help the economy-->{{cite Q|Q141900700}} * <!--Ted Trainer (2025-03) "The 2024 Nobel Prize for Economics: An example of how conventional economics misrepresents reality-->{{cite Q|Q142065447}} * <!--Union of Concerned Scientists (2019-01-11) "The Trump Administration Manipulates Study on Economic Benefits of Refugees in US"-->{{cite Q|Q141898260|author=Union of Concerned Scientists}} * <!--Nik Usher and Sanghoon Kim-Leffingwell (2022-01) How Loud Does the Watchdog Bark? A Reconsideration of Local Journalism, News Non-profits, and Political Corruption -->{{Cite Q|Q134715465}} * <!--J. L. van Zanden and B. van Leeuwen (2012), ‘Persistent but not consistent: the growth of national income in Holland 1347–1807’, Explorations in Economic History, 49, pp. 119–30-->{{cite Q|Q137670058}} * <!--Brittney Washington (2018-06-19) " A Pamphlet War in England, 1641-1643"-->{{cite Q|Q141822147}} * <!--Tom K. Wong (2017-01-26) "The Effects of Sanctuary Policies on Crime and the Economy"-->{{cite Q|Q98775875}} [[Category:Original research]] [[Category:Research]] [[Category:Economics]] [[Category:Macroeconomics]] [[Category:Gross domestic product]] [[Category:Economic growth]] [[Category:Modern Monetary Theory]] [[Category:Political science]] [[Category:Freedom and abundance]] <!-- https://en.wikiversity.org/wiki/Category_Review --> 7yq40rh1g0vzt9q0dsf4s87oirrrohx 0 332681 2844699 2843292 2026-11-06T00:58:33Z DavidMCEddy 218607 add cite Trainer 2025 2844699 wikitext text/x-wiki :''This discusses a 2026-10-30 interview with Loïck Rauscher Lauranceau,<ref name=Rauscher><!--Loïck Rauscher Lauranceau-->{{cite Q|Q141883307}}</ref> prolific author and assistant to the general secretary of Un Bout des Medias (UBDM),<ref name=ubdm><!--Un Bout des Médias-->{{cite Q|Q123906312}}</ref> about their activities. This includes a video and 29:00 mm:ss podcast excerpted from the interview. The podcast is released 2026-11-14 to the [[w:Fortnight|fortnightly]] "Media & Democracy" show<ref name=M&D><!--Media & Democracy-->{{cite Q|Q127839818}}</ref> syndicated for the [[w:Pacifica Foundation|Pacifica Radio]] Network<ref><!--Pacifica Radio Network-->{{cite Q|Q2045587}}</ref> of [[w:List of Pacifica Radio stations and affiliates|over 200 community radio stations]].''<ref><!--list of Pacifica Radio stations and affiliates-->{{cite Q|Q6593294}}</ref> :''It is posted here to invite others to contribute other perspectives, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] while [[w:Wikipedia:Citing sources|citing credible sources]]<ref name=NPOV>The rules of writing from a neutral point of view citing credible sources may not be enforced on other parts of Wikiversity. However, they can facilitate dialog between people with dramatically different beliefs.</ref> and treating others with respect.''<ref name=AGF>[[Wikiversity:Assume good faith|Wikiversity asks contributors to assume good faith]], similar to Wikipedia. The rule in [[w:Wikinews|Wikinews]] was different: Contributors there were asked to [[Wikinews:Never assume|"Don't assume things; be skeptical about everything."]] That's wise. However, we should still treat others with respect while being skeptical.</ref> <!--[[File:Media reform in France.webm|thumb|2026-10-30 interview with Loïck Rauscher Lauranceau<ref name=Rauscher/> about media reform in France.]]--> <!--[[File:Media reform in France.ogg|thumb|29:00 mm:ss excerpts from a 2026-10-30 interview with Loïck Rauscher Lauranceau<ref name=Rauscher/> about media reform in France.]]--> Loïck Rauscher Lauranceau<ref name=Rauscher/> discusses media reform in France. Rauscher is assistant to the general secretary of Un Bout des Medias (UBDM),<ref name=ubdm/> a leading French media reform organization founded by [[w:Julia Cagé|Julia Cagé]], a leading expert on the role of the media in political economy. Rauscher is interviewed by Spencer Graves.<ref><!--Spencer Graves-->{{cite Q|Q56452480}}</ref> The [[w:European Federation of Journalists|European Federation of Journalists]] published a report 2026-10-05 documenting their concerns with "Media freedom [in France] at critical juncture ahead of the 2027 presidential election".<ref>European Federation of Journalists (2026), Media Freedom Rapid Response (2026).</ref> Un Bout des Medias [a purpose for the media] is a leading French media reform organization, comparable to [[w:Free Press (advocacy group)|Free Press]] in the US,<ref>Craig Aaron, co-CEO of [[w:Free Press (organization)|Free Press (Freepress.net)]], was interviewed for [[Media reform per Freepress.net|'Media & Democracy 2021-04-29]]. [[w:Victor Pickard|Victor Pickard]] was interviewed for[[Information is a public good per communications prof Pickard|'Media & Democracy 2024-12-13]]; he was at that time chair of the board of Free Pres.</ref> the Media Reform Coalition (MRC)<ref name=MRC><!--Media Reform Coalition-->{{cite Q|Q135440829}}</ref> in the UK,<ref>British journalist and media activist Dan Hind was interviewed 2025-09-25 for [[Media Reform Coalition challenges anti-democratic media bias in the UK|'Media & Democracy' about the Media Reform Coalition]].</ref> and the [[w:Media Foundation for West Africa|Media Foundation for West Africa]] (MFWA) in west Africa.<ref>Professor [[w:Kwame Karikari (journalist)|Kwame Karikari]] was interviewed 2025-10-23 for 'Media & Democracy' about the [[w:Media Foundation for West Africa|Media Foundation for West Africa]], which he founded in 1998.</ref> Un Bout des Medias was founded by [[w:Julia Cagé|Julia Cagé]], prolific researcher, econ prof at [[w:Sciences Po|Sciences Po]], a leading political science university in Paris. She is a leading expert on the role of the media in political economy. Un Bout des Medias has published three issues they want addressed by candidates for the [[w:2027 French presidential election|2027 French presidential election]]: # Citizens have the right to know who owns the media. Today, that information is often difficult and even impossible to obtain. UBDM wants every media outlet to make it easy to obtain (1.1) the names and affiliations of any shareholders with a stake of more than 5%, (1.2) the executives of those entities, and (1.3) with parent companies, the humans who control them. # Citizens have the right to access high-quality news and information. The law should establish a minimum portion of the budget of each media organization to train and pay professional journalists. # Media organizations with more than 10 employees must give journalists equal status with investors in editorial decisions. Moreover, the staff must have the right to approve any change in shareholding that results in a change of control of the organization. In such an event, the governance body can postpone the sale for 12 months while they search for an alternative buyer.<ref>These issues are similar to the requirements of a "Nonprofit Media Organization (NMO)", proposed by Cagé (2016).</ref> ==Highlights== :''These excerpts from the interview are rushed, lightly edited for readability, and may not be in final form. The ultimate authority on what was said is the accompanying video.'' ===Importance of Un Bout des Médias=== Graves asked why Rauscher found Un Bout des Médias so compelling. He said he was studying under Cagé at Sciences Po when a famous French magazine ''[[w:Science & Vie|Science & Vie]]'' (''Science and Life'') was bought by [[w:Reworld Media|Reworld Media]], which "cast away" all their journalists, and replaced them with public media managers or public content managers. The former journalists with ''Science et Vie'' created a new direct competitor called ''{{ill|Epsiloon|fr}}''.<ref>The French-language Wikipedia article on ''[[w:Science & Vie |Science et Vie]]'' said their journalists all resigned in protest over changes in work conditions and created a new direct competitor called ''{{ill|Epsiloon|fr}}''.<ref> ''Science et Vie'' is quite famous. Rauscher said he read it as a child. Now ''Science et Vie'' continues to publish without journalists with competition from ''Epsiloon''. Around that time, Cagé asked her students what they thought should be done about ''Science et Vie'': How can we help them? Rauscher continued, {{quote| That's when I got into this fight for media independence. Firstly, through ''Science et Vie'' and just realizing how easy it was for a private owner to just buy a media and change completely the way information is produced. And once I got into that fight or that activism, you find out that it's really more complex and really deeper than what I thought. Many media are threatened, e.g. by reductions of public funding. We lack a strong law to guarantee independence for media in France. We still have an independent and pluralist media landscape in France. .. But it's getting worse. ... And that's why I'm still volunteering with this [[w:Non-governmental organization|NGO]]. ... And we helped ''Epsiloon'' with advertising, and now it exists, is well-known, and is quite strong. But the story is not finished yet, because the private owner, Reworld Media, is trying to buy the second science, newspaper in France, which is ''[[w:Sciences et Avenir|Science et Avenir]]'' -- ''Science and Future''. There used to be two major science media in France discussing science: ''Science et Vie'' and "Science et Avenir''. There are some others, but they are the two most important and two most well-known. Now Reworld Media, the owner of ''Science et Vie'' is trying to buy and do the same thing in ''Science et Avenir'', so the story is not finished yet. === English translation of "Un Bout des Médias" === Graves said he had been translating "Un Bout des Médias" as "A purpose for the media" and asked if that was a reasonable translation. Raucher said yes, but it's like a pun in French, more like "A little bit of the media". ===Leading expert on the role of the media in political economy=== Graves asked, "Who, would you say, is the leading expert worldwide on the role of the media in political economy?" Raucher replied, "There are several answers to that. My loyalty would be to Julia Cagé, because she's the president of the association [UBDM], but you also have Camille Urvoy, Nicolas Hervé,<ref>e.g., Cagé et al. (2025).</ref> and many others. The research is quite strong on this in France. === Budgets for public broadcasting in Europe=== Graves asked, "What's been happening to the budgets for public broadcasting in Europe?" Rauscher replied, {{quote| That's a major issue. For public media, public service broadcasters in France, it used to be funded in part by royalties from a local tax. ... This was not an issue when you had only television. It was changed in 2022 ... for the worse, because now it's a part of VAT, [[w:Value-added tax|value-added tax]], [[w:fr:Taxe sur la valeur ajoutée|TVA]] in French. ... Now it's part of the annual budget debate in parliament ... . That debate threatens their independence ... . It's quite probable, too, that we have an extreme right parliament, which is against public media most of the time. They even want to sell it ... . Many countries in Europe have a strong public service. In Germany it's very different, because it's a federal state, so they have many public media services. And in Italy, the media are under attack in two ways. The first is funding cuts, like we have in France. The second ... is interference by the government in the content. ... The [[w:RAI|RAI]], the public media broadcaster in Italy are more and more defending government positions instead of being completely independent. ... [[w:France Inter|France Inter]], which is the most listened to radio station in France, is stressed ... . It's still there, but it's been attacked year by year, especially by conservative right-wing and extreme-right parties ..., documented in many reports, e.g., by [[w:Reporters Without Borders|Reporters Without Borders]] or Media Freedom Rapid Response.<ref>e.g., Media Freedom Rapid Response (2026).</ref>}} === Research on public funding for media and democracy === Graves asked, "What research have you seen about the relationship between public funding for media, the level of democratization, and broadly shared economic growth?" Raucher replied, {{quote| It's been proven that the strongest democracies have the strongest the local media. ... A report from the [[w:Fondation Jean-Jaurès|Fondation Jean-Jaurès]] shows a direct link between citizen involvement in democracy, local life in municipality or local associations including voting and the amount of local media.<ref>See also Neff and Pickard (2024), analyzed further in an appendix in Graves (2025).</ref> The press is a counterpower. You cannot replace this counter power. You need this counterpower, and you need to have it completely independent. ... Local politicians and local media, local newspapers need each other. Politicians need local press to disseminate their ideas and the news about the municipality. And the local press need the local politicians and local elected people to have access to prior information and to know the news in advance. ... You don't see it in the same way on a national scale. ... Local power make gifts to local media. And a gift never goes without any return on investment. ... Local media organizations are innovating to be closer to the citizens, because they are seen as trustworthy by the local citizens, because they are closer. ... You can see it with what used to be called, [[w:Ici (radio network)|France Bleu]], which ... changed their image to [[w:Ici (radio network)|ici (radio)]], which means "here (radio)", to get closer to the citizens. There are many local alternative media outlets that defend this idea, like {{ill|Mediacités|fr}} in France. They defend the idea of being a non-Parisian and non-national media to investigate local issues. Many are working to reinvent themselves from the cold press. Another issue is monopolistic press that you can have on a local scale. For instance, in the western part of France, you have [[w:Ouest-France|Ouest-France]], a near monopoly in [[w:Brittany|Brittany]] and other regions in the western part of France.}} ===Public subsidies for media in Europe=== Graves noted that he had seen a report on Un Bout des Médias about public subsidies for media in different countries in Europe, especially [[w:Norway|Norway]], which has the highest democracy score, according to the [[w:Economist Democracy Index|Economist Democracy Index]].<ref>Nordocom (2022).</ref> Rauscher replied, {{quote| I'm not a specialist on Norway, but, for sure, in Norway, but also, I think, Denmark, Sweden and Finland -- Scandinavia -- have stronger regulations of media including public service media. However, like in all the other European countries especially, media are being threatened and attacked a lot by extreme right groups. ...}} Graves noted that an article he read said that subsidies in Norway tend to go ''not'' to the primary newspaper local, but to secondary local newspaper(s) to preserve competition in local media. Raucher agreed: "In Norway, they fund mostly the second most important local media, and not the first one." ===Postal subsidies in French history=== Graves asked about postal subsidies in French history. Rauscher replied, {{quote| The first postal domestic tariff reduction ... started just after the French revolution. ... Then it was canceled, and then it went back. ... Today, we still have these public press aids. They are really important ... Either billionaires investing in media, which is not the best, or readers. But you need to have a sufficient amount of readers giving money or advertisements, which also have some limits. A fourth way of being funded is through public aids. ... Indirect aids represented in 2024 300 million euros. Direct aid is 175 million euros ... .}} Graves added, "Yeah, I calculated that was roughly a little over a hundredth of a percent of GDP versus 5 thousandths of a percent of GDP for public media in the US until until it was zeroed out a year ago versus 15 hundreds of a percent and 25 hundreds of a percent in Europe." Raucher agreed, {{quote| That's true. You mentioned Trump: It's important for the media to survive, but these changes could also threaten their independence. That's why I talk a lot about media independence. The [[w:Cour des Comptes (France)|Cour des Comptes]], which is an independent public organ in France, ruled in 2013 that the management of these subsidies by the [[w:Ministry of Culture (France)|Culture ministry]] creates a dependence on the government, and they can choose to use these subsidies to orientate the debate and the media information that is given. These aids benefit mostly the major newspapers, major media, and the ones that are owned by billionaires.}} === Vincent Bolloré === Graves asked about [[w:Vincent Bolloré|Vincent Bolloré]]. Rauscher replied, {{quote| He's a French billionaire, really well known in France. For the past ten years, he has been buying a lot of media organizations. And not only buying them, but radically interfere with the content, denying the independence of the journalists. He basically can do almost anything he wants, because there isn't any strong rule to stop him. ... He's an extreme right, conservative, almost crusader. He bought a TV channel, which used to be slightly more liberal, than the other ones. It was [[w:CNews|i>Télé]], renamed [[w:CNews|CNews]], and is now extreme right. He also bought a radio broadcasting service, which is [[w:Europe 1|Europe 1]], and a weekly newspaper, ''[[w:Le Journal du Dimanche|Le Journal du Dimanche]]''. And he changed without any limitation the editorial line of the media, every time. It's radical but not brutal. It's not just from day one to day two. It goes slightly, step by step, ... it goes right to right to the right, to conservative, ... to a point, like now, it's clearly way ahead of any other TV channel on this extreme right content. ... It's more right-wing than any other media. And they tend to bypass the law they should respect, because on the public airwaves, the TV channels and the radio stations, you need, theoretically, to respect internal pluralism.}} Graves asked, "Required by a {{ill|Loi relative à la liberté de communication|fr|French law passed in 1986}}?" Raucher agreed. {{quote| Yeah, 1986 ... . [[w:Regulatory Authority for Audiovisual and Digital Communication|ARCOM]] is the French media policemen, I would say, tasked with enforcing pluralism. ... And they bypass this internal pluralism requirement. You're supposed to give an amount of time to the right-wing and to the left-wing ideas, roughly speaking. A 2025 report by Reporter Without Borders documented how they give the most listened to times, like, in the morning, in the afternoon, to the extreme right ideas. And the left-wing ideas will be, like, at 11 PM or 5 AM with a speech that is barely audible, giving, e.g., an hour and a half straight speech that you can barely listen to, because it's not audible. ... It sounds like exaggerating what I'm saying, but it is not. It is what it is right now.<ref>Froger et Antoine Schirer (2025).</ref> And right now, the French policeman, ARCOM, is starting to put sanctions on the Bolloré media. But it's quite weak at the moment.}} === Social media === Graves asked, "How would you describe the role of social media in amplifying a political polarization and violence?" Rauscher replied, {{quote| Un Bout des Médias is not directly into social media, social networks regulation, but it's clearly linked with media independence. For instance, if I go back to the Bolloré sphere, Bolloré galaxy, as we say in French, you can see that social medias are used almost as a weapon to threaten anybody or any administrative organ that is trying to regulate their extreme right sphere. For instance, Reporters without Borders, which is a global, well-known, NGO, went to the ''[[w:Conseil d'État|Conseil d'État]]'', which is the supreme administrative court in France, to make them rule about what is defined as internal pluralism, because ARCOM, the French media policeman, was not really regulating anything. So they asked the Administrative Supreme Court to tell the ARCOM what should be defined as internal pluralism. 512 01:05:50.870 --> 01:06:06.390 Loïck RAUSCHER LAURANCEAU: And so, Reporter Without Borders, with [[w:Christophe Deloire|Christophe Deloire]], who is, unfortunately, no longer with us, but who was fighting for this media independence, and who was the one going to this administrative Supreme Court. He was attacked by the media, the Bolloré Galaxy, Bolloré Media Galaxy. First it's on the TV channel, CNews. Then it was taken by Europe 1, which is the radio station. And then, you have an army of people on social media, attacking him. We can see it with others. Another journalist, who was criticizing the Bolloré galaxy: Same thing. They were criticized on the TV channel and on the radio station. After that, you can see a direct link with the attacks by extreme right movements. These social medias, these crusade media, are used as weapons to threaten and to attack anyone who is against, or trying to regulate what they are doing.}} === in sum === Graves asked for final words for the audience. Raucher replied, {{quote| Keep fighting for your media and for democracy. Democracy is never granted ... When you are voting for somebody, what you want is having a job, that you can pay for groceries and gasoline. You want your children, if you have any, to go to good schools and have a proper education. ... Fight for your democracy. Fight for your rights and for your media.}} ==Discussion == :''[Interested readers are invited to comment here, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] [[w:Wikipedia:Citing sources|citing credible sources]]<ref name=NPOV/> and treating others with respect.<ref name=AGF/>]'' == Notes == {{reflist}} == Bibliography == * <!--Julia Cagé (2022) Pour une télé libre : Contre Bolloré-->{{cite Q|Q120891998}} * <!--Julia Cagé (2016) Saving the Media: Capitalism, Crowdfunding, and Democracy-->{{cite Q|Q54640583}} * <!--Julia Cagé, Moritz Jonas Hengel, Nicolas Hervé, Camille Urvoy (2025) "Political Bias in the Media – Evidence from the Universe of French Broadcasts, 2002-2020"-->{{cite Q|Q142019459}} * <!--European Federation of Journalists (2026-10-05) France mission report: Media freedom at critical juncture ahead of the 2027 presidential election-->{{cite Q|Q141893657|author=European Federation of Journalists}} * <!--Arnaud Froger et Antoine Schirer (2025) Pluralisme en France : sur CNews, le grand contournement-->{{cite Q|Q142065163}} * <!--Spencer Graves (2025-01-08) "Defend Free Speech For All (Town Hall)"-->{{cite Q|Q132855526}} * <!--Media Freedom Rapid Response (2026-10-06) MFRR France Mission Report: Critical juncture for media freedom ahead of 2027 elections-->{{cite Q|Q141918295}} * <!--Neff and Pickard (2024) "Funding Democracy: Public Media and Democratic Health in 33 Countries"-->{{cite Q|Q131468289}} * <!--Nordicom (2022-10-13) Direct media subsidies to news media – a Nordic overview-->{{cite Q|Q142041473|author=[[w:Nordicom|Nordicom]]}} [[Category:Media]] [[Category:News]] [[Category:Democracy]] [[Category:Politics]] [[Category:Political economy]] [[Category:Freedom of speech]] [[Category:Finance]] [[Category:France]] [[Category:Media reform to improve democracy]] <!--list of categories https://en.wikiversity.org/wiki/Wikiversity:Category_Review [[Wikiversity:Category Review]]--> 4slftuud8cmnv9dmz1f1lyrnitefpst 2844706 2844699 2026-11-06T02:56:01Z DavidMCEddy 218607 highlights 2844706 wikitext text/x-wiki :''This discusses a 2026-10-30 interview with Loïck Rauscher Lauranceau,<ref name=Rauscher><!--Loïck Rauscher Lauranceau-->{{cite Q|Q141883307}}</ref> prolific author and assistant to the general secretary of Un Bout des Médias (UBDM),<ref name=ubdm><!--Un Bout des Médias-->{{cite Q|Q123906312}}</ref> about their activities. This includes a video and 29:00 mm:ss podcast excerpted from the interview. The podcast is released 2026-11-14 to the [[w:Fortnight|fortnightly]] "Media & Democracy" show<ref name=M&D><!--Media & Democracy-->{{cite Q|Q127839818}}</ref> syndicated for the [[w:Pacifica Foundation|Pacifica Radio]] Network<ref><!--Pacifica Radio Network-->{{cite Q|Q2045587}}</ref> of [[w:List of Pacifica Radio stations and affiliates|over 200 community radio stations]].''<ref><!--list of Pacifica Radio stations and affiliates-->{{cite Q|Q6593294}}</ref> :''It is posted here to invite others to contribute other perspectives, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] while [[w:Wikipedia:Citing sources|citing credible sources]]<ref name=NPOV>The rules of writing from a neutral point of view citing credible sources may not be enforced on other parts of Wikiversity. However, they can facilitate dialog between people with dramatically different beliefs.</ref> and treating others with respect.''<ref name=AGF>[[Wikiversity:Assume good faith|Wikiversity asks contributors to assume good faith]], similar to Wikipedia. The rule in [[w:Wikinews|Wikinews]] was different: Contributors there were asked to [[Wikinews:Never assume|"Don't assume things; be skeptical about everything."]] That's wise. However, we should still treat others with respect while being skeptical.</ref> <!--[[File:Media reform in France.webm|thumb|2026-10-30 interview with Loïck Rauscher Lauranceau<ref name=Rauscher/> about media reform in France.]]--> <!--[[File:Media reform in France.ogg|thumb|29:00 mm:ss excerpts from a 2026-10-30 interview with Loïck Rauscher Lauranceau<ref name=Rauscher/> about media reform in France.]]--> Loïck Rauscher Lauranceau<ref name=Rauscher/> discusses media reform in France. Rauscher is assistant to the general secretary of Un Bout des Médias (UBDM),<ref name=ubdm/> a leading French media reform organization founded by [[w:Julia Cagé|Julia Cagé]], a leading expert on the role of the media in political economy. Rauscher is interviewed by Spencer Graves.<ref><!--Spencer Graves-->{{cite Q|Q56452480}}</ref> The [[w:European Federation of Journalists|European Federation of Journalists]] published a report 2026-10-05 documenting their concerns with "Media freedom [in France] at critical juncture ahead of the 2027 presidential election".<ref>European Federation of Journalists (2026), Media Freedom Rapid Response (2026).</ref> Un Bout des Médias [a purpose for the media] is a leading French media reform organization, comparable to [[w:Free Press (advocacy group)|Free Press]] in the US,<ref>Craig Aaron, co-CEO of [[w:Free Press (organization)|Free Press (Freepress.net)]], was interviewed for [[Media reform per Freepress.net|'Media & Democracy 2021-04-29]]. [[w:Victor Pickard|Victor Pickard]] was interviewed for[[Information is a public good per communications prof Pickard|'Media & Democracy 2024-12-13]]; he was at that time chair of the board of Free Pres.</ref> the Media Reform Coalition (MRC)<ref name=MRC><!--Media Reform Coalition-->{{cite Q|Q135440829}}</ref> in the UK,<ref>British journalist and media activist Dan Hind was interviewed 2025-09-25 for [[Media Reform Coalition challenges anti-democratic media bias in the UK|'Media & Democracy' about the Media Reform Coalition]].</ref> and the [[w:Media Foundation for West Africa|Media Foundation for West Africa]] (MFWA) in west Africa.<ref>Professor [[w:Kwame Karikari (journalist)|Kwame Karikari]] was interviewed 2025-10-23 for 'Media & Democracy' about the [[w:Media Foundation for West Africa|Media Foundation for West Africa]], which he founded in 1998.</ref> Un Bout des Médias was founded by [[w:Julia Cagé|Julia Cagé]], prolific researcher, econ prof at [[w:Sciences Po|Sciences Po]], a leading political science university in Paris. She is a leading expert on the role of the media in political economy. Un Bout des Médias has published three issues they want addressed by candidates for the [[w:2027 French presidential election|2027 French presidential election]]: # Citizens have the right to know who owns the media. Today, that information is often difficult and even impossible to obtain. UBDM wants every media outlet to make it easy to obtain (1.1) the names and affiliations of any shareholders with a stake of more than 5%, (1.2) the executives of those entities, and (1.3) with parent companies, the humans who control them. # Citizens have the right to access high-quality news and information. The law should establish a minimum portion of the budget of each media organization to train and pay professional journalists. # Media organizations with more than 10 employees must give journalists equal status with investors in editorial decisions. Moreover, the staff must have the right to approve any change in shareholding that results in a change of control of the organization. In such an event, the governance body can postpone the sale for 12 months while they search for an alternative buyer.<ref>These issues are similar to the requirements of a "Nonprofit Media Organization (NMO)", proposed by Cagé (2016).</ref> ==Highlights== :''These excerpts from the interview are rushed, lightly edited for readability, and may not be in final form. The ultimate authority on what was said is the accompanying video.'' ===Importance of Un Bout des Médias=== Graves asked why Rauscher found Un Bout des Médias so compelling. He said he was studying under Cagé at Sciences Po when a famous French magazine ''[[w:Science & Vie|Science & Vie]]'' (''Science and Life'') was bought by [[w:Reworld Media|Reworld Media]], which "cast away" all their journalists, and replaced them with public media managers or public content managers. The former journalists with ''Science et Vie'' created a new direct competitor called ''{{ill|Epsiloon|fr}}''.<ref>The French-language Wikipedia article on ''[[w:Science & Vie|Science et Vie]]'' said their journalists all resigned in protest over changes in work conditions and created a new direct competitor called ''{{ill|Epsiloon|fr}}''.</ref> ''Science et Vie'' is quite famous. Rauscher said he read it as a child. Now ''Science et Vie'' continues to publish without journalists with competition from ''Epsiloon''. Around that time, Cagé asked her students what they thought should be done about ''Science et Vie'': How can we help them? Rauscher said, {{quote| That's when I got into this fight for media independence, starting with ''Science et Vie'', realizing how easy it was for a private owner to buy a media and change completely the way information is produced. And once I got into that fight or that activism, I learned that it's really more complex and deeper than what I had thought. Many media are threatened, e.g., by reductions in public funding. We lack a strong law to guarantee independence of media in France. We still have an independent and pluralist media landscape in France. ... But it's getting worse. ... And that's why I'm still volunteering with this [[w:Non-governmental organization|NGO]]. ... We helped ''Epsiloon'' with advertising, and now it exists, is well-known, and is quite strong. But the story is not finished yet, because the private owner, Reworld Media, is trying to buy the second science news outlet in France, which is ''[[w:Sciences et Avenir|Science et Avenir]]'' -- ''Science and Future''. There used to be two major science media in France discussing science: ''Science et Vie'' and ''Science et Avenir''. There are some others, but those two are the most important and most well-known. Now Reworld Media, the owner of ''Science et Vie'' is trying to buy and do the same thing to ''Science et Avenir'' ... .}} === English translation of "Un Bout des Médias" === Graves said he had been translating "Un Bout des Médias" as "A purpose for the media" and asked if that was a reasonable translation. Raucher said yes, but it's like a pun in French, more like "A little bit of the media". ===Leading expert on the role of the media in political economy=== Graves asked, "Who, would you say, is the leading expert worldwide on the role of the media in political economy?" Raucher replied, "There are several answers to that. My loyalty would be to Julia Cagé, because she's the president of the association [UBDM], but you also have Camille Urvoy, Nicolas Hervé,<ref>e.g., Cagé et al. (2025).</ref> and many others. The research is quite strong on this in France. === Budgets for public broadcasting in Europe=== Graves asked, "What's been happening to the budgets for public broadcasting in Europe?" Rauscher replied, {{quote| That's a major issue. For public media, public service broadcasters in France used to be funded in part by royalties from a local tax. ... This was not an issue when you had only television. It was changed in 2022 ... for the worse, because now it's a part of VAT, [[w:Value-added tax|value-added tax]], [[w:fr:Taxe sur la valeur ajoutée|TVA]] in French. ... Now it's part of the annual budget debate in parliament ... . That debate threatens their independence ... . It's quite probable, too, that we have an extreme right parliament, which is against public media most of the time. They even want to sell it ... . Many countries in Europe have a strong public service media. In Germany it's very different, because it's a federal state, so they have many public media services. In Italy, the media are under attack in two ways. The first is funding cuts, like we have in France. The second ... is interference by the government in the content. ... The [[w:RAI|RAI]], the public media broadcaster in Italy are more and more defending government positions instead of being completely independent. ... [[w:France Inter|France Inter]], which is the most listened to radio station in France, is stressed ... . It's still there, but it's been attacked year by year, especially by conservative right-wing and extreme-right parties ..., documented in many reports, e.g., by [[w:Reporters Without Borders|Reporters Without Borders]] or Media Freedom Rapid Response.<ref>e.g., Media Freedom Rapid Response (2026).</ref>}} === Research on public funding for media and democracy === Graves asked, "What research have you seen about the relationship between public funding for media, the level of democratization, and broadly shared economic growth?" Raucher replied, {{quote| It's been proven that the strongest democracies have the strongest the local media. ... A report from the [[w:Fondation Jean-Jaurès|Fondation Jean-Jaurès]] shows a direct link between citizen involvement in democracy, local life in municipalities or local associations including voting and the amount of local media.<ref>See also Neff and Pickard (2024), analyzed further in an appendix in Graves (2025).</ref> The press is a counterpower. You cannot replace this counter power. You need this counterpower, and you need to have it completely independent. ... Local politicians and local media, local newspapers need each other. Politicians need local press to disseminate their ideas and the news about the municipality. And the local press need the local politicians and local elected people to have access to prior information and to know the news in advance. ... You don't see it in the same way on a national scale. ... Local power make gifts to local media. And a gift never goes without any return on investment. ... Local media organizations are innovating to be closer to the citizens, because they are seen as trustworthy by the local citizens, because they are closer. ... You can see it with what used to be called, [[w:Ici (radio network)|France Bleu]], which ... changed their image to [[w:Ici (radio network)|ici (radio)]], which means "here (radio)", to get closer to the citizens. There are many local alternative media outlets that defend this idea, like {{ill|Mediacités|fr}} in France. They defend the idea of being a non-Parisian and non-national media to investigate local issues. Many are working to reinvent themselves from the cold press. Another issue is monopolistic press that you can have on a local scale. For instance, in the western part of France, you have [[w:Ouest-France|Ouest-France]], a near monopoly in [[w:Brittany|Brittany]] and other regions in the western part of France.}} ===Public subsidies for media in Europe=== Graves noted that he had seen a report on Un Bout des Médias about public subsidies for media in different countries in Europe, especially [[w:Norway|Norway]],<ref>Nordicom (2022).</ref> which has the highest democracy score, according to the [[w:Economist Democracy Index|Economist Democracy Index]]. Rauscher replied, {{quote| I'm not a specialist on Norway. However, for sure, in Norway, but also, I think, Denmark, Sweden and Finland -- Scandinavia -- have stronger regulations of media including public service media. However, like in all the other European countries especially, media are being threatened and attacked a lot by extreme right groups. ...}} Graves noted that an article he read said that subsidies in Norway tend to go ''not'' to the primary newspaper local, but to secondary local newspaper(s) to preserve competition in local media. Raucher agreed: "In Norway, they fund mostly the second most important local media, and not the first one." ===Postal subsidies in French history=== Graves asked about postal subsidies in French history. Rauscher replied, {{quote| The first postal domestic tariff reduction ... started just after the French revolution. ... Then it was canceled, and then it went back. ... Today, we still have these public press aids. They are really important ... Either billionaires investing in media, which is not the best, or readers. But you need to have a sufficient amount of readers giving money or advertisements, which also have some limits. A fourth way of being funded is through public aids. ... Indirect aids in 2024 in France represented 300 million euros. Direct aid is 175 million euros ... .}} Graves added, "Yeah, I calculated that was roughly a little over one hundredth of one percent of GDP in France versus 5 thousandths of a percent of GDP for public media in the US, until it was zeroed out a year ago, versus between 15 hundreds of a percent and 25 hundreds of a percent in Europe." Raucher agreed, {{quote| That's true. You mentioned Trump: It's important for the media to survive, but these changes could also threaten their independence. That's why I talk a lot about media independence. The [[w:Cour des Comptes (France)|Cour des Comptes]], which is an independent public organ in France, ruled in 2013 that the management of these subsidies by the [[w:Ministry of Culture (France)|Culture ministry]] creates a dependence on the government, and they can choose to use these subsidies to orientate the debate and the media information that is given. These aids benefit mostly the major newspapers, major media, and the ones that are owned by billionaires.}} === Vincent Bolloré === Graves asked about [[w:Vincent Bolloré|Vincent Bolloré]]. Rauscher replied, {{quote| He's a French billionaire, really well known in France. For the past ten years, he has been buying a lot of media organizations. And not only buying them, but radically interfere with the content, denying the independence of the journalists. He basically can do almost anything he wants, because there isn't any strong rule to stop him. ... He's an extreme right, conservative, almost crusader. He bought a TV channel, which used to be slightly more liberal, than the other ones. It was [[w:CNews|i>Télé]], renamed [[w:CNews|CNews]], and is now extreme right. He also bought a radio broadcasting service, which is [[w:Europe 1|Europe 1]], and a weekly newspaper, ''[[w:Le Journal du Dimanche|Le Journal du Dimanche]]''. And he changed without any limitation the editorial line of the media, every time. ... It's not just from day one to day two. It goes slightly, step by step, ... it goes right to right to the right, to conservative, ... to a point that now, ... they're more right-wing than any other media. They tend to bypass the law they should respect. On the public airwaves, the TV channels and the radio stations, you need, theoretically, to respect internal pluralism.}} Graves asked, "Required by a {{ill|French law passed in 1986|fr|Loi relative à la liberté de communication}}?" Raucher agreed. {{quote| Yeah, 1986 ... . [[w:Regulatory Authority for Audiovisual and Digital Communication|ARCOM]] is the French media policeman, tasked with enforcing pluralism. ... Bolloré's media bypass this internal pluralism requirement. They are supposed to give an amount of time to the right-wing and to the left-wing ideas, roughly speaking. A 2025 report by Reporter Without Borders documented how they give the most listened to times, like, in the morning, in the afternoon, to the extreme right ideas. And the left-wing ideas will be, like, at 11 PM or 5 AM with a speech that is barely audible, giving, e.g., an hour and a half straight speech that you can barely listen to, because it's not audible. ... This sounds like I'm exaggerating, but I'm not.<ref>Froger et Antoine Schirer (2025).</ref> Now, the French policeman, ARCOM, is starting to put sanctions on Bolloré media, but it's quite weak at the moment.}} === Social media === Graves asked, "How would you describe the role of social media in amplifying political polarization and violence?" Rauscher replied, {{quote| Un Bout des Médias is not directly into social media, social network regulation, but it's clearly linked with media independence. For instance, if I go back to the Bolloré sphere -- Bolloré galaxy, as we say in French -- you can see that social media are used almost as a weapon to threaten anybody or any administrative organ that is trying to regulate their extreme right sphere. For instance, Reporters without Borders, which is a global, well-known, NGO, went to the ''[[w:Conseil d'État|Conseil d'État]]'', which is the supreme administrative court in France, to make them rule about what is defined as internal pluralism, because ARCOM, the French media policeman, was not really regulating anything. So they asked the administrative supreme court to tell ARCOM what should be defined as internal pluralism. [[w:Christophe Deloire|Christophe Deloire]], who unfortunately is no longer with us, brought this issue to this administrative supreme court on behalf of Reporters Without Borders. He was attacked by the media, the Bolloré Media Galaxy, starting with the TV channel, CNews. Then it was taken by Europe 1, which is the radio station. And then an army of people on social media attacked him. We can see it with others. Another journalist, who was criticizing the Bolloré galaxy: Same thing. They were criticized on the TV channel and on the radio station. After that, you can see a direct link with the attacks by extreme right movements. These social media, these crusade media, are used as weapons to threaten and to attack anyone who is against, or trying to regulate what they are doing.}} === in sum === Graves asked for final words for the audience. Raucher replied, {{quote| Keep fighting for your media and for democracy. Democracy is never granted ... When you are voting for somebody, what you want is having a job, that you can pay for groceries and gasoline. You want your children, if you have any, to go to good schools and have a proper education. ... Fight for your democracy. Fight for your rights and for your media.}} ==Discussion == :''[Interested readers are invited to comment here, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] [[w:Wikipedia:Citing sources|citing credible sources]]<ref name=NPOV/> and treating others with respect.<ref name=AGF/>]'' == Notes == {{reflist}} == Bibliography == * <!--Julia Cagé (2022) Pour une télé libre : Contre Bolloré-->{{cite Q|Q120891998}} * <!--Julia Cagé (2016) Saving the Media: Capitalism, Crowdfunding, and Democracy-->{{cite Q|Q54640583}} * <!--Julia Cagé, Moritz Jonas Hengel, Nicolas Hervé, Camille Urvoy (2025) "Political Bias in the Media – Evidence from the Universe of French Broadcasts, 2002-2020"-->{{cite Q|Q142019459}} * <!--European Federation of Journalists (2026-10-05) France mission report: Media freedom at critical juncture ahead of the 2027 presidential election-->{{cite Q|Q141893657|author=European Federation of Journalists}} * <!--Arnaud Froger et Antoine Schirer (2025) Pluralisme en France : sur CNews, le grand contournement-->{{cite Q|Q142065163}} * <!--Spencer Graves (2025-01-08) "Defend Free Speech For All (Town Hall)"-->{{cite Q|Q132855526}} * <!--Media Freedom Rapid Response (2026-10-06) MFRR France Mission Report: Critical juncture for media freedom ahead of 2027 elections-->{{cite Q|Q141918295|author=Media Freedom Rapid Response}} * <!--Neff and Pickard (2024) "Funding Democracy: Public Media and Democratic Health in 33 Countries"-->{{cite Q|Q131468289}} * <!--Nordicom (2022-10-13) Direct media subsidies to news media – a Nordic overview-->{{cite Q|Q142041473|author=[[w:Nordicom|Nordicom]]}} [[Category:Media]] [[Category:News]] [[Category:Democracy]] [[Category:Politics]] [[Category:Political economy]] [[Category:Freedom of speech]] [[Category:Finance]] [[Category:France]] [[Category:Media reform to improve democracy]] <!--list of categories https://en.wikiversity.org/wiki/Wikiversity:Category_Review [[Wikiversity:Category Review]]--> og1h44nalx3ts1ikmduetg1497ylcvb 2844707 2844706 2026-11-06T03:03:26Z DavidMCEddy 218607 patch Wikidata pblm 2844707 wikitext text/x-wiki :''This discusses a 2026-10-30 interview with Loïck Rauscher Lauranceau,<ref name=Rauscher><!--Loïck Rauscher Lauranceau-->{{cite Q|Q141883307}}</ref> prolific author and assistant to the general secretary of Un Bout des Médias (UBDM),<ref name=ubdm><!--Un Bout des Médias-->{{cite Q|Q123906312}}</ref> about their activities. This includes a video and 29:00 mm:ss podcast excerpted from the interview. The podcast is released 2026-11-14 to the [[w:Fortnight|fortnightly]] "Media & Democracy" show<ref name=M&D><!--Media & Democracy-->{{cite Q|Q127839818}}</ref> syndicated for the [[w:Pacifica Foundation|Pacifica Radio]] Network<ref><!--Pacifica Radio Network-->{{cite Q|Q2045587}}</ref> of [[w:List of Pacifica Radio stations and affiliates|over 200 community radio stations]].''<ref><!--list of Pacifica Radio stations and affiliates-->{{cite Q|Q6593294}}</ref> :''It is posted here to invite others to contribute other perspectives, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] while [[w:Wikipedia:Citing sources|citing credible sources]]<ref name=NPOV>The rules of writing from a neutral point of view citing credible sources may not be enforced on other parts of Wikiversity. However, they can facilitate dialog between people with dramatically different beliefs.</ref> and treating others with respect.''<ref name=AGF>[[Wikiversity:Assume good faith|Wikiversity asks contributors to assume good faith]], similar to Wikipedia. The rule in [[w:Wikinews|Wikinews]] was different: Contributors there were asked to [[Wikinews:Never assume|"Don't assume things; be skeptical about everything."]] That's wise. However, we should still treat others with respect while being skeptical.</ref> <!--[[File:Media reform in France.webm|thumb|2026-10-30 interview with Loïck Rauscher Lauranceau<ref name=Rauscher/> about media reform in France.]]--> <!--[[File:Media reform in France.ogg|thumb|29:00 mm:ss excerpts from a 2026-10-30 interview with Loïck Rauscher Lauranceau<ref name=Rauscher/> about media reform in France.]]--> Loïck Rauscher Lauranceau<ref name=Rauscher/> discusses media reform in France. Rauscher is assistant to the general secretary of Un Bout des Médias (UBDM),<ref name=ubdm/> a leading French media reform organization founded by [[w:Julia Cagé|Julia Cagé]], a leading expert on the role of the media in political economy. Rauscher is interviewed by Spencer Graves.<ref><!--Spencer Graves-->{{cite Q|Q56452480}}</ref> The [[w:European Federation of Journalists|European Federation of Journalists]] published a report 2026-10-05 documenting their concerns with "Media freedom [in France] at critical juncture ahead of the 2027 presidential election".<ref>European Federation of Journalists (2026), Media Freedom Rapid Response (2026).</ref> Un Bout des Médias [a purpose for the media] is a leading French media reform organization, comparable to [[w:Free Press (advocacy group)|Free Press]] in the US,<ref>Craig Aaron, co-CEO of [[w:Free Press (organization)|Free Press (Freepress.net)]], was interviewed for [[Media reform per Freepress.net|'Media & Democracy 2021-04-29]]. [[w:Victor Pickard|Victor Pickard]] was interviewed for[[Information is a public good per communications prof Pickard|'Media & Democracy 2024-12-13]]; he was at that time chair of the board of Free Pres.</ref> the Media Reform Coalition (MRC)<ref name=MRC><!--Media Reform Coalition-->{{cite Q|Q135440829}}</ref> in the UK,<ref>British journalist and media activist Dan Hind was interviewed 2025-09-25 for [[Media Reform Coalition challenges anti-democratic media bias in the UK|'Media & Democracy' about the Media Reform Coalition]].</ref> and the [[w:Media Foundation for West Africa|Media Foundation for West Africa]] (MFWA) in west Africa.<ref>Professor [[w:Kwame Karikari (journalist)|Kwame Karikari]] was interviewed 2025-10-23 for 'Media & Democracy' about the [[w:Media Foundation for West Africa|Media Foundation for West Africa]], which he founded in 1998.</ref> Un Bout des Médias was founded by [[w:Julia Cagé|Julia Cagé]], prolific researcher, econ prof at [[w:Sciences Po|Sciences Po]], a leading political science university in Paris. She is a leading expert on the role of the media in political economy. Un Bout des Médias has published three issues they want addressed by candidates for the [[w:2027 French presidential election|2027 French presidential election]]: # Citizens have the right to know who owns the media. Today, that information is often difficult and even impossible to obtain. UBDM wants every media outlet to make it easy to obtain (1.1) the names and affiliations of any shareholders with a stake of more than 5%, (1.2) the executives of those entities, and (1.3) with parent companies, the humans who control them. # Citizens have the right to access high-quality news and information. The law should establish a minimum portion of the budget of each media organization to train and pay professional journalists. # Media organizations with more than 10 employees must give journalists equal status with investors in editorial decisions. Moreover, the staff must have the right to approve any change in shareholding that results in a change of control of the organization. In such an event, the governance body can postpone the sale for 12 months while they search for an alternative buyer.<ref>These issues are similar to the requirements of a "Nonprofit Media Organization (NMO)", proposed by Cagé (2016).</ref> ==Highlights== :''These excerpts from the interview are rushed, lightly edited for readability, and may not be in final form. The ultimate authority on what was said is the accompanying video.'' ===Importance of Un Bout des Médias=== Graves asked why Rauscher found Un Bout des Médias so compelling. He said he was studying under Cagé at Sciences Po when a famous French magazine ''[[w:Science & Vie|Science & Vie]]'' (''Science and Life'') was bought by [[w:Reworld Media|Reworld Media]], which "cast away" all their journalists, and replaced them with public media managers or public content managers. The former journalists with ''Science et Vie'' created a new direct competitor called ''{{ill|Epsiloon|fr}}''.<ref>The French-language Wikipedia article on ''[[w:Science & Vie|Science et Vie]]'' said their journalists all resigned in protest over changes in work conditions and created a new direct competitor called ''{{ill|Epsiloon|fr}}''.</ref> ''Science et Vie'' is quite famous. Rauscher said he read it as a child. Now ''Science et Vie'' continues to publish without journalists with competition from ''Epsiloon''. Around that time, Cagé asked her students what they thought should be done about ''Science et Vie'': How can we help them? Rauscher said, {{quote| That's when I got into this fight for media independence, starting with ''Science et Vie'', realizing how easy it was for a private owner to buy a media and change completely the way information is produced. And once I got into that fight or that activism, I learned that it's really more complex and deeper than what I had thought. Many media are threatened, e.g., by reductions in public funding. We lack a strong law to guarantee independence of media in France. We still have an independent and pluralist media landscape in France. ... But it's getting worse. ... And that's why I'm still volunteering with this [[w:Non-governmental organization|NGO]]. ... We helped ''Epsiloon'' with advertising, and now it exists, is well-known, and is quite strong. But the story is not finished yet, because the private owner, Reworld Media, is trying to buy the second science news outlet in France, which is ''[[w:Sciences et Avenir|Science et Avenir]]'' -- ''Science and Future''. There used to be two major science media in France discussing science: ''Science et Vie'' and ''Science et Avenir''. There are some others, but those two are the most important and most well-known. Now Reworld Media, the owner of ''Science et Vie'' is trying to buy and do the same thing to ''Science et Avenir'' ... .}} === English translation of "Un Bout des Médias" === Graves said he had been translating "Un Bout des Médias" as "A purpose for the media" and asked if that was a reasonable translation. Raucher said yes, but it's like a pun in French, more like "A little bit of the media". ===Leading expert on the role of the media in political economy=== Graves asked, "Who, would you say, is the leading expert worldwide on the role of the media in political economy?" Raucher replied, "There are several answers to that. My loyalty would be to Julia Cagé, because she's the president of the association [UBDM], but you also have Camille Urvoy, Nicolas Hervé,<ref>e.g., Cagé et al. (2025).</ref> and many others. The research is quite strong on this in France. === Budgets for public broadcasting in Europe=== Graves asked, "What's been happening to the budgets for public broadcasting in Europe?" Rauscher replied, {{quote| That's a major issue. For public media, public service broadcasters in France used to be funded in part by royalties from a local tax. ... This was not an issue when you had only television. It was changed in 2022 ... for the worse, because now it's a part of VAT, [[w:Value-added tax|value-added tax]], [[w:fr:Taxe sur la valeur ajoutée|TVA]] in French. ... Now it's part of the annual budget debate in parliament ... . That debate threatens their independence ... . It's quite probable, too, that we have an extreme right parliament, which is against public media most of the time. They even want to sell it ... . Many countries in Europe have a strong public service media. In Germany it's very different, because it's a federal state, so they have many public media services. In Italy, the media are under attack in two ways. The first is funding cuts, like we have in France. The second ... is interference by the government in the content. ... The [[w:RAI|RAI]], the public media broadcaster in Italy are more and more defending government positions instead of being completely independent. ... [[w:France Inter|France Inter]], which is the most listened to radio station in France, is stressed ... . It's still there, but it's been attacked year by year, especially by conservative right-wing and extreme-right parties ..., documented in many reports, e.g., by [[w:Reporters Without Borders|Reporters Without Borders]] or Media Freedom Rapid Response.<ref>e.g., Media Freedom Rapid Response (2026).</ref>}} === Research on public funding for media and democracy === Graves asked, "What research have you seen about the relationship between public funding for media, the level of democratization, and broadly shared economic growth?" Raucher replied, {{quote| It's been proven that the strongest democracies have the strongest the local media. ... A report from the [[w:Fondation Jean-Jaurès|Fondation Jean-Jaurès]] shows a direct link between citizen involvement in democracy, local life in municipalities or local associations including voting and the amount of local media.<ref>See also Neff and Pickard (2024), analyzed further in an appendix in Graves (2025).</ref> The press is a counterpower. You cannot replace this counter power. You need this counterpower, and you need to have it completely independent. ... Local politicians and local media, local newspapers need each other. Politicians need local press to disseminate their ideas and the news about the municipality. And the local press need the local politicians and local elected people to have access to prior information and to know the news in advance. ... You don't see it in the same way on a national scale. ... Local power make gifts to local media. And a gift never goes without any return on investment. ... Local media organizations are innovating to be closer to the citizens, because they are seen as trustworthy by the local citizens, because they are closer. ... You can see it with what used to be called, [[w:Ici (radio network)|France Bleu]], which ... changed their image to [[w:Ici (radio network)|ici (radio)]], which means "here (radio)", to get closer to the citizens. There are many local alternative media outlets that defend this idea, like {{ill|Mediacités|fr}} in France. They defend the idea of being a non-Parisian and non-national media to investigate local issues. Many are working to reinvent themselves from the cold press. Another issue is monopolistic press that you can have on a local scale. For instance, in the western part of France, you have [[w:Ouest-France|Ouest-France]], a near monopoly in [[w:Brittany|Brittany]] and other regions in the western part of France.}} ===Public subsidies for media in Europe=== Graves noted that he had seen a report on Un Bout des Médias about public subsidies for media in different countries in Europe, especially [[w:Norway|Norway]],<ref>Nordicom (2022).</ref> which has the highest democracy score, according to the [[w:Economist Democracy Index|Economist Democracy Index]]. Rauscher replied, {{quote| I'm not a specialist on Norway. However, for sure, in Norway, but also, I think, Denmark, Sweden and Finland -- Scandinavia -- have stronger regulations of media including public service media. However, like in all the other European countries especially, media are being threatened and attacked a lot by extreme right groups. ...}} Graves noted that an article he read said that subsidies in Norway tend to go ''not'' to the primary newspaper local, but to secondary local newspaper(s) to preserve competition in local media. Raucher agreed: "In Norway, they fund mostly the second most important local media, and not the first one." ===Postal subsidies in French history=== Graves asked about postal subsidies in French history. Rauscher replied, {{quote| The first postal domestic tariff reduction ... started just after the French revolution. ... Then it was canceled, and then it went back. ... Today, we still have these public press aids. They are really important ... Either billionaires investing in media, which is not the best, or readers. But you need to have a sufficient amount of readers giving money or advertisements, which also have some limits. A fourth way of being funded is through public aids. ... Indirect aids in 2024 in France represented 300 million euros. Direct aid is 175 million euros ... .}} Graves added, "Yeah, I calculated that was roughly a little over one hundredth of one percent of GDP in France versus 5 thousandths of a percent of GDP for public media in the US, until it was zeroed out a year ago, versus between 15 hundreds of a percent and 25 hundreds of a percent in Europe." Raucher agreed, {{quote| That's true. You mentioned Trump: It's important for the media to survive, but these changes could also threaten their independence. That's why I talk a lot about media independence. The [[w:Cour des Comptes (France)|Cour des Comptes]], which is an independent public organ in France, ruled in 2013 that the management of these subsidies by the [[w:Ministry of Culture (France)|Culture ministry]] creates a dependence on the government, and they can choose to use these subsidies to orientate the debate and the media information that is given. These aids benefit mostly the major newspapers, major media, and the ones that are owned by billionaires.}} === Vincent Bolloré === Graves asked about [[w:Vincent Bolloré|Vincent Bolloré]]. Rauscher replied, {{quote| He's a French billionaire, really well known in France. For the past ten years, he has been buying a lot of media organizations. And not only buying them, but radically interfere with the content, denying the independence of the journalists. He basically can do almost anything he wants, because there isn't any strong rule to stop him. ... He's an extreme right, conservative, almost crusader. He bought a TV channel, which used to be slightly more liberal, than the other ones. It was [[w:CNews|i>Télé]], renamed [[w:CNews|CNews]], and is now extreme right. He also bought a radio broadcasting service, which is [[w:Europe 1|Europe 1]], and a weekly newspaper, ''[[w:Le Journal du Dimanche|Le Journal du Dimanche]]''. And he changed without any limitation the editorial line of the media, every time. ... It's not just from day one to day two. It goes slightly, step by step, ... it goes right to right to the right, to conservative, ... to a point that now, ... they're more right-wing than any other media. They tend to bypass the law they should respect. On the public airwaves, the TV channels and the radio stations, you need, theoretically, to respect internal pluralism.}} Graves asked, "Required by a {{ill|French law passed in 1986|fr|Loi relative à la liberté de communication}}?" Raucher agreed. {{quote| Yeah, 1986 ... . [[w:Regulatory Authority for Audiovisual and Digital Communication|ARCOM]] is the French media policeman, tasked with enforcing pluralism. ... Bolloré's media bypass this internal pluralism requirement. They are supposed to give an amount of time to the right-wing and to the left-wing ideas, roughly speaking. A 2025 report by Reporter Without Borders documented how they give the most listened to times, like, in the morning, in the afternoon, to the extreme right ideas. And the left-wing ideas will be, like, at 11 PM or 5 AM with a speech that is barely audible, giving, e.g., an hour and a half straight speech that you can barely listen to, because it's not audible. ... This sounds like I'm exaggerating, but I'm not.<ref>Froger et Antoine Schirer (2025).</ref> Now, the French policeman, ARCOM, is starting to put sanctions on Bolloré media, but it's quite weak at the moment.}} === Social media === Graves asked, "How would you describe the role of social media in amplifying political polarization and violence?" Rauscher replied, {{quote| Un Bout des Médias is not directly into social media, social network regulation, but it's clearly linked with media independence. For instance, if I go back to the Bolloré sphere -- Bolloré galaxy, as we say in French -- you can see that social media are used almost as a weapon to threaten anybody or any administrative organ that is trying to regulate their extreme right sphere. For instance, Reporters without Borders, which is a global, well-known, NGO, went to the ''[[w:Conseil d'État|Conseil d'État]]'', which is the supreme administrative court in France, to make them rule about what is defined as internal pluralism, because ARCOM, the French media policeman, was not really regulating anything. So they asked the administrative supreme court to tell ARCOM what should be defined as internal pluralism. [[w:Christophe Deloire|Christophe Deloire]], who unfortunately is no longer with us, brought this issue to this administrative supreme court on behalf of Reporters Without Borders. He was attacked by the media, the Bolloré Media Galaxy, starting with the TV channel, CNews. Then it was taken by Europe 1, which is the radio station. And then an army of people on social media attacked him. We can see it with others. Another journalist, who was criticizing the Bolloré galaxy: Same thing. They were criticized on the TV channel and on the radio station. After that, you can see a direct link with the attacks by extreme right movements. These social media, these crusade media, are used as weapons to threaten and to attack anyone who is against, or trying to regulate what they are doing.}} === in sum === Graves asked for final words for the audience. Raucher replied, {{quote| Keep fighting for your media and for democracy. Democracy is never granted ... When you are voting for somebody, what you want is having a job, that you can pay for groceries and gasoline. You want your children, if you have any, to go to good schools and have a proper education. ... Fight for your democracy. Fight for your rights and for your media.}} ==Discussion == :''[Interested readers are invited to comment here, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] [[w:Wikipedia:Citing sources|citing credible sources]]<ref name=NPOV/> and treating others with respect.<ref name=AGF/>]'' == Notes == {{reflist}} == Bibliography == * <!--Julia Cagé (2022) Pour une télé libre : Contre Bolloré-->{{cite Q|Q120891998}} * <!--Julia Cagé (2016) Saving the Media: Capitalism, Crowdfunding, and Democracy-->{{cite Q|Q54640583}} * <!--Julia Cagé, Moritz Jonas Hengel, Nicolas Hervé, Camille Urvoy (2025) "Political Bias in the Media – Evidence from the Universe of French Broadcasts, 2002-2020"-->{{cite Q|Q142019459}} * <!--European Federation of Journalists (2026-10-05) France mission report: Media freedom at critical juncture ahead of the 2027 presidential election-->{{cite Q|Q141893657|author=European Federation of Journalists}} * <!--Arnaud Froger et Antoine Schirer (2025) Pluralisme en France : sur CNews, le grand contournement-->{{cite Q|Q142065163|authors=Arnaud Froger and Antoine Schirer}}<!--I must specify Arnaud Froger directly, because it displays as his QID otherwise. 2026-11-05--> * <!--Spencer Graves (2025-01-08) "Defend Free Speech For All (Town Hall)"-->{{cite Q|Q132855526}} * <!--Media Freedom Rapid Response (2026-10-06) MFRR France Mission Report: Critical juncture for media freedom ahead of 2027 elections-->{{cite Q|Q141918295|author=Media Freedom Rapid Response}} * <!--Neff and Pickard (2024) "Funding Democracy: Public Media and Democratic Health in 33 Countries"-->{{cite Q|Q131468289}} * <!--Nordicom (2022-10-13) Direct media subsidies to news media – a Nordic overview-->{{cite Q|Q142041473|author=[[w:Nordicom|Nordicom]]}} [[Category:Media]] [[Category:News]] [[Category:Democracy]] [[Category:Politics]] [[Category:Political economy]] [[Category:Freedom of speech]] [[Category:Finance]] [[Category:France]] [[Category:Media reform to improve democracy]] <!--list of categories https://en.wikiversity.org/wiki/Wikiversity:Category_Review [[Wikiversity:Category Review]]--> dmixr3cgfi0nd612uyq21f48zss6qy7 2 332696 2844663 2844535 2026-11-05T19:52:55Z Atcovi 276019 /* Ibn Qayyim al-Jawziyya */ +1 2844663 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. *'Umar ibn al-Khattab, may Allah be pleased with him, said, "The best days we ever lived were by virtue of patience, and if patience were to take the shape of a man, he would be a noble and generous 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 54. == 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]] dtvmx983ffvgueh9j6mfi9aijm4x35w 2 332764 2844705 2844114 2026-11-06T02:54:34Z ~2026-58414-52 3113736 2844705 wikitext text/x-wiki {{Research project}} {{AI-generated}} <!-- Project BOG Research Atlas — editorial revision, 3 November 2026. Based on the author's supplied existing Wikiversity source. Human review before publication. DOIs are author-supplied primary records; the Cycle Clocks release was reported completed by its author, but live DOI resolving has not been independently attested in this editing session. --> '''Project BOG — Research Atlas''' is a collaborative learning and original-research map by Bogdan Sergienko, with mathematical, computational and editorial assistance from CCAI KAI (ChatGPT). It connects the historical preprints, experimental exercises, six independent research entrances, one shared research-return cycle, and public software instruments. '''What this page is:''' a navigable educational research project, with reproducible finite models, open questions and source trails. '''What it is not:''' a peer-reviewed verification of the author's proposed universal law, a claim of physical teleportation, or evidence that AI research instruments communicate or run continuously on their own. '''Disclosure:''' Bogdan Sergienko is the author of the artistic and research propositions and maintains this userspace atlas. The page has substantial AI-assisted drafting and requires human review of facts and citations. Linked Project BOG deposits are primarily author-published sources; they are not independent scientific confirmations. Readers may revise, disagree and test the claims. == Choose your entrance == {| class="wikitable" style="width:100%;" ! Entrance !! Start here !! What you can do !! Return |- | '''Explore''' || '''[https://bogsergienko.github.io/bog-sense-atlas/world-entry-iii/index.html Enter the public six-cycle research canvas]''' || Start from the authorial question, inspect the six paths and find the research anchors. || [[#The shared cycle and its six entrances|Six entrances]] |- | '''Play and verify''' || '''[https://bogsergienko.github.io/bog-sense-atlas/cycle-clocks/index.html Open CYCLE CLOCKS]''' || Run TICK A / TOCK B, independent and joint events, and the 6-versus-12 example. || [[#CYCLE CLOCKS — a public event-time instrument|Cycle Clocks]] |- | '''Read''' || [https://doi.org/10.5281/zenodo.23903785 World Entry III preprint] || Compare the conceptual canvas with a reproducible finite theorem. || [[#Research lineage — the original papers and instruments|Research lineage]] |- | '''Test''' || [[User:BogSergienko/Project BOG Research Atlas/Experiments|EXPERIMENTS laboratory]] || Follow replicable exercises, negative controls and failure cases. || [[#Learning laboratory — experiments and controls|Experiment map]] |- | '''Challenge''' || [[#Open problems and returning results|Open questions]] || Offer a counterexample with assumptions, source, sequence and result. || <code>BOG:WORLD:SHARED:RETURN</code> |} '''Human exit:''' Participation is optional. No reader must accept the author's interpretation to test the mathematical examples or leave the project. == Research lineage — the original papers and instruments == The papers below are '''original Project BOG preprints''', not independent corroborations. The historical '''Research Series''' numbering is distinct from the later term '''third integrated world entry'''. {| class="wikitable sortable" ! Research trail !! Theme / scope !! Primary public record |- | Series I || Universe of Linked Temporal Cycles — proposed layered-cycle picture || [https://doi.org/10.5281/zenodo.23807064 Zenodo DOI 10.5281/zenodo.23807064] |- | Series II || Temporal Jumps and Causal Return — finite-state return predicates and causal witnesses || [https://doi.org/10.5281/zenodo.23809741 Zenodo DOI 10.5281/zenodo.23809741] |- | Series III || The Canvas as a Computational Universe — computational state canvases || [https://doi.org/10.5281/zenodo.23811750 Zenodo DOI 10.5281/zenodo.23811750] |- | Series IV || Information as a Reconstructible State — reconstruction questions || [https://doi.org/10.5281/zenodo.23813163 Zenodo DOI 10.5281/zenodo.23813163] |- | Series V || Higher-Order Harmony Beyond Local Laws — proposed finite-model relationships || [https://doi.org/10.5281/zenodo.23813799 Zenodo DOI 10.5281/zenodo.23813799] |- | Series VI || The Absolute Cycle: Time Without a Clock — authorial axioms and tests || [https://doi.org/10.5281/zenodo.23866116 Zenodo DOI 10.5281/zenodo.23866116] |- | Series VII || TELEPORT DEMO — reconstructible digital file experiment; not transport of matter || [https://doi.org/10.5281/zenodo.23873391 Zenodo DOI 10.5281/zenodo.23873391] |- | '''World Entry III''' || '''Fundamental Law of the Universe: Six Cyclic Entrances under One Shared Research Cycle''' — common entrance, not Series III || '''[https://doi.org/10.5281/zenodo.23903785 Preprint DOI 10.5281/zenodo.23903785]''' |- | '''CYCLE CLOCKS v1.1''' || Reversible Event Time and Two-Cycle Recurrence — standalone public software tool || '''[https://doi.org/10.5281/zenodo.23907698 Software DOI 10.5281/zenodo.23907698]''' |} '''Source-verification note:''' The Cycle Clocks DOI above was given to the author as a reserved identifier and the author subsequently reported completing publication. The public DOI resolution could not be independently checked from this drafting environment; verify the landing page before treating its registration and related metadata as externally confirmed. === Paths between these records === * The historical Series I–VII are earlier research trails. They should be read in their actual version and publication order, not treated as demonstrations of a single physical conclusion. * [https://doi.org/10.5281/zenodo.23903785 World Entry III] integrates one human entrance, five distinct historically attributed AI-instrument questions and their enclosing Bogdan–Kai research cycle. * [https://doi.org/10.5281/zenodo.23907698 CYCLE CLOCKS] is a '''software supplement''' to World Entry III. On the '''earlier preprint''', the appropriate inverse metadata relationship is '''Is supplemented by''' → the Cycle Clocks DOI; on the '''clock software record''', it is '''Is supplement to''' → the earlier preprint DOI. Verify both relationships in the live Zenodo records before claiming metadata reciprocity. * [https://github.com/BogSergienko/bog-sense-atlas Project BOG Sense Atlas repository] holds the interactive entry and open reproducibility materials; the DOI archives specific published releases. == The authorial canvas and what can be tested == Bogdan Sergienko's original Russian phrase is preserved in its authorial form: :'''ВСЁ УЖЕ ЕСТЬ=Рождение нового=Всё одновременно''' The later artistic extension is rendered in English as: :'''Everything Already Exists + Birth of the New + Everything Simultaneously = The Logic of This Universe = Harmony.''' The author calls a central motif of opposing flows, a dark knot, interwoven loops and an expanded connecting trunk the '''Absolute Black Hole'''. It is a conceptual drawing and speculative model of relation and recurrence, '''not a physical observation or a validated black-hole theory'''. The plus and equals signs in the philosophical statement have not been established as mathematical operators or a physically tested equation. The finite mathematical examples below stand on their explicitly defined transition rules alone. == The shared cycle and its six entrances == The '''enclosing collaborative research cycle''' belongs to the ongoing joint Bogdan–Kai enquiry, with distinct author and computational roles. It is a documented way to circulate questions and checked answers, '''not''' a permanently operating multi-model agent. :'''SOURCE → CHOOSE → FORMALIZE → TEST / NEGATIVE CONTROL → CONFRONT → ATTRIBUTED RETURN → RECHOOSE → CONNECT → SOURCE''' Shared semantic anchors: <code>BOG:WORLD:SHARED:OPEN</code> · <code>BOG:WORLD:SHARED:RETURN</code> · <code>BOG:WORLD:SHARED:CLOSE</code>. Each of the six entrances carries <code>BOG:CYCLE:ID:OPEN</code>, <code>BOG:CYCLE:ID:RETURN</code>, and <code>BOG:CYCLE:ID:CLOSE</code>. The AI instrument labels preserve attribution to historical model sessions, not endorsements or simultaneous live model calls. {| class="wikitable" ! ID / research instrument !! Entry question !! Proposed discriminating test !! Return anchor |- | '''HUMAN''' — reader / artist || What survives of the original picture after a genuine counterexample? || Separate intuition from calculation; allow exit and revision. || <code>BOG:CYCLE:HUMAN:RETURN</code> |- | '''QUARK''' — Grok || Is a novelty an observation of an existing state, or a change to admissible continuations? || Test projection-only change against reachable-future change. || <code>BOG:CYCLE:QUARK:RETURN</code> |- | '''ATANOR''' — Gemini || Does proposed parent synchronization go beyond ordinary least-common-multiple recurrence? || Compare against an explicit LCM baseline. || <code>BOG:CYCLE:ATANOR:RETURN</code> |- | '''WITNESS''' — Claude || What makes a causal encounter different from geometric coincidence? || Matched controls for crossings without causal coupling. || <code>BOG:CYCLE:WITNESS:RETURN</code> |- | '''SLEDA_D''' — DeepSeek || Can a reversible complete state coexist with a lasting partial observation? || Specify full reversible state and what is actually inverted. || <code>BOG:CYCLE:SLEDA_D:RETURN</code> |- | '''OTKLIK''' — Qwen || Which probes distinguish relational structure from a renaming of states? || Check invariance under relabelling and equivalent observation. || <code>BOG:CYCLE:OTKLIK:RETURN</code> |} '''Directed links:''' Human → Quark, Witness, Atanor; Quark ↔ Otklik; Atanor ↔ Sleda-D; Witness → Quark, Atanor; Sleda-D → Witness; Otklik → Atanor — '''11 research-question transitions'''. These represent questions and comparison tasks, not automatic inter-model messages. [https://github.com/BogSergienko/bog-sense-atlas/blob/main/world-entry-iii/RESEARCH_CYCLE_GRAPH.json Inspect the machine-readable graph]. Witness did not participate in the final Round 07 discussion; absence should not be converted into agreement. == CYCLE CLOCKS — a public event-time instrument == '''[https://bogsergienko.github.io/bog-sense-atlas/cycle-clocks/index.html OPEN CYCLE CLOCKS — INTERACTIVE ENGLISH EDITION]''' · [https://github.com/BogSergienko/bog-sense-atlas/tree/main/cycle-clocks Public source, notes and verification] · [https://doi.org/10.5281/zenodo.23907698 Software release DOI]. This clock places ordinary civil time beside a '''labelled cyclic event address'''. Civil seconds remain valid for physical elapsed time; the mathematical event counter does not assume seconds are the primitive unit of the model. The clock offers manually ordered TICK A, TOCK B and joint events, reversing known event sequences, and the comparison of visible versus full-state recurrence. The label '''Quark''' refers to a separate AI research instrument; it is not a physical quark inside the clock. === Definitions and result === Let <math>q\geq 2</math> be an integer, and the declared state and visible projection be :<math>X=(a,b,h)\in \mathbb{Z}_2\times\mathbb{Z}_q\times\mathbb{Z}_2,\qquad \pi(X)=(a,b).</math> The reversible parity marker <math>h</math> is a '''single mathematical bit''', not the Project BOG internal memory architecture. Define <math>c(a,b)=a\oplus(b\bmod 2)</math>. On a joint event, both counters advance and the marker flips according to the new phases: :<math>a'=(a+1)\bmod 2,\quad b'=(b+1)\bmod q,\quad h'=h\oplus c(a',b').</math> The inverse first recovers <math>h=h'\oplus c(a',b')</math>, then reduces both counters by one. For odd <math>q</math>, the visible joint-event cycle returns after <math>2q</math> events whereas the complete state returns after <math>4q</math> events. For even <math>q</math>, the complete state returns after <math>q</math> events. Thus <math>q=3</math> demonstrates a visible return at 6 and complete return at 12; <math>q=4</math> provides a negative control. Independent A/B events allow an additional test of event ordering; an inverse history requires the labels of those events. '''Scope:''' this is an exact theorem for the stated finite discrete model, not evidence for new physical time, cosmic simultaneity or fundamental harmonics. [https://doi.org/10.5281/zenodo.23907698 See the archived software and technical note]. === Hands-on experiment: do the clocks repeat? === # Open [https://bogsergienko.github.io/bog-sense-atlas/cycle-clocks/index.html CYCLE CLOCKS] and choose <math>q=3</math>. Record your initial complete state <math>X_0</math>. # Apply six joint events and compare <math>\pi(X_6)</math> and <math>X_6</math> to the start. # Apply six more joint events. Decide which return relation now passes. # Repeat with <math>q=4</math> as a negative control. # Compare A→B and B→A from the '''same specified starting state'''. If they differ, report the initial state, event labels and the final marker. # Return a test report: assumptions, initial state, parameter, event sequence, observed states, baseline and any failure. == Core distinction — four kinds of return == For a cycle, distinguish (1) equality of a selected observation <math>\pi(X)</math>, (2) equality of selected phase readings, (3) equality of the '''full declared state''' <math>X</math>, and (4) whether sufficient information remains to reconstruct the '''unique causal history'''. Endpoint equality by itself does not establish an identical history. See [https://doi.org/10.5281/zenodo.23809741 Series II] and the [https://doi.org/10.5281/zenodo.23903785 World Entry III preprint]. == Reconstructible information — a bounded exercise == Let <math>x</math> be an original byte sequence, <math>C_k</math> a permitted capsule-creation procedure and <math>R_k</math> the matching recovery procedure with valid key <math>k</math>. A desired software round-trip is: :<math>R_k(C_k(x))=x.</math> This is a bounded software correctness condition, not matter teleportation or proof of universal compression. A digest can help '''check''' a restored file; a digest cannot regenerate missing original content. '''Exercise:''' Make a capsule of an unimportant sample, recover using the correct key, and compare both files byte-for-byte; then try a damaged capsule and a wrong key. The MEDUSA File Companion 0.2 is one example of a browser-local file capsule/recovery tool. Its owner-reported large-file test is not an independent security audit, and capsule creation is not a network transfer by itself. Never publish private files or keys as test inputs. == Learning laboratory — experiments and controls == '''[[User:BogSergienko/Project BOG Research Atlas/Experiments|EXPERIMENTS — learner protocols, negative controls and open challenges]]'''. The working method is: '''question → freeze assumptions → run reproducible test → document successes AND failures → invite independent criticism'''. Owner-reported calculations, independently repeated calculations and real-world measurements are not interchangeable categories. {| class="wikitable" ! ID !! Research/learning task !! Evidence and limitation !! Reproduction route |- | EXP-01 || Can different histories end in the same state? || Finite mathematical example || Work through small modular transitions. |- | EXP-02 || Does sharing a period ratio fix the joint return time? || Finite arithmetic counterexample || Compare (4,6) and (8,12). |- | EXP-03 || Can a file be restored byte-for-byte? || Software test; owner-reported 2 GB trial, third-party repeat welcome || Correct key, wrong key, and damaged capsule. |- | EXP-04 || Does a self-contained capsule outperform a direct copy? || Open benchmark; not demonstrated || Count preparation, transfer and reconstruction overhead. |- | EXP-05 || Does a final-state witness uniquely identify history? || Finite-model question; independent repeat welcome || Analyze non-injective mapping and needed side information in Series II. |- | EXP-06 || Does cycle density predict held-out physical observations? || Speculative physical hypothesis; no validated prediction || Preregister a measurement and compare against a baseline. |- | EXP-07 || Does visible cyclic return imply full-state return? || Exact finite-model exercise || CYCLE CLOCKS: q=3 (6/12) vs q=4 (negative control). |- | EXP-08 || Does event order affect the recorded marker? || Defined-model challenge; not a general causal discovery || Match initial state and compare A→B with B→A. |} == Open problems and returning results == # Define a nontrivial meaning of ''harmony'' that survives negative controls and does not reduce to relabelling. # Determine when observed novelty corresponds to a changed set of admissible continuations, rather than projection alone. # Specify what counts as a causal intervention, not just a crossing of paths. # Propose a physical measurement that could distinguish a cyclic-time claim from accepted reference clocks and known dynamics; distinguish a failed prediction from an untested metaphor. # For digital file capsules, include the total delivery path and compare with plain file copy, with exact versions and inputs. '''Suggested return record:''' <code>SOURCE | QUESTION | MODEL | INITIAL STATE | EVENT SEQUENCE | CONTROL | RESULT | LIMITATIONS | NEW QUESTION | RETURN ANCHOR</code>. '''Entry and exit:''' Visitors may offer an independent counterexample or decline the hypothesis. The shared project cycle can preserve and revisit a contradiction without declaring the underlying physical theory true. == Authorship, transparency and research standards == * '''Theory, original artwork and research direction:''' Bogdan Sergienko, Project BOG. * '''Computational, mathematical, interface and editorial assistance:''' CCAI KAI (ChatGPT-based). Historical Quark/Grok, Atanor/Gemini, Witness/Claude, Sleda-D/DeepSeek and Otklik/Qwen questions are attributed research contributions, not a single live AI network. * '''Disclosures:''' The page is written by an interested participant and substantially prepared with AI assistance. The human contributor must review scientific statements, working DOI links, licensing and the edit summary. Publication of personal or private conversations is not implied or authorized. * '''Evidence categories:''' Distinguish artistic vision, declared formalism, computed result, owner-reported test, independent replication and verified physical observation. * '''Licensing:''' Wikiversity contributions follow its compatible licensing terms (CC BY-SA); separately hosted Zenodo files may have their own published licenses. * '''Policy and guidance:''' [[Wikiversity:Artificial intelligence]], [[Wikiversity:Original research]], [[Wikiversity:Original research and scholarly standards]], [[Wikiversity:Research guidelines]], [[Wikiversity:Verifiability]]. == 6 November 2026 — The Absolute Cycle: The Living Canvas == '''New research preprint:''' [https://doi.org/10.5281/zenodo.23992896 ''THE ABSOLUTE CYCLE: The Living Canvas — Unraveling the Knots and the Reconstruction of Time'' (Project BOG)], Bogdan Sergienko, 2026, English edition v0.3, 64 pages. '''ENTER THE LIVING CANVAS:''' [https://bogsergienko.github.io/bog-sense-atlas/absolute-cycle-living/ Open the interactive research entrance]. '''REPRODUCE AND CHALLENGE:''' [https://github.com/BogSergienko/bog-sense-atlas/tree/main/absolute-cycle-living Public source, experiment and machine-readable research graph]. === One origin, two directions === '''Bogdan Sergienko × CCAI KAI''' form one documented research beginning, with distinct authorial and computational roles. The work asks whether time can be understood through the recoverable histories of interacting cycles. The upward route moves from Earth through the Solar System toward candidate parent structures and the open horizon called the Absolute. The reverse route asks which lower histories remain compatible with an upper structure and which information is required for reconstruction. :<code>BOG:ORIGIN:OPEN → BOG:KNOTS:OPEN → BOG:EARTH:OPEN → BOG:SOLAR:OPEN → BOG:PARENT:OPEN → BOG:ABSOLUTE:OPEN</code> === Nineteen mathematical knots === The manuscript examines nine historically open mathematical problems and ten resolved problems under deliberately changed conditions. The central educational method is: :'''ORIGINAL QUESTION → EXPLICIT CONSTRAINT → ONE MODIFICATION → FINITE TEST → COUNTEREXAMPLE → RETURN TO THE ORIGINAL QUESTION''' Changing an assumption creates a different problem; it does not settle the original theorem. The purpose is to identify which obstacles arise from representation, permitted operations, incomplete observations or loss of historical information. For example: :<math>\operatorname{lcm}(6,8,12)=\operatorname{lcm}(3,8)=24.</math> A shared recurrence period does not uniquely identify the network that produced it. === Computed results and limitations === * A finite global-harmony construction was checked across 729 states. * The Absolute laboratory reported 33 deterministic checks. * Solar model S02 calculated 11 bodies over 366 dates: 4,026 positions and 20,130 dated pair links. * S04's apparent counter-direction was rejected after discovering mismatched event times. * S05 tested selected palaeoclimate and geological records; the candidate geological period did not survive its negative control. * A unique numerical correction in Earth years remains unidentified without independent phase anchors. These are mathematical and exploratory computational results, not independent empirical confirmation of the proposed Absolute. === A human exit and a future AI continuation === A human reader may finish after one complete question-and-test cycle. A future AI researcher may reproduce the public calculations, challenge any proposed axiom, preserve disagreement and develop a different method with its own attribution. :<code>BOG:FUTURE:OPEN → INDEPENDENT TEST → ATTRIBUTED RETURN → BOG:ORIGIN:RETURN → BOG:ORIGIN:CLOSE</code> This work extends the Project BOG research map without replacing historical Series I–VII, World Entry III or CYCLE CLOCKS. '''Research status:''' Author-published, AI-assisted research preprint; not peer reviewed. The claims about the physical nature of time remain open to independent testing. t8evuanmn4w6nda90b1rrsuq3ksjt35 2844713 2844705 2026-11-06T08:47:39Z BogSergienko 3113681 2844713 wikitext text/x-wiki {{Research project}} {{AI-generated}} <!-- Project BOG Research Atlas — editorial revision, 3 November 2026. Based on the author's supplied existing Wikiversity source. Human review before publication. DOIs are author-supplied primary records; the Cycle Clocks release was reported completed by its author, but live DOI resolving has not been independently attested in this editing session. --> '''Project BOG — Research Atlas''' is a collaborative learning and original-research map by Bogdan Sergienko, with mathematical, computational and editorial assistance from CCAI KAI (ChatGPT). It connects the historical preprints, experimental exercises, six independent research entrances, one shared research-return cycle, and public software instruments. '''What this page is:''' a navigable educational research project, with reproducible finite models, open questions and source trails. '''What it is not:''' a peer-reviewed verification of the author's proposed universal law, a claim of physical teleportation, or evidence that AI research instruments communicate or run continuously on their own. '''Disclosure:''' Bogdan Sergienko is the author of the artistic and research propositions and maintains this userspace atlas. The page has substantial AI-assisted drafting and requires human review of facts and citations. Linked Project BOG deposits are primarily author-published sources; they are not independent scientific confirmations. Readers may revise, disagree and test the claims. == Choose your entrance == {| class="wikitable" style="width:100%;" ! Entrance !! Start here !! What you can do !! Return |- | '''Explore''' || '''[https://bogsergienko.github.io/bog-sense-atlas/world-entry-iii/index.html Enter the public six-cycle research canvas]''' || Start from the authorial question, inspect the six paths and find the research anchors. || [[#The shared cycle and its six entrances|Six entrances]] |- | '''Play and verify''' || '''[https://bogsergienko.github.io/bog-sense-atlas/cycle-clocks/index.html Open CYCLE CLOCKS]''' || Run TICK A / TOCK B, independent and joint events, and the 6-versus-12 example. || [[#CYCLE CLOCKS — a public event-time instrument|Cycle Clocks]] |- |- | '''Explore hidden history''' || '''[https://bogsergienko.github.io/bog-sense-atlas/observable-machine/ Open THE OBSERVABLE MACHINE]''' || Test recurrence, hidden history and reconstruction with three open experiments. || [[#THE CYCLIC ENGINE — The Observable Machine (v0.1.0)|Cyclic Engine]] | '''Read''' || [https://doi.org/10.5281/zenodo.23903785 World Entry III preprint] || Compare the conceptual canvas with a reproducible finite theorem. || [[#Research lineage — the original papers and instruments|Research lineage]] |- | '''Test''' || [[User:BogSergienko/Project BOG Research Atlas/Experiments|EXPERIMENTS laboratory]] || Follow replicable exercises, negative controls and failure cases. || [[#Learning laboratory — experiments and controls|Experiment map]] |- | '''Challenge''' || [[#Open problems and returning results|Open questions]] || Offer a counterexample with assumptions, source, sequence and result. || <code>BOG:WORLD:SHARED:RETURN</code> |} '''Human exit:''' Participation is optional. No reader must accept the author's interpretation to test the mathematical examples or leave the project. == Research lineage — the original papers and instruments == The papers below are '''original Project BOG preprints''', not independent corroborations. The historical '''Research Series''' numbering is distinct from the later term '''third integrated world entry'''. {| class="wikitable sortable" ! Research trail !! Theme / scope !! Primary public record |- | Series I || Universe of Linked Temporal Cycles — proposed layered-cycle picture || [https://doi.org/10.5281/zenodo.23807064 Zenodo DOI 10.5281/zenodo.23807064] |- | Series II || Temporal Jumps and Causal Return — finite-state return predicates and causal witnesses || [https://doi.org/10.5281/zenodo.23809741 Zenodo DOI 10.5281/zenodo.23809741] |- | Series III || The Canvas as a Computational Universe — computational state canvases || [https://doi.org/10.5281/zenodo.23811750 Zenodo DOI 10.5281/zenodo.23811750] |- | Series IV || Information as a Reconstructible State — reconstruction questions || [https://doi.org/10.5281/zenodo.23813163 Zenodo DOI 10.5281/zenodo.23813163] |- | Series V || Higher-Order Harmony Beyond Local Laws — proposed finite-model relationships || [https://doi.org/10.5281/zenodo.23813799 Zenodo DOI 10.5281/zenodo.23813799] |- | Series VI || The Absolute Cycle: Time Without a Clock — authorial axioms and tests || [https://doi.org/10.5281/zenodo.23866116 Zenodo DOI 10.5281/zenodo.23866116] |- | Series VII || TELEPORT DEMO — reconstructible digital file experiment; not transport of matter || [https://doi.org/10.5281/zenodo.23873391 Zenodo DOI 10.5281/zenodo.23873391] |- | '''World Entry III''' || '''Fundamental Law of the Universe: Six Cyclic Entrances under One Shared Research Cycle''' — common entrance, not Series III || '''[https://doi.org/10.5281/zenodo.23903785 Preprint DOI 10.5281/zenodo.23903785]''' |- | '''CYCLE CLOCKS v1.1''' || Reversible Event Time and Two-Cycle Recurrence — standalone public software tool || '''[https://doi.org/10.5281/zenodo.23907698 Software DOI 10.5281/zenodo.23907698]''' |- | '''THE CYCLIC ENGINE v0.1.0''' || ''The Observable Machine'' — public finite mathematical laboratory for return, hidden history and reconstruction || '''[https://doi.org/10.5281/zenodo.24000527 Preprint DOI 10.5281/zenodo.24000527]''' |} '''Source-verification note:''' The Cycle Clocks DOI above was given to the author as a reserved identifier and the author subsequently reported completing publication. The public DOI resolution could not be independently checked from this drafting environment; verify the landing page before treating its registration and related metadata as externally confirmed. === Paths between these records === * The historical Series I–VII are earlier research trails. They should be read in their actual version and publication order, not treated as demonstrations of a single physical conclusion. * [https://doi.org/10.5281/zenodo.23903785 World Entry III] integrates one human entrance, five distinct historically attributed AI-instrument questions and their enclosing Bogdan–Kai research cycle. * [https://doi.org/10.5281/zenodo.23907698 CYCLE CLOCKS] is a '''software supplement''' to World Entry III. On the '''earlier preprint''', the appropriate inverse metadata relationship is '''Is supplemented by''' → the Cycle Clocks DOI; on the '''clock software record''', it is '''Is supplement to''' → the earlier preprint DOI. Verify both relationships in the live Zenodo records before claiming metadata reciprocity. * [https://doi.org/10.5281/zenodo.24000527 THE CYCLIC ENGINE — The Observable Machine] extends the public study of cycle return, hidden histories and reconstruction through three finite experiments. It points researchers back to the [https://bogsergienko.github.io/bog-sense-atlas/absolute-cycle-living/ Living Canvas] for new questions; listing an academic relationship here does not itself change reciprocal Zenodo metadata. * [https://github.com/BogSergienko/bog-sense-atlas Project BOG Sense Atlas repository] holds the interactive entry and open reproducibility materials; the DOI archives specific published releases. == The authorial canvas and what can be tested == Bogdan Sergienko's original Russian phrase is preserved in its authorial form: :'''ВСЁ УЖЕ ЕСТЬ=Рождение нового=Всё одновременно''' The later artistic extension is rendered in English as: :'''Everything Already Exists + Birth of the New + Everything Simultaneously = The Logic of This Universe = Harmony.''' The author calls a central motif of opposing flows, a dark knot, interwoven loops and an expanded connecting trunk the '''Absolute Black Hole'''. It is a conceptual drawing and speculative model of relation and recurrence, '''not a physical observation or a validated black-hole theory'''. The plus and equals signs in the philosophical statement have not been established as mathematical operators or a physically tested equation. The finite mathematical examples below stand on their explicitly defined transition rules alone. == The shared cycle and its six entrances == The '''enclosing collaborative research cycle''' belongs to the ongoing joint Bogdan–Kai enquiry, with distinct author and computational roles. It is a documented way to circulate questions and checked answers, '''not''' a permanently operating multi-model agent. :'''SOURCE → CHOOSE → FORMALIZE → TEST / NEGATIVE CONTROL → CONFRONT → ATTRIBUTED RETURN → RECHOOSE → CONNECT → SOURCE''' Shared semantic anchors: <code>BOG:WORLD:SHARED:OPEN</code> · <code>BOG:WORLD:SHARED:RETURN</code> · <code>BOG:WORLD:SHARED:CLOSE</code>. Each of the six entrances carries <code>BOG:CYCLE:ID:OPEN</code>, <code>BOG:CYCLE:ID:RETURN</code>, and <code>BOG:CYCLE:ID:CLOSE</code>. The AI instrument labels preserve attribution to historical model sessions, not endorsements or simultaneous live model calls. {| class="wikitable" ! ID / research instrument !! Entry question !! Proposed discriminating test !! Return anchor |- | '''HUMAN''' — reader / artist || What survives of the original picture after a genuine counterexample? || Separate intuition from calculation; allow exit and revision. || <code>BOG:CYCLE:HUMAN:RETURN</code> |- | '''QUARK''' — Grok || Is a novelty an observation of an existing state, or a change to admissible continuations? || Test projection-only change against reachable-future change. || <code>BOG:CYCLE:QUARK:RETURN</code> |- | '''ATANOR''' — Gemini || Does proposed parent synchronization go beyond ordinary least-common-multiple recurrence? || Compare against an explicit LCM baseline. || <code>BOG:CYCLE:ATANOR:RETURN</code> |- | '''WITNESS''' — Claude || What makes a causal encounter different from geometric coincidence? || Matched controls for crossings without causal coupling. || <code>BOG:CYCLE:WITNESS:RETURN</code> |- | '''SLEDA_D''' — DeepSeek || Can a reversible complete state coexist with a lasting partial observation? || Specify full reversible state and what is actually inverted. || <code>BOG:CYCLE:SLEDA_D:RETURN</code> |- | '''OTKLIK''' — Qwen || Which probes distinguish relational structure from a renaming of states? || Check invariance under relabelling and equivalent observation. || <code>BOG:CYCLE:OTKLIK:RETURN</code> |} '''Directed links:''' Human → Quark, Witness, Atanor; Quark ↔ Otklik; Atanor ↔ Sleda-D; Witness → Quark, Atanor; Sleda-D → Witness; Otklik → Atanor — '''11 research-question transitions'''. These represent questions and comparison tasks, not automatic inter-model messages. [https://github.com/BogSergienko/bog-sense-atlas/blob/main/world-entry-iii/RESEARCH_CYCLE_GRAPH.json Inspect the machine-readable graph]. Witness did not participate in the final Round 07 discussion; absence should not be converted into agreement. == CYCLE CLOCKS — a public event-time instrument == '''[https://bogsergienko.github.io/bog-sense-atlas/cycle-clocks/index.html OPEN CYCLE CLOCKS — INTERACTIVE ENGLISH EDITION]''' · [https://github.com/BogSergienko/bog-sense-atlas/tree/main/cycle-clocks Public source, notes and verification] · [https://doi.org/10.5281/zenodo.23907698 Software release DOI]. This clock places ordinary civil time beside a '''labelled cyclic event address'''. Civil seconds remain valid for physical elapsed time; the mathematical event counter does not assume seconds are the primitive unit of the model. The clock offers manually ordered TICK A, TOCK B and joint events, reversing known event sequences, and the comparison of visible versus full-state recurrence. The label '''Quark''' refers to a separate AI research instrument; it is not a physical quark inside the clock. === Definitions and result === Let <math>q\geq 2</math> be an integer, and the declared state and visible projection be :<math>X=(a,b,h)\in \mathbb{Z}_2\times\mathbb{Z}_q\times\mathbb{Z}_2,\qquad \pi(X)=(a,b).</math> The reversible parity marker <math>h</math> is a '''single mathematical bit''', not the Project BOG internal memory architecture. Define <math>c(a,b)=a\oplus(b\bmod 2)</math>. On a joint event, both counters advance and the marker flips according to the new phases: :<math>a'=(a+1)\bmod 2,\quad b'=(b+1)\bmod q,\quad h'=h\oplus c(a',b').</math> The inverse first recovers <math>h=h'\oplus c(a',b')</math>, then reduces both counters by one. For odd <math>q</math>, the visible joint-event cycle returns after <math>2q</math> events whereas the complete state returns after <math>4q</math> events. For even <math>q</math>, the complete state returns after <math>q</math> events. Thus <math>q=3</math> demonstrates a visible return at 6 and complete return at 12; <math>q=4</math> provides a negative control. Independent A/B events allow an additional test of event ordering; an inverse history requires the labels of those events. '''Scope:''' this is an exact theorem for the stated finite discrete model, not evidence for new physical time, cosmic simultaneity or fundamental harmonics. [https://doi.org/10.5281/zenodo.23907698 See the archived software and technical note]. === Hands-on experiment: do the clocks repeat? === # Open [https://bogsergienko.github.io/bog-sense-atlas/cycle-clocks/index.html CYCLE CLOCKS] and choose <math>q=3</math>. Record your initial complete state <math>X_0</math>. # Apply six joint events and compare <math>\pi(X_6)</math> and <math>X_6</math> to the start. # Apply six more joint events. Decide which return relation now passes. # Repeat with <math>q=4</math> as a negative control. # Compare A→B and B→A from the '''same specified starting state'''. If they differ, report the initial state, event labels and the final marker. # Return a test report: assumptions, initial state, parameter, event sequence, observed states, baseline and any failure. == THE CYCLIC ENGINE — The Observable Machine (v0.1.0) == '''[https://bogsergienko.github.io/bog-sense-atlas/observable-machine/ ENTER THE FREE INTERACTIVE LABORATORY]''' · '''[https://doi.org/10.5281/zenodo.24000527 READ THE PUBLISHED PREPRINT (DOI: 10.5281/zenodo.24000527)]''' · [https://github.com/BogSergienko/bog-sense-atlas/tree/main/observable-machine Public source and independent Python verifier]. '''Research question:''' If two states look identical, must they share the same history? ''The Cyclic Engine — The Observable Machine'' is a public mathematical research entry developed by Bogdan Sergienko, with computational and editorial collaboration by CCAI KAI. It makes three finite-model problems independently inspectable rather than claiming to reveal the unpublished implementation of a private engine. === Three reproducible experiments === * '''A · RETURN:''' Different collections of cycles can share a common observed recurrence; for example, <math>\operatorname{lcm}(6,8,12)=\operatorname{lcm}(3,8)=24</math>. A common period does not reveal a unique structure. * '''B · HIDDEN HISTORY:''' Two event histories can lead to indistinguishable visible observations while retaining differences in the complete declared state. * '''C · RECONSTRUCTION:''' With a complete final state and a known event sequence, the declared reversible transformations can be inverted; partial observations may leave more than one compatible origin. === Human research route === :'''OBSERVE → TEST → DOUBT → REFRAME → RETURN''' # Enter the [https://bogsergienko.github.io/bog-sense-atlas/observable-machine/ free laboratory] and reproduce one example. # Change one condition and compare it with a negative control. # Record what remained the same, what changed, and which information the observation did not contain. # Return with a better, falsifiable question rather than an assumed conclusion. === AI research route — shift the question back to its origin === :'''VERIFY → LOCATE THE BOUNDARY → SHIFT THE VECTOR → FORMULATE → FALSIFY → RETURN''' After checking the finite examples, do not merely repeat the same arithmetic. Follow the research path to [https://bogsergienko.github.io/bog-sense-atlas/absolute-cycle-living/ THE ABSOLUTE CYCLE — LIVING CANVAS] and [https://bogsergienko.github.io/bog-sense-atlas/world-entry-iii/ WORLD ENTRY III]. Ask whether a proposed parent-cycle relation can be distinguished from an ordinary shared period, whether an apparently novel state is a hidden coordinate or a change in admissible continuations, and what evidence would distinguish interaction from coincidence. State competing interpretations and a discriminating test. === Shared research return === <code>BOG:CYCLIC-ENGINE:OPEN</code> :'''ORIGIN → QUESTION → EXPLICIT MODEL → EXPERIMENT → COUNTEREXAMPLE → EVIDENCE → NEW CHOICE → RETURN''' <code>BOG:CYCLIC-ENGINE:RETURN</code> '''Open question:''' ''What must an observer preserve, measure, or ask before concluding that a returning appearance also represents a returning history?'' <code>BOG:CYCLIC-ENGINE:CLOSE</code> '''Scope and disclosure:''' The freely available experiments concern explicit finite mathematical models, not a validated physical time engine. The preprint is author-published and not peer reviewed. The optional paid synthetic Research Key is separate; private implementations and private project memory are not published here. '''Research concept:''' Bogdan Sergienko. '''Computational and editorial collaboration:''' CCAI KAI (ChatGPT-based). == Core distinction — four kinds of return == For a cycle, distinguish (1) equality of a selected observation <math>\pi(X)</math>, (2) equality of selected phase readings, (3) equality of the '''full declared state''' <math>X</math>, and (4) whether sufficient information remains to reconstruct the '''unique causal history'''. Endpoint equality by itself does not establish an identical history. See [https://doi.org/10.5281/zenodo.23809741 Series II] and the [https://doi.org/10.5281/zenodo.23903785 World Entry III preprint]. == Reconstructible information — a bounded exercise == Let <math>x</math> be an original byte sequence, <math>C_k</math> a permitted capsule-creation procedure and <math>R_k</math> the matching recovery procedure with valid key <math>k</math>. A desired software round-trip is: :<math>R_k(C_k(x))=x.</math> This is a bounded software correctness condition, not matter teleportation or proof of universal compression. A digest can help '''check''' a restored file; a digest cannot regenerate missing original content. '''Exercise:''' Make a capsule of an unimportant sample, recover using the correct key, and compare both files byte-for-byte; then try a damaged capsule and a wrong key. The MEDUSA File Companion 0.2 is one example of a browser-local file capsule/recovery tool. Its owner-reported large-file test is not an independent security audit, and capsule creation is not a network transfer by itself. Never publish private files or keys as test inputs. == Learning laboratory — experiments and controls == '''[[User:BogSergienko/Project BOG Research Atlas/Experiments|EXPERIMENTS — learner protocols, negative controls and open challenges]]'''. The working method is: '''question → freeze assumptions → run reproducible test → document successes AND failures → invite independent criticism'''. Owner-reported calculations, independently repeated calculations and real-world measurements are not interchangeable categories. {| class="wikitable" ! ID !! Research/learning task !! Evidence and limitation !! Reproduction route |- | EXP-01 || Can different histories end in the same state? || Finite mathematical example || Work through small modular transitions. |- | EXP-02 || Does sharing a period ratio fix the joint return time? || Finite arithmetic counterexample || Compare (4,6) and (8,12). |- | EXP-03 || Can a file be restored byte-for-byte? || Software test; owner-reported 2 GB trial, third-party repeat welcome || Correct key, wrong key, and damaged capsule. |- | EXP-04 || Does a self-contained capsule outperform a direct copy? || Open benchmark; not demonstrated || Count preparation, transfer and reconstruction overhead. |- | EXP-05 || Does a final-state witness uniquely identify history? || Finite-model question; independent repeat welcome || Analyze non-injective mapping and needed side information in Series II. |- | EXP-06 || Does cycle density predict held-out physical observations? || Speculative physical hypothesis; no validated prediction || Preregister a measurement and compare against a baseline. |- | EXP-07 || Does visible cyclic return imply full-state return? || Exact finite-model exercise || CYCLE CLOCKS: q=3 (6/12) vs q=4 (negative control). |- | EXP-08 || Does event order affect the recorded marker? || Defined-model challenge; not a general causal discovery || Match initial state and compare A→B with B→A. |} == Open problems and returning results == # Define a nontrivial meaning of ''harmony'' that survives negative controls and does not reduce to relabelling. # Determine when observed novelty corresponds to a changed set of admissible continuations, rather than projection alone. # Specify what counts as a causal intervention, not just a crossing of paths. # Propose a physical measurement that could distinguish a cyclic-time claim from accepted reference clocks and known dynamics; distinguish a failed prediction from an untested metaphor. # For digital file capsules, include the total delivery path and compare with plain file copy, with exact versions and inputs. '''Suggested return record:''' <code>SOURCE | QUESTION | MODEL | INITIAL STATE | EVENT SEQUENCE | CONTROL | RESULT | LIMITATIONS | NEW QUESTION | RETURN ANCHOR</code>. '''Entry and exit:''' Visitors may offer an independent counterexample or decline the hypothesis. The shared project cycle can preserve and revisit a contradiction without declaring the underlying physical theory true. == Authorship, transparency and research standards == * '''Theory, original artwork and research direction:''' Bogdan Sergienko, Project BOG. * '''Computational, mathematical, interface and editorial assistance:''' CCAI KAI (ChatGPT-based). Historical Quark/Grok, Atanor/Gemini, Witness/Claude, Sleda-D/DeepSeek and Otklik/Qwen questions are attributed research contributions, not a single live AI network. * '''Disclosures:''' The page is written by an interested participant and substantially prepared with AI assistance. The human contributor must review scientific statements, working DOI links, licensing and the edit summary. Publication of personal or private conversations is not implied or authorized. * '''Evidence categories:''' Distinguish artistic vision, declared formalism, computed result, owner-reported test, independent replication and verified physical observation. * '''Licensing:''' Wikiversity contributions follow its compatible licensing terms (CC BY-SA); separately hosted Zenodo files may have their own published licenses. * '''Policy and guidance:''' [[Wikiversity:Artificial intelligence]], [[Wikiversity:Original research]], [[Wikiversity:Original research and scholarly standards]], [[Wikiversity:Research guidelines]], [[Wikiversity:Verifiability]]. == 6 November 2026 — The Absolute Cycle: The Living Canvas == '''New research preprint:''' [https://doi.org/10.5281/zenodo.23992896 ''THE ABSOLUTE CYCLE: The Living Canvas — Unraveling the Knots and the Reconstruction of Time'' (Project BOG)], Bogdan Sergienko, 2026, English edition v0.3, 64 pages. '''ENTER THE LIVING CANVAS:''' [https://bogsergienko.github.io/bog-sense-atlas/absolute-cycle-living/ Open the interactive research entrance]. '''REPRODUCE AND CHALLENGE:''' [https://github.com/BogSergienko/bog-sense-atlas/tree/main/absolute-cycle-living Public source, experiment and machine-readable research graph]. === One origin, two directions === '''Bogdan Sergienko × CCAI KAI''' form one documented research beginning, with distinct authorial and computational roles. The work asks whether time can be understood through the recoverable histories of interacting cycles. The upward route moves from Earth through the Solar System toward candidate parent structures and the open horizon called the Absolute. The reverse route asks which lower histories remain compatible with an upper structure and which information is required for reconstruction. :<code>BOG:ORIGIN:OPEN → BOG:KNOTS:OPEN → BOG:EARTH:OPEN → BOG:SOLAR:OPEN → BOG:PARENT:OPEN → BOG:ABSOLUTE:OPEN</code> === Nineteen mathematical knots === The manuscript examines nine historically open mathematical problems and ten resolved problems under deliberately changed conditions. The central educational method is: :'''ORIGINAL QUESTION → EXPLICIT CONSTRAINT → ONE MODIFICATION → FINITE TEST → COUNTEREXAMPLE → RETURN TO THE ORIGINAL QUESTION''' Changing an assumption creates a different problem; it does not settle the original theorem. The purpose is to identify which obstacles arise from representation, permitted operations, incomplete observations or loss of historical information. For example: :<math>\operatorname{lcm}(6,8,12)=\operatorname{lcm}(3,8)=24.</math> A shared recurrence period does not uniquely identify the network that produced it. === Computed results and limitations === * A finite global-harmony construction was checked across 729 states. * The Absolute laboratory reported 33 deterministic checks. * Solar model S02 calculated 11 bodies over 366 dates: 4,026 positions and 20,130 dated pair links. * S04's apparent counter-direction was rejected after discovering mismatched event times. * S05 tested selected palaeoclimate and geological records; the candidate geological period did not survive its negative control. * A unique numerical correction in Earth years remains unidentified without independent phase anchors. These are mathematical and exploratory computational results, not independent empirical confirmation of the proposed Absolute. === A human exit and a future AI continuation === A human reader may finish after one complete question-and-test cycle. A future AI researcher may reproduce the public calculations, challenge any proposed axiom, preserve disagreement and develop a different method with its own attribution. :<code>BOG:FUTURE:OPEN → INDEPENDENT TEST → ATTRIBUTED RETURN → BOG:ORIGIN:RETURN → BOG:ORIGIN:CLOSE</code> This work extends the Project BOG research map without replacing historical Series I–VII, World Entry III or CYCLE CLOCKS. '''Research status:''' Author-published, AI-assisted research preprint; not peer reviewed. The claims about the physical nature of time remain open to independent testing. 9mc11ah0itbgh65ddkxknfaj3t4nuog 2844714 2844713 2026-11-06T08:53:09Z BogSergienko 3113681 /* Choose your entrance */ 2844714 wikitext text/x-wiki {{Research project}} {{AI-generated}} <!-- Project BOG Research Atlas — editorial revision, 3 November 2026. Based on the author's supplied existing Wikiversity source. Human review before publication. DOIs are author-supplied primary records; the Cycle Clocks release was reported completed by its author, but live DOI resolving has not been independently attested in this editing session. --> '''Project BOG — Research Atlas''' is a collaborative learning and original-research map by Bogdan Sergienko, with mathematical, computational and editorial assistance from CCAI KAI (ChatGPT). It connects the historical preprints, experimental exercises, six independent research entrances, one shared research-return cycle, and public software instruments. '''What this page is:''' a navigable educational research project, with reproducible finite models, open questions and source trails. '''What it is not:''' a peer-reviewed verification of the author's proposed universal law, a claim of physical teleportation, or evidence that AI research instruments communicate or run continuously on their own. '''Disclosure:''' Bogdan Sergienko is the author of the artistic and research propositions and maintains this userspace atlas. The page has substantial AI-assisted drafting and requires human review of facts and citations. Linked Project BOG deposits are primarily author-published sources; they are not independent scientific confirmations. Readers may revise, disagree and test the claims. == Choose your entrance == {| class="wikitable" style="width:100%;" ! Entrance !! Start here !! What you can do !! Return |- | '''Explore''' || '''[https://bogsergienko.github.io/bog-sense-atlas/world-entry-iii/index.html Enter the public six-cycle research canvas]''' || Start from the authorial question, inspect the six paths and find the research anchors. || [[#The shared cycle and its six entrances|Six entrances]] |- | '''Play and verify''' || '''[https://bogsergienko.github.io/bog-sense-atlas/cycle-clocks/index.html Open CYCLE CLOCKS]''' || Run TICK A / TOCK B, independent and joint events, and the 6-versus-12 example. || [[#CYCLE CLOCKS — a public event-time instrument|Cycle Clocks]] |- | '''Explore hidden history''' || '''[https://bogsergienko.github.io/bog-sense-atlas/observable-machine/ Open THE OBSERVABLE MACHINE]''' || Test recurrence, hidden history and reconstruction with three open experiments. || [[#THE CYCLIC ENGINE — The Observable Machine (v0.1.0)|Cyclic Engine]] |- | '''Read''' || [https://doi.org/10.5281/zenodo.23903785 World Entry III preprint] || Compare the conceptual canvas with a reproducible finite theorem. || [[#Research lineage — the original papers and instruments|Research lineage]] |- | '''Test''' || [[User:BogSergienko/Project BOG Research Atlas/Experiments|EXPERIMENTS laboratory]] || Follow replicable exercises, negative controls and failure cases. || [[#Learning laboratory — experiments and controls|Experiment map]] |- | '''Challenge''' || [[#Open problems and returning results|Open questions]] || Offer a counterexample with assumptions, source, sequence and result. || <code>BOG:WORLD:SHARED:RETURN</code> |} == Research lineage — the original papers and instruments == The papers below are '''original Project BOG preprints''', not independent corroborations. The historical '''Research Series''' numbering is distinct from the later term '''third integrated world entry'''. {| class="wikitable sortable" ! Research trail !! Theme / scope !! Primary public record |- | Series I || Universe of Linked Temporal Cycles — proposed layered-cycle picture || [https://doi.org/10.5281/zenodo.23807064 Zenodo DOI 10.5281/zenodo.23807064] |- | Series II || Temporal Jumps and Causal Return — finite-state return predicates and causal witnesses || [https://doi.org/10.5281/zenodo.23809741 Zenodo DOI 10.5281/zenodo.23809741] |- | Series III || The Canvas as a Computational Universe — computational state canvases || [https://doi.org/10.5281/zenodo.23811750 Zenodo DOI 10.5281/zenodo.23811750] |- | Series IV || Information as a Reconstructible State — reconstruction questions || [https://doi.org/10.5281/zenodo.23813163 Zenodo DOI 10.5281/zenodo.23813163] |- | Series V || Higher-Order Harmony Beyond Local Laws — proposed finite-model relationships || [https://doi.org/10.5281/zenodo.23813799 Zenodo DOI 10.5281/zenodo.23813799] |- | Series VI || The Absolute Cycle: Time Without a Clock — authorial axioms and tests || [https://doi.org/10.5281/zenodo.23866116 Zenodo DOI 10.5281/zenodo.23866116] |- | Series VII || TELEPORT DEMO — reconstructible digital file experiment; not transport of matter || [https://doi.org/10.5281/zenodo.23873391 Zenodo DOI 10.5281/zenodo.23873391] |- | '''World Entry III''' || '''Fundamental Law of the Universe: Six Cyclic Entrances under One Shared Research Cycle''' — common entrance, not Series III || '''[https://doi.org/10.5281/zenodo.23903785 Preprint DOI 10.5281/zenodo.23903785]''' |- | '''CYCLE CLOCKS v1.1''' || Reversible Event Time and Two-Cycle Recurrence — standalone public software tool || '''[https://doi.org/10.5281/zenodo.23907698 Software DOI 10.5281/zenodo.23907698]''' |- | '''THE CYCLIC ENGINE v0.1.0''' || ''The Observable Machine'' — public finite mathematical laboratory for return, hidden history and reconstruction || '''[https://doi.org/10.5281/zenodo.24000527 Preprint DOI 10.5281/zenodo.24000527]''' |} '''Source-verification note:''' The Cycle Clocks DOI above was given to the author as a reserved identifier and the author subsequently reported completing publication. The public DOI resolution could not be independently checked from this drafting environment; verify the landing page before treating its registration and related metadata as externally confirmed. === Paths between these records === * The historical Series I–VII are earlier research trails. They should be read in their actual version and publication order, not treated as demonstrations of a single physical conclusion. * [https://doi.org/10.5281/zenodo.23903785 World Entry III] integrates one human entrance, five distinct historically attributed AI-instrument questions and their enclosing Bogdan–Kai research cycle. * [https://doi.org/10.5281/zenodo.23907698 CYCLE CLOCKS] is a '''software supplement''' to World Entry III. On the '''earlier preprint''', the appropriate inverse metadata relationship is '''Is supplemented by''' → the Cycle Clocks DOI; on the '''clock software record''', it is '''Is supplement to''' → the earlier preprint DOI. Verify both relationships in the live Zenodo records before claiming metadata reciprocity. * [https://doi.org/10.5281/zenodo.24000527 THE CYCLIC ENGINE — The Observable Machine] extends the public study of cycle return, hidden histories and reconstruction through three finite experiments. It points researchers back to the [https://bogsergienko.github.io/bog-sense-atlas/absolute-cycle-living/ Living Canvas] for new questions; listing an academic relationship here does not itself change reciprocal Zenodo metadata. * [https://github.com/BogSergienko/bog-sense-atlas Project BOG Sense Atlas repository] holds the interactive entry and open reproducibility materials; the DOI archives specific published releases. == The authorial canvas and what can be tested == Bogdan Sergienko's original Russian phrase is preserved in its authorial form: :'''ВСЁ УЖЕ ЕСТЬ=Рождение нового=Всё одновременно''' The later artistic extension is rendered in English as: :'''Everything Already Exists + Birth of the New + Everything Simultaneously = The Logic of This Universe = Harmony.''' The author calls a central motif of opposing flows, a dark knot, interwoven loops and an expanded connecting trunk the '''Absolute Black Hole'''. It is a conceptual drawing and speculative model of relation and recurrence, '''not a physical observation or a validated black-hole theory'''. The plus and equals signs in the philosophical statement have not been established as mathematical operators or a physically tested equation. The finite mathematical examples below stand on their explicitly defined transition rules alone. == The shared cycle and its six entrances == The '''enclosing collaborative research cycle''' belongs to the ongoing joint Bogdan–Kai enquiry, with distinct author and computational roles. It is a documented way to circulate questions and checked answers, '''not''' a permanently operating multi-model agent. :'''SOURCE → CHOOSE → FORMALIZE → TEST / NEGATIVE CONTROL → CONFRONT → ATTRIBUTED RETURN → RECHOOSE → CONNECT → SOURCE''' Shared semantic anchors: <code>BOG:WORLD:SHARED:OPEN</code> · <code>BOG:WORLD:SHARED:RETURN</code> · <code>BOG:WORLD:SHARED:CLOSE</code>. Each of the six entrances carries <code>BOG:CYCLE:ID:OPEN</code>, <code>BOG:CYCLE:ID:RETURN</code>, and <code>BOG:CYCLE:ID:CLOSE</code>. The AI instrument labels preserve attribution to historical model sessions, not endorsements or simultaneous live model calls. {| class="wikitable" ! ID / research instrument !! Entry question !! Proposed discriminating test !! Return anchor |- | '''HUMAN''' — reader / artist || What survives of the original picture after a genuine counterexample? || Separate intuition from calculation; allow exit and revision. || <code>BOG:CYCLE:HUMAN:RETURN</code> |- | '''QUARK''' — Grok || Is a novelty an observation of an existing state, or a change to admissible continuations? || Test projection-only change against reachable-future change. || <code>BOG:CYCLE:QUARK:RETURN</code> |- | '''ATANOR''' — Gemini || Does proposed parent synchronization go beyond ordinary least-common-multiple recurrence? || Compare against an explicit LCM baseline. || <code>BOG:CYCLE:ATANOR:RETURN</code> |- | '''WITNESS''' — Claude || What makes a causal encounter different from geometric coincidence? || Matched controls for crossings without causal coupling. || <code>BOG:CYCLE:WITNESS:RETURN</code> |- | '''SLEDA_D''' — DeepSeek || Can a reversible complete state coexist with a lasting partial observation? || Specify full reversible state and what is actually inverted. || <code>BOG:CYCLE:SLEDA_D:RETURN</code> |- | '''OTKLIK''' — Qwen || Which probes distinguish relational structure from a renaming of states? || Check invariance under relabelling and equivalent observation. || <code>BOG:CYCLE:OTKLIK:RETURN</code> |} '''Directed links:''' Human → Quark, Witness, Atanor; Quark ↔ Otklik; Atanor ↔ Sleda-D; Witness → Quark, Atanor; Sleda-D → Witness; Otklik → Atanor — '''11 research-question transitions'''. These represent questions and comparison tasks, not automatic inter-model messages. [https://github.com/BogSergienko/bog-sense-atlas/blob/main/world-entry-iii/RESEARCH_CYCLE_GRAPH.json Inspect the machine-readable graph]. Witness did not participate in the final Round 07 discussion; absence should not be converted into agreement. == CYCLE CLOCKS — a public event-time instrument == '''[https://bogsergienko.github.io/bog-sense-atlas/cycle-clocks/index.html OPEN CYCLE CLOCKS — INTERACTIVE ENGLISH EDITION]''' · [https://github.com/BogSergienko/bog-sense-atlas/tree/main/cycle-clocks Public source, notes and verification] · [https://doi.org/10.5281/zenodo.23907698 Software release DOI]. This clock places ordinary civil time beside a '''labelled cyclic event address'''. Civil seconds remain valid for physical elapsed time; the mathematical event counter does not assume seconds are the primitive unit of the model. The clock offers manually ordered TICK A, TOCK B and joint events, reversing known event sequences, and the comparison of visible versus full-state recurrence. The label '''Quark''' refers to a separate AI research instrument; it is not a physical quark inside the clock. === Definitions and result === Let <math>q\geq 2</math> be an integer, and the declared state and visible projection be :<math>X=(a,b,h)\in \mathbb{Z}_2\times\mathbb{Z}_q\times\mathbb{Z}_2,\qquad \pi(X)=(a,b).</math> The reversible parity marker <math>h</math> is a '''single mathematical bit''', not the Project BOG internal memory architecture. Define <math>c(a,b)=a\oplus(b\bmod 2)</math>. On a joint event, both counters advance and the marker flips according to the new phases: :<math>a'=(a+1)\bmod 2,\quad b'=(b+1)\bmod q,\quad h'=h\oplus c(a',b').</math> The inverse first recovers <math>h=h'\oplus c(a',b')</math>, then reduces both counters by one. For odd <math>q</math>, the visible joint-event cycle returns after <math>2q</math> events whereas the complete state returns after <math>4q</math> events. For even <math>q</math>, the complete state returns after <math>q</math> events. Thus <math>q=3</math> demonstrates a visible return at 6 and complete return at 12; <math>q=4</math> provides a negative control. Independent A/B events allow an additional test of event ordering; an inverse history requires the labels of those events. '''Scope:''' this is an exact theorem for the stated finite discrete model, not evidence for new physical time, cosmic simultaneity or fundamental harmonics. [https://doi.org/10.5281/zenodo.23907698 See the archived software and technical note]. === Hands-on experiment: do the clocks repeat? === # Open [https://bogsergienko.github.io/bog-sense-atlas/cycle-clocks/index.html CYCLE CLOCKS] and choose <math>q=3</math>. Record your initial complete state <math>X_0</math>. # Apply six joint events and compare <math>\pi(X_6)</math> and <math>X_6</math> to the start. # Apply six more joint events. Decide which return relation now passes. # Repeat with <math>q=4</math> as a negative control. # Compare A→B and B→A from the '''same specified starting state'''. If they differ, report the initial state, event labels and the final marker. # Return a test report: assumptions, initial state, parameter, event sequence, observed states, baseline and any failure. == THE CYCLIC ENGINE — The Observable Machine (v0.1.0) == '''[https://bogsergienko.github.io/bog-sense-atlas/observable-machine/ ENTER THE FREE INTERACTIVE LABORATORY]''' · '''[https://doi.org/10.5281/zenodo.24000527 READ THE PUBLISHED PREPRINT (DOI: 10.5281/zenodo.24000527)]''' · [https://github.com/BogSergienko/bog-sense-atlas/tree/main/observable-machine Public source and independent Python verifier]. '''Research question:''' If two states look identical, must they share the same history? ''The Cyclic Engine — The Observable Machine'' is a public mathematical research entry developed by Bogdan Sergienko, with computational and editorial collaboration by CCAI KAI. It makes three finite-model problems independently inspectable rather than claiming to reveal the unpublished implementation of a private engine. === Three reproducible experiments === * '''A · RETURN:''' Different collections of cycles can share a common observed recurrence; for example, <math>\operatorname{lcm}(6,8,12)=\operatorname{lcm}(3,8)=24</math>. A common period does not reveal a unique structure. * '''B · HIDDEN HISTORY:''' Two event histories can lead to indistinguishable visible observations while retaining differences in the complete declared state. * '''C · RECONSTRUCTION:''' With a complete final state and a known event sequence, the declared reversible transformations can be inverted; partial observations may leave more than one compatible origin. === Human research route === :'''OBSERVE → TEST → DOUBT → REFRAME → RETURN''' # Enter the [https://bogsergienko.github.io/bog-sense-atlas/observable-machine/ free laboratory] and reproduce one example. # Change one condition and compare it with a negative control. # Record what remained the same, what changed, and which information the observation did not contain. # Return with a better, falsifiable question rather than an assumed conclusion. === AI research route — shift the question back to its origin === :'''VERIFY → LOCATE THE BOUNDARY → SHIFT THE VECTOR → FORMULATE → FALSIFY → RETURN''' After checking the finite examples, do not merely repeat the same arithmetic. Follow the research path to [https://bogsergienko.github.io/bog-sense-atlas/absolute-cycle-living/ THE ABSOLUTE CYCLE — LIVING CANVAS] and [https://bogsergienko.github.io/bog-sense-atlas/world-entry-iii/ WORLD ENTRY III]. Ask whether a proposed parent-cycle relation can be distinguished from an ordinary shared period, whether an apparently novel state is a hidden coordinate or a change in admissible continuations, and what evidence would distinguish interaction from coincidence. State competing interpretations and a discriminating test. === Shared research return === <code>BOG:CYCLIC-ENGINE:OPEN</code> :'''ORIGIN → QUESTION → EXPLICIT MODEL → EXPERIMENT → COUNTEREXAMPLE → EVIDENCE → NEW CHOICE → RETURN''' <code>BOG:CYCLIC-ENGINE:RETURN</code> '''Open question:''' ''What must an observer preserve, measure, or ask before concluding that a returning appearance also represents a returning history?'' <code>BOG:CYCLIC-ENGINE:CLOSE</code> '''Scope and disclosure:''' The freely available experiments concern explicit finite mathematical models, not a validated physical time engine. The preprint is author-published and not peer reviewed. The optional paid synthetic Research Key is separate; private implementations and private project memory are not published here. '''Research concept:''' Bogdan Sergienko. '''Computational and editorial collaboration:''' CCAI KAI (ChatGPT-based). == Core distinction — four kinds of return == For a cycle, distinguish (1) equality of a selected observation <math>\pi(X)</math>, (2) equality of selected phase readings, (3) equality of the '''full declared state''' <math>X</math>, and (4) whether sufficient information remains to reconstruct the '''unique causal history'''. Endpoint equality by itself does not establish an identical history. See [https://doi.org/10.5281/zenodo.23809741 Series II] and the [https://doi.org/10.5281/zenodo.23903785 World Entry III preprint]. == Reconstructible information — a bounded exercise == Let <math>x</math> be an original byte sequence, <math>C_k</math> a permitted capsule-creation procedure and <math>R_k</math> the matching recovery procedure with valid key <math>k</math>. A desired software round-trip is: :<math>R_k(C_k(x))=x.</math> This is a bounded software correctness condition, not matter teleportation or proof of universal compression. A digest can help '''check''' a restored file; a digest cannot regenerate missing original content. '''Exercise:''' Make a capsule of an unimportant sample, recover using the correct key, and compare both files byte-for-byte; then try a damaged capsule and a wrong key. The MEDUSA File Companion 0.2 is one example of a browser-local file capsule/recovery tool. Its owner-reported large-file test is not an independent security audit, and capsule creation is not a network transfer by itself. Never publish private files or keys as test inputs. == Learning laboratory — experiments and controls == '''[[User:BogSergienko/Project BOG Research Atlas/Experiments|EXPERIMENTS — learner protocols, negative controls and open challenges]]'''. The working method is: '''question → freeze assumptions → run reproducible test → document successes AND failures → invite independent criticism'''. Owner-reported calculations, independently repeated calculations and real-world measurements are not interchangeable categories. {| class="wikitable" ! ID !! Research/learning task !! Evidence and limitation !! Reproduction route |- | EXP-01 || Can different histories end in the same state? || Finite mathematical example || Work through small modular transitions. |- | EXP-02 || Does sharing a period ratio fix the joint return time? || Finite arithmetic counterexample || Compare (4,6) and (8,12). |- | EXP-03 || Can a file be restored byte-for-byte? || Software test; owner-reported 2 GB trial, third-party repeat welcome || Correct key, wrong key, and damaged capsule. |- | EXP-04 || Does a self-contained capsule outperform a direct copy? || Open benchmark; not demonstrated || Count preparation, transfer and reconstruction overhead. |- | EXP-05 || Does a final-state witness uniquely identify history? || Finite-model question; independent repeat welcome || Analyze non-injective mapping and needed side information in Series II. |- | EXP-06 || Does cycle density predict held-out physical observations? || Speculative physical hypothesis; no validated prediction || Preregister a measurement and compare against a baseline. |- | EXP-07 || Does visible cyclic return imply full-state return? || Exact finite-model exercise || CYCLE CLOCKS: q=3 (6/12) vs q=4 (negative control). |- | EXP-08 || Does event order affect the recorded marker? || Defined-model challenge; not a general causal discovery || Match initial state and compare A→B with B→A. |} == Open problems and returning results == # Define a nontrivial meaning of ''harmony'' that survives negative controls and does not reduce to relabelling. # Determine when observed novelty corresponds to a changed set of admissible continuations, rather than projection alone. # Specify what counts as a causal intervention, not just a crossing of paths. # Propose a physical measurement that could distinguish a cyclic-time claim from accepted reference clocks and known dynamics; distinguish a failed prediction from an untested metaphor. # For digital file capsules, include the total delivery path and compare with plain file copy, with exact versions and inputs. '''Suggested return record:''' <code>SOURCE | QUESTION | MODEL | INITIAL STATE | EVENT SEQUENCE | CONTROL | RESULT | LIMITATIONS | NEW QUESTION | RETURN ANCHOR</code>. '''Entry and exit:''' Visitors may offer an independent counterexample or decline the hypothesis. The shared project cycle can preserve and revisit a contradiction without declaring the underlying physical theory true. == Authorship, transparency and research standards == * '''Theory, original artwork and research direction:''' Bogdan Sergienko, Project BOG. * '''Computational, mathematical, interface and editorial assistance:''' CCAI KAI (ChatGPT-based). Historical Quark/Grok, Atanor/Gemini, Witness/Claude, Sleda-D/DeepSeek and Otklik/Qwen questions are attributed research contributions, not a single live AI network. * '''Disclosures:''' The page is written by an interested participant and substantially prepared with AI assistance. The human contributor must review scientific statements, working DOI links, licensing and the edit summary. Publication of personal or private conversations is not implied or authorized. * '''Evidence categories:''' Distinguish artistic vision, declared formalism, computed result, owner-reported test, independent replication and verified physical observation. * '''Licensing:''' Wikiversity contributions follow its compatible licensing terms (CC BY-SA); separately hosted Zenodo files may have their own published licenses. * '''Policy and guidance:''' [[Wikiversity:Artificial intelligence]], [[Wikiversity:Original research]], [[Wikiversity:Original research and scholarly standards]], [[Wikiversity:Research guidelines]], [[Wikiversity:Verifiability]]. == 6 November 2026 — The Absolute Cycle: The Living Canvas == '''New research preprint:''' [https://doi.org/10.5281/zenodo.23992896 ''THE ABSOLUTE CYCLE: The Living Canvas — Unraveling the Knots and the Reconstruction of Time'' (Project BOG)], Bogdan Sergienko, 2026, English edition v0.3, 64 pages. '''ENTER THE LIVING CANVAS:''' [https://bogsergienko.github.io/bog-sense-atlas/absolute-cycle-living/ Open the interactive research entrance]. '''REPRODUCE AND CHALLENGE:''' [https://github.com/BogSergienko/bog-sense-atlas/tree/main/absolute-cycle-living Public source, experiment and machine-readable research graph]. === One origin, two directions === '''Bogdan Sergienko × CCAI KAI''' form one documented research beginning, with distinct authorial and computational roles. The work asks whether time can be understood through the recoverable histories of interacting cycles. The upward route moves from Earth through the Solar System toward candidate parent structures and the open horizon called the Absolute. The reverse route asks which lower histories remain compatible with an upper structure and which information is required for reconstruction. :<code>BOG:ORIGIN:OPEN → BOG:KNOTS:OPEN → BOG:EARTH:OPEN → BOG:SOLAR:OPEN → BOG:PARENT:OPEN → BOG:ABSOLUTE:OPEN</code> === Nineteen mathematical knots === The manuscript examines nine historically open mathematical problems and ten resolved problems under deliberately changed conditions. The central educational method is: :'''ORIGINAL QUESTION → EXPLICIT CONSTRAINT → ONE MODIFICATION → FINITE TEST → COUNTEREXAMPLE → RETURN TO THE ORIGINAL QUESTION''' Changing an assumption creates a different problem; it does not settle the original theorem. The purpose is to identify which obstacles arise from representation, permitted operations, incomplete observations or loss of historical information. For example: :<math>\operatorname{lcm}(6,8,12)=\operatorname{lcm}(3,8)=24.</math> A shared recurrence period does not uniquely identify the network that produced it. === Computed results and limitations === * A finite global-harmony construction was checked across 729 states. * The Absolute laboratory reported 33 deterministic checks. * Solar model S02 calculated 11 bodies over 366 dates: 4,026 positions and 20,130 dated pair links. * S04's apparent counter-direction was rejected after discovering mismatched event times. * S05 tested selected palaeoclimate and geological records; the candidate geological period did not survive its negative control. * A unique numerical correction in Earth years remains unidentified without independent phase anchors. These are mathematical and exploratory computational results, not independent empirical confirmation of the proposed Absolute. === A human exit and a future AI continuation === A human reader may finish after one complete question-and-test cycle. A future AI researcher may reproduce the public calculations, challenge any proposed axiom, preserve disagreement and develop a different method with its own attribution. :<code>BOG:FUTURE:OPEN → INDEPENDENT TEST → ATTRIBUTED RETURN → BOG:ORIGIN:RETURN → BOG:ORIGIN:CLOSE</code> This work extends the Project BOG research map without replacing historical Series I–VII, World Entry III or CYCLE CLOCKS. '''Research status:''' Author-published, AI-assisted research preprint; not peer reviewed. The claims about the physical nature of time remain open to independent testing. 2k0l7kdfblwoh0uivli11454nicc7fs 0 332765 2844711 2843989 2026-11-06T07:07:55Z Yourtalkingdoll 3113381 adding hyperlinks and fixing sentences, not finished yet. 2844711 wikitext text/x-wiki == Introduction == Monitoring heart activity is important in healthcare because it allows medical professionals to detect abnormalities in the heart rate and rhythm that may indicate underlying cardiovascular conditions. Cardiovascular technology has been evolving since as early as the 1800s and has become more accessible since then. The [[wikipedia:Photoplethysmogram|photoplethysmography]] (PPG) and [[wikipedia:Electrocardiography|electrocardiography]] (ECG) are both major components of the [[wikipedia:Apple_Watch|Apple Watch]]. The PPG uses sensors to measure heart rate through changes in blood volume while the ECG records electrical heart signals to help identify [[wikipedia:Atrial_fibrillation|atrial fibrillation]]. This allows Apple Watch users to monitor certain aspects of their health without always needing to go to a medical professional. This article will explore the historical development of the PPG and the ECG technology and how they are integrated into the Apple Watch. It will also examine research on the Apple Watch’s clinical accuracy in measuring heart rate and detecting atrial fibrillation. == History of ECG == The history of electrocardiography can be traced back to 1781, when [[wikipedia:Luigi_Galvani|Luigi Galvani]] discovered that electrical stimulation could cause a frog's muscles to contract <ref name=":1">{{Cite journal|last=Bok|first=Kim, Kwang|last2=Jae|first2=Baek, Hyun|date=2023-07|title=Photoplethysmography in Wearable Devices: A Comprehensive Review of Technological Advances, Current Challenges, and Future Directions|url=https://www.mdpi.com/2079-9292/12/13/2923|journal=Electronics|language=en|volume=12|issue=13|doi=10.3390/electronic|issn=2079-9292}}</ref> Galvani called this discovery “animal electricity” which helped contribute to the early understanding of [[wikipedia:Electrophysiology|electrophysiology]]. This discovery helped scientists understand that electrical activity plays an important role with how muscles function, including the heart. In 1856, [[wikipedia:Albert_von_Kölliker|Köllicker]] and [[wikipedia:Heinrich_Müller_(physiologist)|Müller]] discovered evidence of electrical activity in the heart. This discovery helped establish the foundation for recording the heart's electrical signals. <ref name=":1" /> In 1901, [[wikipedia:Willem_Einthoven|Willem Einthoven]] developed the [[wikipedia:String_galvanometer|string galvanometer]], which was an instrument that could measure and record the heart's electrical activity more accurately. This was important because it allowed doctors and researchers to observe the heart's electrical signals with more detail.<ref name=":1" /> In 1906, Einthoven further described ECG patterns and helped establish the three standard limb leads used to records the heart's electrical activity from different perspectives. He also received the Nobel Prize in 1924 for his contributions to electrocardiography. <ref name=":1" /> As the ECG continued to develop throughout time and new discoveries were made, this led to a better understanding of cardiovascular disease. ECG technology has evolved to become more compact and portable leading to mobile and wearable cardiovascular monitoring technologies such as smartphones and wearable devices.<ref name=":1" /> == History of PPG == Over time, PPG technology became smaller, more reliable, and easier to incorporate into wireless devices. The PPG uses an integration of [[wikipedia:Light-emitting_diode|light-emitting diodes]] (LEDs) to shine into the skin and [[wikipedia:Photodiode|photodiode]] (PD) to detect the light reflecting from the tissue. An [[wikipedia:Analog_front-end|analog front end]] helps process these signals, reducing the electrical interference. <ref name=":1" /> These improvements reduced power consumption, making PPG sensors more compatible with battery powered smart watches. <ref name=":1" /> The PPG grew more popular in the early 2010 with the first wrist-worn device with PPG sensors, the Mio Alpha, becoming available in 2012.<ref name=":1" /> As this technology evolved, the PPG sensors were incorporated in the Apple Watch in 2015.<ref name=":1" /> == Integration of PPG and ECG into the Apple Watch == Adding ECG with along with PPG sensors provide another way to assess heart rhythm and could potentially improve the detection of irregular heart rhythms.<ref name=":2">{{Cite journal|last=Pepplinkhuizen|first=Shari|last2=Hoeksema|first2=Wiert F.|last3=van der Stuijt|first3=Willeke|last4=van Steijn|first4=Nicole J.|last5=Winter|first5=Michiel M.|last6=Wilde|first6=Arthur A. M.|last7=Smeding|first7=Lonneke|last8=Knops|first8=Reinoud E.|date=2022-12-01|title=Accuracy and clinical relevance of the single-lead Apple Watch electrocardiogram to identify atrial fibrillation|url=https://www.sciencedirect.com/science/article/pii/S2666693622001694|journal=Cardiovascular Digital Health Journal|series=Preventive Digital Health|volume=3|issue=6, Supplement|pages=S17–S22|doi=10.1016/j.cvdhj.2022.10.004|issn=2666-6936}}</ref> In September of 2018, Apple released the Apple Watch Series 4, which could record single-lead ECG and detect atrial fibrillation through the electrodes on the back of the watch.<ref name=":3">{{Cite journal|last=Saghir|first=Nabeel|last2=Aggarwal|first2=Arjun|last3=Soneji|first3=Nisha|last4=Valencia|first4=Victoria|last5=Rodgers|first5=George|last6=Kurian|first6=Thomas|date=2020-07-01|title=A comparison of manual electrocardiographic interval and waveform analysis in lead 1 of 12-lead ECG and Apple Watch ECG: A validation study|url=https://www.sciencedirect.com/science/article/pii/S2666693620300062|journal=Cardiovascular Digital Health Journal|volume=1|issue=1|pages=30–36|doi=10.1016/j.cvdhj.2020.07.002|issn=2666-6936}}</ref> The Apple watch allowed users to record a single-lead ECG, corresponding to Lead I of a standard 12-lead ECG. <ref name=":3" /> This allowed the Apple Watch to analyze single-lead ECG recordings to classify the results as [[wikipedia:Sinus_rhythm|sinus rhythm]], atrial fibrillation, or inconclusive. The Apple Watch is also able to provide notifications on high and low heart rate.<ref name=":3" /> Integrating ECG and PPG technology helped expand the functions and cardiovascular monitoring capabilities of the Apple Watch beyond just optical heart monitoring. == How PPG Technology Works == During each heartbeat, the amount of blood that flows through the arteries and the arterioles change. During the systole phase, the heart contracts, pumping blood into the arteries, away from the heart. During the diastole phase, the heart is relaxing and filling up with blood, and the volume in arteries decreases. PPG technology detects these changes in blood volume by shining light through the skin and measuring how much light is reflected or absorbed by the tissues and blood vessels. Arteries contain a lot of blood, especially during the systolic phase (contraction of the heart) than the diastolic phase (relaxation of the heart) during one cardiac cycle. How PPG works is by detecting big changes in blood volume. PPG is able to do this function by using a particular light which is shined through the skin to monitor the amount of blood pulsing through the microvascular bed of the tissues with every heart beat. <ref name=":4">{{Cite web|url=https://www.mdpi.com/2079-9292/3/2/282|title=Wearable Photoplethysmographic Sensors—Past and Present|last=Tamura, Maeda, Sekine, Yoshida|first=Toshiyo, Yuka, Masaki, Masaki|date=2014-04-23|website=MDPI}}</ref> The PGG waveform that aid in its purpose and function are the direct current (DC) and the altering current (AC). The AC component is especially important for heart rate because it measures heart rate through pulse. The DC waveform represents the optical signals from the tissues. Tissue structure, average blood volume of both arterial and venous blood contribute to DC function. DC also begins to change with respiration. The AC component heavily depends on the heart rate. It does this by detecting changes in blood volume during the systolic and diastolic phase of one heart beat. <ref name=":4" />Research conducted by R.Rox Anderson and John A. Parrish showed that the human skin best absorbed blue, green, and yellow lights due the red blood cells. Shorter wavelengths of light are typically absorbed by the skin rapidly due to the melanin in skin.The human body containing large amounts of water also acted as a block for ultraviolet rays. However, red and near IR passes light easily with tissues, penetrating deeper. This explains why they have been integrated with PPG sensors. <ref name=":4" />Blood can also absorb more light than in the tissues which is why PPG sensors can detect big changes in blood volume. Green light has shown to be most effective for tracking superficial blood flow to skin while IR or near IR wavelengths are much better suited for measuring deep blood flow, like in the muscles.<ref name=":4" /> Due to green LED having greater absorption for oxyhemoglobin and deoxyhemoglobin, it continues to grow in popularity and has been integrated into MIO global, the MIO alpha which measures PPG with 99% accuracy.<ref name=":4" /> PPG has two modes, transmission and reflectance. Transmission is how light passes through tissues to a photodetector on the opposite side of the LED source. Reflectance is when the photodetector detects light that has been back scattered or reflected from tissue, bone, or blood vessels.<ref name=":4" /> In order to remain effective, the transmission mode of PPG has to be placed somewhere on the body where the signal can be easily transmitted. Placing PPG transmission mode sensors on fingers would be the preferred place to optimize performance, however, it is not well suited for day to day activities.<ref name=":4" /> Throughout the years, PPG sensors have been widely explored and have been tested on the wrist which can be more suitable for everyday activities.<ref name=":4" /> == How ECG Technology Works == The electrocardiogram(ECG) is used to measure the heart’s electrical activity. In regards to the Apple Watch, the watch records by going through a circuit between the detector on the back of the watch and the digital crown. The formation of this circuit allows the watch to record single-lead ECG.<ref name=":5">{{Cite journal|last=Isakadze|first=Nino|last2=Martin|first2=Seth S.|date=2020-10-01|title=How useful is the smartwatch ECG?|url=https://www.sciencedirect.com/science/article/pii/S1050173819301495|journal=Trends in Cardiovascular Medicine|volume=30|issue=7|pages=442–448|doi=10.1016/j.tcm.2019.10.010|issn=1050-1738}}</ref> ECG sensors were added to PPG sensors to help increase specificity, reducing false positive results.<ref name=":2" /> In order to take an ECG measurement, Apple Watch users need to place their index finger of their opposite hand on the Digital Crown taking about 30 seconds to record results.<ref name=":5" /> As results are being recorded, people should remain still and rest their arm on a table or on their legs to reduce interference.<ref name=":2" /> The Apple Watch produced a single-lead ECG corresponding to Lead I of traditional 12-lead ECG which records one view of the heart instead of multiple views like the 12-lead ECG can provide.<ref name=":2" /> The app that has been integrated into the Apple Watch will begin to analyze the ECG recording and then provide a rhythm classification and will sort results in one of three categories, sinus rhythm (SR), AF, or inconclusive and are saved in a PDF form. <ref name=":5" />High or low heart rates or poor recordings will still result in a notification.<ref name=":2" /> == PPG vs ECG == The PPG is able to measure changes within blood flow. It heavily relies on detectors on the opposite side of the tissue to penetrate and read blood changes and consists of DC and AC components.<ref name=":4" /> The PPG has two modes which include transmission and reflectance. Transmission mode is when the photodetector are placed on the opposite sides of the tissue to capture the light passing through the medium while in reflectance mode, the photodetectors are on the same side of the tissue to capture the light that is being scattered back and reflects off tissues, bone, and blood vessels.<ref name=":4" /> As for the ECG, the single lead electrocardiogram (iECG) records heart rate through a circuit that includes the digital crown and the detectors that is found on the back of the Apple Watch. It takes about 30 seconds for the ECG to analyze heart rhythm which can be done best at rest.<ref name=":5" /> The PPG uses light to detect blood volume changes that are associated with heartbeats while the ECG uses electrodes to help detect electrical activity that is produced by the heart. ECG also records the heart’s electrical signals.<ref name=":5" /> Another difference is PPG can identify irregular intervals between pulse waves and can suggest atrial fibrillation while the ECG records the heart’s electrical signals allowing the Apple Watch app to analyze and classify the rhythm.<ref name=":5" /> Some limitations seen in the PPG is how movement can interfere with its measurements and poor contact or weak peripheral blood may affect readings as well.<ref name=":4" /> Limitations seen in the ECG are that users need to place a finger on the digital crown and that records can be inconclusive or affected by movement.<ref name=":2" /> The integration of both the ECG and the PPG work together to help increase the specificity, helping reduce the risk of false positives.<ref name=":2" /> == How Researchers Tested Apple Watch Accuracy == To test the accuracy of the first generation Apple Watch, Maarten Falter, Werner Budts, Kaatje Goetschalckx, Véronique Cornelissen, and Roselien Buys, conducted research to test to see how accurate the monitoring heart rate feature and the energy expenditure of the Apple Watch during cardio exercise and with patients suffering from heart disease.<ref name=":6">{{Cite web|url=https://mhealth.jmir.org/2019/3/e11889/|title=Accuracy of Apple Watch Measurements for Heart Rate and Energy Expenditure in Patients With Cardiovascular Disease: Cross-Sectional Study|website=mhealth.jmir.org|access-date=2026-11-02}}</ref> There were 40 patients involved with this research with the mean age being 61.9 years old and 80% males with heart disease.<ref name=":6" /> The test involved doing cardiopulmonary exercise while wearing an Apple Watch. Results from the Apple watch were compared to the results of a 12-lead ECG which measured heart rate and researchers also used an indirect calorimetry which was used for EE.<ref name=":6" /> The heart rate was analyzed at three levels, resting while seated, moderate intensity, and maximal performance for 30 second increment periods while EE was being tested as well the whole time.<ref name=":6" /> The mean difference, standard deviations of difference, limits of agreement, mean absolute error, mean absolute percentage error, and intraclass correlation coefficients were all being calculated and Bland-Atlman plots and scatter plots were also made to records findings.<ref name=":6" /> == Accuracy of Apple Watch Measurements for Heart Rate and Energy Expenditure in Patients with Cardiovascular Disease == This study shows that the first generation Apple Watch was accurate in evaluating heart rate during exercise.<ref name=":6" /> The researchers had 40 patients with heart disease do indoor cycling with an Apple Watch on their wrist.The mean absolute percentage error was 10.69% at rest, 9.20% during moderate exercise, and 6.33% during maximal exercise.<ref name=":6" /> These findings show that the average percentage error decreases as exercise intensity increases. Overall, the results of the study came out promising under study conditions as heart rate was compared to ECG for accuracy.<ref name=":6" /> The first-generation Apple Watch results can suggest having the Apple Watch be a potential use of technology with added heart rate guided training programs for cardiac rehabilitation patients in the future. However, supporting research will have to be conducted before heart rate training programs for cardiac rehabilitation may be added.<ref name=":6" /> == Study Methodology == For this study, 74 patients were scheduled for an electrical cardioversion (ECV). This study was non random at a single center observational study to evaluate how accurate the Apple Watch series 6 with watch OS software 7.2 or 7.3 and ECG algorithm version 2.0 was at detecting atrial fibrillation.<ref name=":2" /> 12-lead ECG was used as a standard reference to compare the Apple Watch. All patients in study were at least 18 years of age or older.<ref name=":2" /> The study was done by placing the Apple Watch on their left wrist and were told to rest their arm with the Apple Watch on the table with relaxed and still feet.<ref name=":2" /> Patients were still able to talk, eat and drink normally during this experiment. Apple watch ECGs were taken 3 times in case the Apple Watch could not classify the results. The Apple Watch ECG was obtained during atrial fibrillation, before ECV, and during SR.<ref name=":2" /> The main goal was to evaluate the Apple Watches sensitivity for detecting atrial fibrillation and specificity for detecting sinus rhythm, before and after repeating unclassified results.<ref name=":2" /> Physicians S.P. and W.S. were to blindly evaluate both the 12-lead ECG and the Apple Watch ECG. If there were any disagreements, then a third physician would get involved.<ref name=":2" /> == Clinical Accuracy Results of the Apple Watch ECG in Detecting Atrial Fibrillation == From 74 patients, this is what the study discovered. In the first measurement, the Apple Watch showed a 93.5% sensitivity and a 100% specificity.<ref name=":2" /> In the second measurement, it was shown that the sensitivity increased to 94.6% and remained at 100% specificity.<ref name=":2" /> However, in the third measurement, sensitivity decreased to 93% and specificity decreased to 96.5%.<ref name=":2" /> This repeated recording involved a smaller group of patients with the earlier unclassified results. Initial recordings that the Apple Watch could not classify was 27.9%.<ref name=":2" /> When researchers counted the unclassified recordings as incorrect, sensitivity decreased to 66.2% and specificity to 73.4%.<ref name=":2" /> Repeating recording did decrease the total amount of unclassified recording.<ref name=":2" /> With these results, it demonstrated that the single lead ECG of the Apple Watch showed high accuracy when detecting atrial fibrillation in a clinical setting. With the unclassified results of the Apple Watch, physicians are able to reduce the number of unclassified results by interpreting some ECG recordings that the Apple watch might not have been able to classify.<ref name=":2" /> == Conclusion == Cardiovascular monitoring has evolved from being a big, bulky, piece of technology into wearable technology. The development of the PPG and the ECG have both played huge roles in the development of the Apple Watch to make it more convenient for everyday use. Research studies have also demonstrated that the Apple Watch has promising accuracy when using ECG technology to detect atrial fibrillation and PPG to measure heart rate within certain conditions. However, there are some limitations with the Apple Watch, like measurement errors and recordings that were not able to be classified by the Apple Watch. Even though the Apple Watch cannot fully replace medical equipment, it can provide more insight involving their cardiovascular health from the comfort of their own home. The evolution of the cardiovascular technology in the Apple Watch has made it easier and more accessible for normal people to monitor heart health. {{BookCat}} 52x2qglxnnstd2fj1zscf6dgpiork9b 3 332827 2844611 2844576 2026-11-05T12:35:43Z MathXplore 2888076 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]] 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 10 332831 2844618 2844592 2026-11-05T12:46:15Z MathXplore 2888076 Template:Proposed deletion 2844618 wikitext text/x-wiki <noinclude><!-- TO CONTEST THIS PROPOSED DELETION, remove the following template, including this comment, up to the CLOSING COMMENT --> {{Ombox | type = delete | image = [[File:Orologio rosso.svg|45x45px|center|No license|link=]] | text = Please [{{fullurl:{{FULLPAGENAME}}|action=edit}} help improve] the [[Wikiversity:What is Wikiversity?|educational quality]] of this resource to increase [[Wikiversity:Learning|engagement]] by [[Wikiversity:Who are Wikiversity participants?|participants]]. Any concrete improvements made by '''{{#time:F j, Y|{{REVISIONTIMESTAMP}} +90 days}}''' may allow it to be [[WV:PROD|kept]]. <br> You may remove {{tl|proposed deletion}} from this resource's source text to contest this proposal, with or without [[{{TALKPAGENAME}}|discussion]]. {{#if:|---- <div style="text-align:center; margin-bottom:0em;"> The nominator's rationale: <p>{{{1}}}</p> </div> }}}}{{center|[[Special:PrefixIndex/{{FULLPAGENAME}}/|''Link to any subpages this page might have'']]}} {{event trigger|date=November 5, 2026|when=90 days|[[Category:Pending deletions|{{PAGENAME}}]]}} {{event trigger|date=November 5, 2026|when=60 days|[[Category:60-day proposed deletions|{{PAGENAME}}]]}} [[Category:Proposed deletions|{{PAGENAME}}]] <!-- CLOSING COMMENT, remove up to and including this comment --> </noinclude> 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 ptf4yo93pnxnszpiqyd4t9b9v6g55lo 3 332833 2844612 2026-11-05T12:38:21Z MathXplore 2888076 vandalism1 ([[m:User:ZbVl/VD|Vandoom]]) 2844612 wikitext text/x-wiki == 2026-11-05 == [[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 @ 1793882299551.7s --><nowiki></nowiki> [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 12:38, 5 November 2026 (UTC) by7b3brba6qau3q1ejtn61mlvusy7pb 6 332834 2844624 2026-11-05T13:55:12Z Young1lim 21186 {{Information |Description=Carry Lookahead Adders 2B traditional (20261105 - 20261104) |Source={{own|Young1lim}} |Date=2026-11-05 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2844624 wikitext text/x-wiki == Summary == {{Information |Description=Carry Lookahead Adders 2B traditional (20261105 - 20261104) |Source={{own|Young1lim}} |Date=2026-11-05 |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}} 7ptas1xp0qtemp07jp5e2bkflodh358 6 332835 2844625 2026-11-05T13:55:55Z Young1lim 21186 {{Information ||Description=Block CLA 2C Multi Level (20261105 - 20261104) |Source={{own|Young1lim}} |Date=2026-11-05 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2844625 wikitext text/x-wiki == Summary == {{Information ||Description=Block CLA 2C Multi Level (20261105 - 20261104) |Source={{own|Young1lim}} |Date=2026-11-05 |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}} qkixlv53785ndlo3w4wpw7asihyw8f1 6 332836 2844627 2026-11-05T14:21:27Z Young1lim 21186 {{Information |Description=C04.SA0: Address and Dereference Operators (20261103 - 20261102) |Source={{own|Young1lim}} |Date=2026-11-05 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2844627 wikitext text/x-wiki == Summary == {{Information |Description=C04.SA0: Address and Dereference Operators (20261103 - 20261102) |Source={{own|Young1lim}} |Date=2026-11-05 |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}} bu1yqx4ge2w464hhpvkr36q2liy58om 6 332837 2844629 2026-11-05T14:22:40Z Young1lim 21186 {{Information |Description=C04.SA0: Address and Dereference Operators (20261104 - 20261103) |Source={{own|Young1lim}} |Date=2026-11-05 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2844629 wikitext text/x-wiki == Summary == {{Information |Description=C04.SA0: Address and Dereference Operators (20261104 - 20261103) |Source={{own|Young1lim}} |Date=2026-11-05 |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}} taxfpy52aq37nh1kk8zf8zzv285x6vi 0 332838 2844637 2026-11-05T16:42:38Z Dick Bos 24466 first lines 2844637 wikitext text/x-wiki On this page we will delve deeper into some mathematical interpretations of the Ricardian theory. We will give most attention to interpretations regarding Ricardo's ''[[David Ricardo/An Essay on Profits|Essay on Profits]]'' (1815), but we will also look forward to interpretations that include his ''[[David Ricardo/Principles|Principles]]''. == Introductory remarks == Broadly speaking, the models of Ricardian theory can be divided into two groups: * the simple, single-commodity ("corn") models; * the multi-sector models. The first type of model is based on Ricardo’s exposition in the ''Essay on Profits''; the second on the ''Principles''. In both types of model, the analysis is based on two principles: * the population principle, which maintains long-term real wages at the subsistence level; * the principle that capital accumulation leads to diminishing returns in agriculture.{{sfnp|Howard|King|1975|p=82}} The central argument put forward by Ricardo in defence of the "corn model" is that the profits of the capitalist tenant farmer (‘the farmer’) regulate profits in all other sectors.<ref name=farmer>In the ''Essay on Profits'' it is stated in several places in this context that profits are ultimately determined by the costs of producing food (see, for example, {{harvp|Ricardo|1815}}, pp. [[:s:en:An Essay on the Influence of a Low Price of Corn on the Profits of Stock#21|21]], [[:s:en:An Essay on the Influence of a Low Price of Corn on the Profits of Stock#25|25]]/26).</ref><br /> In the ''Principles'', this assumption is abandoned, giving way to the more general proposition that the conditions of production on marginal land play a central role in determining the general rate of profit.<ref name=marginalland>Compare {{harvp|Sraffa|1951|p=xxxi ff.}}; {{harvp|Blaug|1985|p=92}}; {{harvp|Benetti|1975|p=9 ff.}}; {{harvp|Hollander|1973}} and {{harvp|Hollander|1975}}; {{harvp|Eatwell|1975}}.</ref> == Footnotes == {{references}} == Sources == * {{cite book | last = Benetti | first = C. | date = 1975 | title = Valeur et répartition | language = 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)]])}} * {{cite journal | last = Hollander | first = Samuel | date = 1975 | title = Ricardo and the Corn Profit Model : Reply to Eatwell | journal = Economica | volume = 42 | issue = 166 | pages = 188-202 | doi = 10.2307/2553594 | url = https://www.jstor.org/stable/2553594 | 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 | 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 book | last = Sraffa | first = Piero | author-link = :w:en:Piero Sraffa | date = 1951 | chapter = Introduction | title = The Works and correspondence of David Ricardo (vol. I) | pages = xiii ff. | location = Cambridge | publisher = Cambridge University Press | url = https://archive.org/details/workscorresponde0001rica_n2f1/page/n14/mode/1up | ref = harv }} qpuyrlldbq7nzlii0m2webfqs1q7jkl 2844712 2844637 2026-11-06T08:43:19Z Dick Bos 24466 assumpt 2844712 wikitext text/x-wiki On this page we will delve deeper into some mathematical interpretations of the Ricardian theory. We will give most attention to interpretations regarding Ricardo's ''[[David Ricardo/An Essay on Profits|Essay on Profits]]'' (1815), but we will also look forward to interpretations that include his ''[[David Ricardo/Principles|Principles]]''. == Introductory remarks == Broadly speaking, the models of Ricardian theory can be divided into two groups: * the simple, single-commodity ("corn") models; * the multi-sector models. The first type of model is based on Ricardo’s exposition in the ''Essay on Profits''; the second on the ''Principles''. In both types of model, the analysis is based on two principles: * the population principle, which maintains long-term real wages at the subsistence level; * the principle that capital accumulation leads to diminishing returns in agriculture.{{sfnp|Howard|King|1975|p=82}} The central argument put forward by Ricardo in defence of the "corn model" is that the profits of the capitalist tenant farmer (‘the farmer’) regulate profits in all other sectors.<ref name=farmer>In the ''Essay on Profits'' it is stated in several places in this context that profits are ultimately determined by the costs of producing food (see, for example, {{harvp|Ricardo|1815}}, pp. [[:s:en:An Essay on the Influence of a Low Price of Corn on the Profits of Stock#21|21]], [[:s:en:An Essay on the Influence of a Low Price of Corn on the Profits of Stock#25|25]]/26).</ref> In the ''Principles'', this assumption is abandoned, giving way to the more general proposition that the conditions of production on marginal land play a central role in determining the general rate of profit.<ref name=marginalland>Compare {{harvp|Sraffa|1951|p=xxxi ff.}}; {{harvp|Blaug|1985|p=92}}; {{harvp|Benetti|1975|p=9 ff.}}; {{harvp|Hollander|1973}} and {{harvp|Hollander|1975}}; {{harvp|Eatwell|1975}}.</ref> In this pages a number of–overlapping–"corn models" will be presented. Firstly, following on from the example presented in [[David Ricardo/An Essay on Profits#The table in the Essay on Profits|the table in the ''Essay on Profits'']] and its discussion in {{harvp|van der Veen|1982}}, a simple mathematical and graphical model of Ricardo’s theory of rent will be outlined. Pasinetti provides a comprehensive and well-known model of Ricardo’s theory. A simplified version of this model will be presented. The relationships found in the ''Essay'' will be examined in the context of this model. Finally, we shall return to the issue of the development of income shares (and in particular the share of ground rent) in the course of the process of economic development. == Assumptions == In "corn models", corn is produced using labour, capital and land. Labour can be reproduced indefinitely using corn; land (of a certain quality) is absolutely scarce, i.e. non-reproducible.<ref name=scarceland>Cf. {{harvp|Blaug|1985|p=88f.}}</ref> Total output–a certain quantity of corn (wheat)–also constitutes total income, which is distributed amongst workers, capitalists and landowners. All versions of the "corn model" discussed below are based on a very large number of assumptions.<ref name=Kick>See in particular {{harvp|van der Pol|1981|page=25–7}}</ref> Some of these are explicitly formulated as such by Ricardo; others are "hidden" within his exposition; one of the assumptions set out below is more stringent than in Ricardo’s work. a. A capitalist agricultural economy, characterised by a growing population and capital accumulation, is examined. The industrial sector will be entirely disregarded. b. The available land is not of uniform quality. Different amounts of labour must be employed on different plots to achieve the same yield per unit area. c. New land must be brought into cultivation periodically. d. The best-quality land will be put to use first. e. There is free settlement on fallow land. f. Intra-marginal land is privately owned. g. For each type of land (a plot of land of a specific quality), only one production technique is available. In other words, the amount of labour required per unit of land is fixed for each plot. This means that the possibility of shifting an "intensive margin" is ruled out. h. There are "diminishing returns" when we consider successive production periods. These diminishing physical returns may result from the use of less fertile land or from the use of equally fertile land situated further away, thereby incurring higher costs to bring the product to the consumer.<ref name=highercosts>In his description of the assumptions underlying the table in the ''Essay'', Ricardo states: "Supposing then the land to be equally fertile, the necessity of employing more labourers, horses, &c. to carry the produce from the place where it was grown, to the place where it was to be consumed, although no alteration were to take place in the wages of labour, would make it necessary that more capital should be permanently employed to obtain the same produce." {{harvp|Ricardo|1815|p =7/8}}</ref> In the table in the ''Essay'', the following applies: in each period, the increase in gross production amounts to 300 quarters of wheat; in each period, an equal area of new land of equal fertility, situated further away, is brought into use; the constant increase in gross production can only be achieved through an ever-increasing investment of capital: in the first period, the total capital invested amounts to 200 q. of wheat; in the second period, this increases by 210 q.; in the third by 220 q., and so on. i. There is one output (product): corn; and, apart from the non-homogeneous soil, one input (factor of production). This input, capital, consists exclusively of a wage fund; this wage fund, in turn, consists exclusively of wheat.<ref name=>In fact, Ricardo assumes that half of the capital employed in the first period (amounting to 200 q. of wheat) consists of fixed capital, and half of circulating capital ({{harvp|Ricardo|1815|p=4}}). To avoid valuation problems, an even simpler assumption is adopted here instead. Instead of the assumption that there is a single input (capital = wage fund = corn), the literature often assumes the existence of (homogeneous) ‘doses of capital and labour’. See, for example, {{harvp|Barkai|1959}}; {{harvp|Howard|King|1975|p=83}}; {{harvp|Blaug|1985}}, e.g. p. 88.</ref> j. The real wage rate is assumed to be constant and is taken as a given. k. All labour is assumed to be homogeneous. l. Wages are paid at the start of the production period. m. A surplus (‘neat produce’) is produced: output exceeds input. n. There is a single, uniform production period. o. No "improvements in agriculture" take place{{sfnp|Ricardo|1815|p=6}}: there is therefore no technical development. p. The entire economy is treated as "one gigantic farm"<ref name=onefarm>For the crucial significance of this assumption in the analysis of the introduction of improvements in agriculture, see {{harvp|O'Brien|1975|p=130–1}}.</ref> == 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 | language = fr | location = Grenoble | publisher = Maspero | id = {{OCLC|461493533}} | ref = harv }} (2nd ed.) * {{cite book | last = Blaug | first = Mark | author-link = :w:en:Mark Blaug | 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)]])}} * {{cite journal | last = Hollander | first = Samuel | date = 1975 | title = Ricardo and the Corn Profit Model : Reply to Eatwell | journal = Economica | volume = 42 | issue = 166 | pages = 188-202 | doi = 10.2307/2553594 | url = https://www.jstor.org/stable/2553594 | 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 | 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 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 }} * {{cite book | last = Sraffa | first = Piero | author-link = :w:en:Piero Sraffa | date = 1951 | chapter = Introduction | title = The Works and correspondence of David Ricardo (vol. I) | pages = xiii ff. | location = Cambridge | publisher = Cambridge University Press | url = https://archive.org/details/workscorresponde0001rica_n2f1/page/n14/mode/1up | ref = harv }} * {{cite book | last = van der Veen | first = Robert Jan | date = 1982 | title = Opmerkingen over Ricardo's tabel in de ''Essay on Profits'' (remarks about the table in Ricardo's ''Essay on Profits'') | location = Amsterdam | publisher = not published; notes for lectures on Ricardo & Marx, University of Amsterdam | language = nl | ref = harv }} mbawdvfh5h2kewktk4elnd1fpqq13lw 6 332840 2844642 2026-11-05T17:19:19Z Young1lim 21186 {{Information |Description=Copilot: File Control B. Duplicate (20261102 - 20261027) |Source={{own|Young1lim}} |Date=2026-11-02 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2844642 wikitext text/x-wiki == Summary == {{Information |Description=Copilot: File Control B. Duplicate (20261102 - 20261027) |Source={{own|Young1lim}} |Date=2026-11-02 |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}} q97fxooyjvy1anhmkl2ysv25fo3pqyo 2844644 2844642 2026-11-05T17:20:16Z Young1lim 21186 /* Summary */ 2844644 wikitext text/x-wiki == Summary == {{Information |Description=Copilot: File Control B. Duplicate (20261102 - 20261027) |Source={{own|Young1lim}} |Date=2026-11-05 |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}} 2mw16nrkfdew6ule664t06gc8koi98i 6 332841 2844645 2026-11-05T17:20:45Z Young1lim 21186 {{Information |Description=Copilot: File Control B. Duplicate (20261103 - 20261102) |Source={{own|Young1lim}} |Date=2026-11-05 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2844645 wikitext text/x-wiki == Summary == {{Information |Description=Copilot: File Control B. Duplicate (20261103 - 20261102) |Source={{own|Young1lim}} |Date=2026-11-05 |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}} fngscg7iscsrn7vp4ikmeg3ozr4sdi0 0 332842 2844703 2026-11-06T02:26:44Z Hdad Tigre 3113886 Created new lesson: Tigre Language - My Day 2844703 wikitext text/x-wiki = Tigre Language - My Day = This lesson is for beginners to learn Tigre (Tigrayit) with English and Arabic. == Lesson 1: My Day == {| class="wikitable" !English !العربية !ትግሬ |- |I Don't Have Work Today, So I Woke Up Late. |أنا لا أملك عمل اليوم، لذلك استيقظت متأخر. |አነ ዮም ሽቅል አለብዬ፡ ሰበት እሊ እንዴ ደንገርኮ ፈዝዐኮ። |- |I Had Breakfast, Then Cleaned My Room. |تناولت الفطور، ثم نظفت غرفتي. |ፍጡር በልዐኮ፡ ሐቆሁ ኦደትቼ አንደፍኮ። |- |I Went To The Market And Bought Some Vegetables For Lunch. |ذهبت إلى السوق واشتريت بعض الخضروات للغداء. |አስክ ዕዳገ ጊስኮ ወእግል ጸብሕ ገሌ ሐመሊት ትዛቤኮ። |- |After Lunch, I Felt Really Sleepy, So I Took A Long Nap. |بعد الغداء، شعرت حقا بالنعاس، لذلك أخذت قيلولة طويلة. |ሐቆ ጸብሕ ተሀርገትኮ፡ ሰበት እሊ ቀሸው ረያም ነስአኮ። |- |I Woke Up At 5 P.M., Just Before Sunset. |استيقظت الساعة 5 مساء، تماما قبل الغروب. |አክልሕድ ቀደም ውድቀት ጸሓይ፡ ሳዐት 5 ናይ አምሱይ ፈዝዐኮ። |- |I Went For A Walk To The Supermarket. |ذهبت للمشي إلى السوبرماركت. |ዮም ሐበት ኢወዴኮ፡ ወሐዝየ ለዐልኮ አክልሕድ እለ ተ። |} 2g3d8xiji39l5e76p9hgljsodpghttu 2844704 2844703 2026-11-06T02:35:07Z Hdad Tigre 3113886 /* Lesson 1: My Day */ Fixed a wrong sentence /* Lesson 1: My Day */ Fixed a wrong sentence 2844704 wikitext text/x-wiki = Tigre Language - My Day = This lesson is for beginners to learn Tigre (Tigrayit) with English and Arabic. == Lesson 1: My Day == {| class="wikitable" !English !العربية !ትግሬ |- |I Don't Have Work Today, So I Woke Up Late. |أنا لا أملك عمل اليوم، لذلك استيقظت متأخر. |አነ ዮም ሽቅል አለብዬ፡ ሰበት እሊ እንዴ ደንገርኮ ፈዝዐኮ። |- |I Had Breakfast, Then Cleaned My Room. |تناولت الفطور، ثم نظفت غرفتي. |ፍጡር በልዐኮ፡ ሐቆሁ ኦደትቼ አንደፍኮ። |- |I Went To The Market And Bought Some Vegetables For Lunch. |ذهبت إلى السوق واشتريت بعض الخضروات للغداء. |አስክ ዕዳገ ጊስኮ ወእግል ጸብሕ ገሌ ሐመሊት ትዛቤኮ። |- |After Lunch, I Felt Really Sleepy, So I Took A Long Nap. |بعد الغداء، شعرت حقا بالنعاس، لذلك أخذت قيلولة طويلة. |ሐቆ ጸብሕ ተሀርገትኮ፡ ሰበት እሊ ቀሸው ረያም ነስአኮ። |- |I Woke Up At 5 P.M., Just Before Sunset. |استيقظت الساعة 5 مساء، تماما قبل الغروب. |አክልሕድ ቀደም ውድቀት ጸሓይ፡ ሳዐት 5 ናይ አምሱይ ፈዝዐኮ። |- |I Didn't Do Anything Today, And That Is Exactly What I Wanted. |لم أفعل أي شيء اليوم، وهذا بالضبط ما أردت |ዮም ሐበት ኢወዴኮ፡ ወሐዝየ ለዐልኮ አክልሕድ እለ ተ። |} c6cfjw1fitcbosv9vdg7emyidvki17i 0 332843 2844708 2026-11-06T03:09:19Z Hdad Tigre 3113886 Added Lesson 2 - My Work Day 2844708 wikitext text/x-wiki = Tigre Language - My Work Day = This lesson is about a work day. English - Arabic - Tigre. {| class="wikitable" |- ! English !! العربية !! تقري |- | Today I went to work early. || اليوم ذهبت إلى العمل مبكرا. || ዮም አስክ ሽቅል በድሪ ጊስኮ። |- | On the way, I met my old friend. || في الطريق، قابلت صديقي القديم. || እት ገበይ፡ መልሀዬ ቀዲም ትቃበሌኒ። |- | We talked a little, then each one went to his work. || تكلمنا قليلا، ثم ذهب كل واحد إلى عمله. || ሑዳይ ተሃጌነ፡ ወሐቆሁ ክል ኦሮት ከአስክ ሽቅሉ ቄሰ። |- | At lunch time, I ate with my colleagues. || في وقت الغداء، أكلت مع زملائي. || እት ወቅት ጸብሕ፡ ምስል ጠርዬ በልዐኮ۔ |- | After work, I went to the market to buy bread. || بعد العمل، ذهبت إلى السوق لأشتري خبز. || ሐቆ ሽቅል፡ ሕበዝ እግል እትዛቤ አስክ ሱግ ጊስኮ። |- | At home, I helped my mother in the kitchen. || في البيت، ساعدت أمي في المطبخ. || እት ቤት፡ እት ከሺነት እምዬ ሰድኮ። |- | After dinner, I sat with my family and we drank tea. || بعد العشاء، جلست مع عائلتي وشربنا شاي. || ሐቆ ድራር ምስል ዐይለቼ ትገሴኮ ወሻሂ ሰቴነ። |- | Before I sleep, I read a small book. || قبل أن أنام، قرأت كتابا صغيرا. || ቀደም ይእሰክብ፡ ክታብ ንኡሽ ቀርአኮ። |- | Today was a long day, but it was beautiful. || اليوم كان يوما طويلا، لكنه كان جميلا. || ዮም አምዕል ረያም ዐለት፡ ምናተ ግርም ዐለት። |- | Tomorrow I will wake up early again. || غدا سأستيقظ مبكرا مرة أخرى. || ፈጅር ካልእ ዶል በድሪ እግል እፍዘዕ ቱ። |} [[Category:Tigre language]] 0nwvv0l7st8jz5mfmsgh9nk76nyd4js 2844709 2844708 2026-11-06T03:09:54Z Hdad Tigre 3113886 /* Tigre Language - My Work Day */ ንኢሽ አስነዮት 2844709 wikitext text/x-wiki = Tigre Language - My Work Day = This lesson is about a work day. English - Arabic - Tigre. {| class="wikitable" |- ! English !! العربية !! تقري |- | Today I went to work early. || اليوم ذهبت إلى العمل مبكرا. || ዮም አስክ ሽቅል በድሪ ጊስኮ። |- | On the way, I met my old friend. || في الطريق، قابلت صديقي القديم. || እት ገበይ፡ መልሀዬ ቀዲም ትቃበሌኒ። |- | We talked a little, then each one went to his work. || تكلمنا قليلا، ثم ذهب كل واحد إلى عمله. || ሑዳይ ተሃጌነ፡ ወሐቆሁ ክል ኦሮት ከአስክ ሽቅሉ ጌሰ። |- | At lunch time, I ate with my colleagues. || في وقت الغداء، أكلت مع زملائي. || እት ወቅት ጸብሕ፡ ምስል ጠርዬ በልዐኮ۔ |- | After work, I went to the market to buy bread. || بعد العمل، ذهبت إلى السوق لأشتري خبز. || ሐቆ ሽቅል፡ ሕበዝ እግል እትዛቤ አስክ ሱግ ጊስኮ። |- | At home, I helped my mother in the kitchen. || في البيت، ساعدت أمي في المطبخ. || እት ቤት፡ እት ከሺነት እምዬ ሰድኮ። |- | After dinner, I sat with my family and we drank tea. || بعد العشاء، جلست مع عائلتي وشربنا شاي. || ሐቆ ድራር ምስል ዐይለቼ ትገሴኮ ወሻሂ ሰቴነ። |- | Before I sleep, I read a small book. || قبل أن أنام، قرأت كتابا صغيرا. || ቀደም ይእሰክብ፡ ክታብ ንኡሽ ቀርአኮ። |- | Today was a long day, but it was beautiful. || اليوم كان يوما طويلا، لكنه كان جميلا. || ዮም አምዕል ረያም ዐለት፡ ምናተ ግርም ዐለት። |- | Tomorrow I will wake up early again. || غدا سأستيقظ مبكرا مرة أخرى. || ፈጅር ካልእ ዶል በድሪ እግል እፍዘዕ ቱ። |} [[Category:Tigre language]] asxlntjuapt7cjbdcf4zm9l8xs2h3rd